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\section{Introduction}

Let $\alpha$ be an irrational number with strong Diophantine type $\gamma$; that is,
\[ 0< \liminf_{q \to \infty} \| q \alpha \| q^{\gamma} < \infty , \]
where $\| \cdot \|$ denotes distance from the nearest integer. Let $X_1, X_2, \ldots$ be independent, identically distributed (i.i.d.) integer-valued random variables, and put $S_k= \sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $\P (|X_1|>t)\sim ct^{-\beta}$, $0<\beta<2$,
in our previous paper \cite{BB2} we investigated the order of magnitude of the discrepancy $D_N(S_k \alpha)$ of the first $N$ terms of the sequence $\{S_k \alpha\}$, where $\{ \cdot \}$ denotes fractional part.
Under some additional regularity conditions (including $\E X_1 =0$ in the case $1< \beta <2$) we proved that
\begin{equation}\label{AA}
D_N (S_k \alpha ) = O \left( \sqrt{\frac{\log \log N}{N}} \log N \right), \quad D_N (S_k \alpha ) = \Omega \left( \sqrt{\frac{\log \log N}{N}} \right) \quad \text{a.s.}
\end{equation}
if $\beta \gamma \le 2$, and
 \begin{equation} \label{AA2} D_N (S_k \alpha ) = O \left( \left( \frac{\log \log N}{N} \right)^{1/(\beta \gamma)} \right), \quad D_N (S_k \alpha ) = \Omega \left( \frac{1}{N^{1/(\beta \gamma)}} \right) \quad \text{a.s.}
 \end{equation}
if $\beta \gamma >2$. Here a.s. (almost surely) means that a given relation holds with probability $1$. These relations show the surprising fact that the asymptotic behavior of $D_N(S_k\alpha)$ changes substantially at the critical value $\gamma= 2/\beta$. A similar result holds in the case $\E X_1^2<\infty$, $\E X_1=0$ (which corresponds formally to $\beta=2$)  when the critical value is $\gamma=1$, and in the case $\E |X_1| <\infty$, $\E X_1 \ne 0$, when the critical value is $\gamma=2$. (In the latter case the result was proved only for some special distributions.)
Note that there is a gap between the upper and lower bounds in (\ref{AA}) and (\ref{AA2}), and one purpose of the present paper is to remove the factor $\log N$ in the upper bound of (\ref{AA}), yielding a precise law of the iterated logarithm (LIL) for $D_N (S_k \alpha)$ in the case $\beta\gamma<1$. Under the same condition we will also prove that $N^{-1/2} D_N (S_k \alpha)$ has a nondegenerate limit distribution. In fact, we will prove the functional version of these results describing the asymptotic behavior of a large class of functionals of the sequence $\{S_k \alpha\}$, including the $L^p$-discrepancy $D_N^{(p)} (S_k \alpha )$, $1 \le p \le \infty$. Recall that $D_N^{(p)}(S_k \alpha)$ is defined as the $L^p ([0,1])$-norm of $F_N(t)-t$, where
\[ F_N (t) = \frac{1}{N} \sum_{k=1}^N I_{[0,t)} (\{ S_k \alpha \} ), \qquad 0 \le t \le 1 \]
is the empirical distribution function, and that the discrepancy can also be expressed in terms of $F_N$ as $D_N (S_k \alpha ) = \sup_{0 \le s<t \le 1} |(F_N(t)-t)-(F_N(s)-s)|$. We define $F_N$, $D_N^{(p)} (a_k)$ and $D_N (a_k)$ for an arbitrary sequence of reals $a_k$ analogously.

From a purely probabilistic point of view $\{S_k \alpha\}$ is a Markov chain with countable state space, but we can also view it on the state space $[0,1)$; in the latter case its stationary distribution is the uniform distribution on $[0,1)$. According to a classical result of L\'evy \cite{L}, the distribution of $\{S_k \alpha\}$ converges weakly to the uniform distribution as $k\to \infty$ if and only if $X_1$ is nondegenerate (i.e.\ attains more than one integers with positive probability). Since the limit distribution is continuous, we then have
\[ \psi_\alpha(k):= \sup_{0\le t \le 1} |\P (\{S_k \alpha\} \le t)-t| \longrightarrow 0 \ \, \text{as} \ \, k\to\infty . \]
Note that $D_N^{(\infty)}(S_k \alpha)$ is the empirical analogue of $\psi_{\alpha}(k)$. The order of magnitude of $\psi_{\alpha}(k)$ was investigated by Schatte \cite{SCH2}, Diaconis \cite{DI}, Su \cite{SU}, Hensley and Su \cite{HS}; improving their results, in \cite{BB} we showed that
\[ \psi_\alpha (k) =O(k^{-1/(\beta \gamma)}) . \]
Moreover, under mild additional regularity conditions on the distribution of $X_1$ this estimate is sharp. This shows that, in contrast to $D_N(S_k \alpha)$, there is no change of behavior of $\psi_{\alpha}(k)$ at $\beta \gamma=2$. Sharp results for the remainder term in the central limit theorem (CLT) and the corresponding Edgeworth expansion for i.i.d.\ sums (without taking the fractional part) under Diophantine conditions were given by Bobkov \cite{BOB1}, \cite{BOB2}.

By the previous result, condition $\beta \gamma<1$ amounts to the same as
$$ \psi_\alpha (k) \ll k^{-(1+\delta)} \quad  (\delta>0),$$
and in this paper we will extend our results on $D_N(S_k\alpha)$ to the discrepancy $D_N (S_k)$ with any i.i.d.\ sequence $X_1, X_2, \dots$ satisfying $\psi (k) \ll k^{-(1+\delta)}$ with some $\delta >0$, where
\begin{equation}\label{psi}
\psi (k): = \sup_{t \in [0,1]} |\P (\{ S_k\} \le t) -t| .
\end{equation}
As in the arithmetic case, we will also prove the functional version of our results, describing the asymptotics of general functionals of the sequence $\{S_k\}$. The LIL for $D_N(S_k)$ was proved by Schatte \cite{SCH3} under the assumption $\psi(k)\ll k^{-(5/2+\delta)}$, $\delta>0$; he also claimed, without proof, that the result remains valid under $\psi(k)\ll k^{-(2+\delta)}$, $\delta>0$. We believe that our condition $\psi (k) \ll k^{-(1+\delta)}$, $\delta >0$ is sharp, but its optimality remains open. For earlier results on the discrepancy of $D_N(S_k)$ in the case of absolutely continuous  $X_1$ (when $\psi(k)\to 0$ exponentially fast), see Bazarova et al. \cite{BBR}, Berkes and Raseta \cite{BR}.

To formulate our results, let $X_1, X_2, \dots$ be i.i.d.\ real-valued random variables, let $S_k=\sum_{j=1}^k X_j$, and let $\psi (k)$ be as in \eqref{psi}. Let $\mathcal{F}$ denote the set of all $1$-periodic functions $f: \R \to \R$ such that $f$ is of bounded variation on $[0,1]$ and $\int_0^1 f(x) \, \mathrm{d}x=0$. For any $f,g \in \mathcal{F}$ set
\begin{equation}\label{C}
C(f,g) = \E f(U) g(U) + \sum_{k=1}^{\infty} \E f(U)g(U+S_k) + \sum_{k=1}^{\infty} \E g(U)f(U+S_k),
\end{equation}
where $U$ is a random variable uniformly distributed on $[0,1)$, independent of $X_1, X_2, \dots$.
\begin{thm}\label{theorem1}
Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $\delta >0$. Then for any $f,g \in \mathcal{F}$ the series in \eqref{C} are absolutely convergent, and $C(f,g)$ is bilinear, symmetric and positive semidefinite in $f,g$; moreover, $C(f,f)=0$ if and only if $f=0$ a.e. For any $f \in \mathcal{F}$ the sum $\sum_{k=1}^N f(S_k)$ satisfies the central limit theorem
\begin{equation}\label{CLT}
\frac{\sum_{k=1}^N f(S_k)}{\sqrt{N}} \overset{d}{\longrightarrow} \mathcal{N} \left( 0,\sigma^2 \right)
\end{equation}
and the law of the iterated logarithm
\[ \limsup_{N \to \infty} \frac{\sum_{k=1}^N f(S_k)}{\sqrt{2N \log \log N}} =\sigma \quad \mathrm{a.s.} \]
with $\sigma = \sqrt{C(f,f)}$.
\end{thm}
\noindent Relation \eqref{CLT} means convergence in distribution to the mean zero normal distribution of variance $\sigma^2$ if $\sigma >0$, and to the constant zero if $\sigma =0$.

Next, let
\[ F_N (t) = \frac{1}{N} \sum_{k=1}^N I_{[0,t)} (\{ S_k \} ), \qquad 0 \le t \le 1 \]
denote the empirical distribution function of the sequence $\{S_k\}$. Further, for any interval $J \subseteq [0,1)$ of length $\lambda (J)$ let $f_J(x)=I_J(\{ x \})-\lambda (J)$ denote the centered indicator of $J$ extended with period $1$, and put $\Gamma (s,t) = C(f_{[0,s)}, f_{[0,t)})$, $s,t \in [0,1]$. The following two results describe the asymptotic behavior of $F_N(t)$ in the distributional and a.s.\ sense. For basic facts about functional limit theorems we refer to Billingsley \cite{BIL}.
\begin{thm}\label{theorem2} Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $\delta >0$.
Then the function $\Gamma (s,t)$ is symmetric, positive semidefinite and continuous on the unit square, and
\begin{equation}\label{donsker}
\sqrt{N} (F_N(t) - t) \overset{D[0,1]}{\longrightarrow} K(t) \quad  \text{as} \quad N \to \infty ,
\end{equation}
where $K(t)$ is a mean zero Gaussian process on $[0,1]$ with covariance function $\Gamma (s,t)$.
\end{thm}
\noindent Relation (\ref{donsker}) expresses weak convergence in the Skorohod space $D[0,1]$, and implies that the distribution of any functional $\Psi$ of the process $\sqrt{N} (F_N(t) - t)$, $0\le t \le 1$ converges weakly to the distribution of $\Psi (K(t))$, provided that $\Psi$ is continuous in the $D[0, 1]$ topology. This gives the limit distribution of a large class of functionals of the sequence $\{S_k\}$. Since the discrepancy and the $L^p$-discrepancy are continuous functionals, we immediately obtain
\[ \begin{split} \sqrt{N} D_N(S_k) &\overset{d}{\longrightarrow} \sup_{0\le s<t \le 1} |K(t)-K(s)| , \\ \sqrt{N} D_N^{(p)}(S_k) &\overset{d}{\longrightarrow} \| K \|_{L^p([0,1])} \qquad (1 \le p \le \infty ) . \end{split} \]
Note that $\Gamma (0,0)=\Gamma (1,1)=0$. From Theorem \ref{theorem1} it follows easily that $\Gamma$ is (strictly) positive definite on the open interval $(0,1)$; in particular, the limit distributions in the previous formula are nondegenerate.

The next result is the LIL analogue of Theorem \ref{theorem2}.
\begin{thm}\label{theorem3}
Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $\delta >0$. Then with probability $1$, the sequence of functions
\[ \frac{N (F_N(t)-t)}{\sqrt{2 N \log \log N}}, \quad 0\le t \le 1 \]
is relatively compact in the Skorohod space $D[0,1]$, and its class of limit functions in the $D[0,1]$ topology is identical with the closed unit ball $B(\Gamma )$ of the reproducing kernel Hilbert space determined by the covariance function $\Gamma (s,t)$.
\end{thm}
\noindent As a consequence, with probability $1$ we have
\[ \begin{split} \limsup_{N\to\infty} \frac{N D_N(S_k)}{\sqrt{2 N \log \log N}} &= \sup_{y \in B(\Gamma )} \sup_{0 \le s<t\le 1} |y(t)-y(s)|, \\ \limsup_{N\to\infty} \frac{N D_N^{(p)}(S_k)}{\sqrt{2 N \log \log N}} &= \sup_{y\in B(\Gamma )} \|y\|_{L^p([0,1])} \qquad (1 \le p \le \infty ) . \end{split} \]
Note that the value of these limsups are positive and finite. Indeed, finiteness is a consequence of the fact that $B(\Gamma)$ is compact in the supremum norm (see the proof of Theorem \ref{theorem3}), whereas positivity follows from Theorem \ref{theorem1} with $f=f_{[0,t)}$ for some $0<t<1$. Again, Theorem \ref{theorem3} yields laws of the iterated logarithm for a large class of functionals of the process $\sqrt{N} (F_N(t)-t)$, $0\le t \le 1$; see Strassen \cite{STR}.

The connection of reproducing kernel Hilbert spaces with the functional LIL for Gaussian processes was first observed by Oodaira \cite{OO}; analogous results for mixing processes and lacunary series were obtained by Philipp \cite{PH}. The latter paper provides a good introduction to the probabilistic aspects of reproducing kernel Hilbert spaces for the reader not familiar with this concept. The same paper also provides the background for the chaining techniques required in the proof of our theorems. For the general theory of reproducing kernel Hilbert spaces we refer to Aronszajn \cite{A}.

If $X_1$ is uniformly distributed on $[0,1)$, then it is not difficult to see that $\{ S_k \}$ is an i.i.d.\ sequence of uniformly distributed random variables. In this simplest case we have $C(f,g)=\E f(U)g(U)$, and consequently $\Gamma (s,t) = \min \{ s,t \} -st$ is the covariance function of the Brownian bridge. The corresponding functional CLT and functional LIL are due to Donsker \cite{DO} and Finkelstein \cite{F}.

In general, $\{S_k \}$ is a sequence of weakly dependent random variables, and there exist in the literature functional CLT's and LIL's for the empirical distribution function for such sequences, but their assumptions are considerably more restrictive than those for the central limit theorem. For example, for $\varphi$-mixing sequences with mixing rate $\varphi_n$, a functional CLT for the empirical distribution function holds under $\sum_{n=1}^\infty n^2 \varphi_n^{1/2}<\infty$  (see Billingsley \cite[p.\ 197, Theorem 22.1]{BIL}), while the (ordinary) CLT for partial sums holds under $\sum_{n=1}^\infty \varphi_n^{1/2}<\infty$. Also, most results in the literature deals with stationary processes, while $\{S_k \}$ is nonstationary.

\section{Couplings and the Kolmogorov metric}

We start with two useful observations on the relationship between the Kolmogorov metric, functions of bounded variation and couplings. The total variation of a function $f$ on $[0,1]$ is denoted by $V(f)$, and the $L^p([0,1])$-norm by $\| f \|_p$.
\begin{lem}\label{randomkoksma}
For any $[0,1]$-valued random variables $X$ and $Y$, and any function $f: [0,1] \to \mathbb{R}$ of bounded variation,
\[ \left| \E f(X)- \E f(Y) \right| \le V(f) \sup_{0 \le t \le 1} \left| \P ( X \le t) -\P (Y \le t) \right| . \]
\end{lem}
\noindent In the special case when the distribution of $X$ is finitely supported with equal weights and $Y$ is uniformly distributed on $[0,1]$, this is known as Koksma's inequality \cite[Chapter 2, Theorem 5.1]{KN}. In particular, for any real-valued random variables $X$ and $Y$, and any $f \in \mathcal{F}$ we have
\begin{equation}\label{maintool}
\left| \E f (X) - \E f (Y) \right| \le V(f) \sup_{0 \le t \le 1} \left| \P ( \{ X \} \le t) -\P (\{Y \} \le t) \right| .
\end{equation}
The claim formally follows from integration by parts; however, as one-sided continuity of $f$ is not assumed, integration by parts strictly speaking cannot be applied.
\begin{proof}[Proof of Lemma \ref{randomkoksma}] First, let $g:[0,1] \to \R$ be a nondecreasing function with $g(0)=0$. Note that for any $x \in [0,g(1)]$ the set $L(x)=\{ y \in [0,1] : g(y) > x \}$ is an interval of the form $[t,1]$ or $(t,1]$ for some $0 \le t \le 1$. Therefore
\[ \begin{split} \left| \E g(X) -\E g(Y) \right| &= \left| \int_0^{g(1)} \left( \P (g(X) >x) - \P (g(Y) >x) \right) \, \mathrm{d}x \right| \\ &= \left| \int_0^{g(1)} \left( \P ( X \in L(x)) - \P (Y \in L(x)) \right) \, \mathrm{d}x \right| \\ &\le g(1) \sup_{0 \le t \le 1} \left| \P (X \le t) -\P (Y \le t) \right| . \end{split} \]
Here $g(1)=V(g)$. Since the left hand side and $V(g)$ are both invariant under adding a constant to $g$, we have
\begin{equation}\label{nondecreasing}
\left| \E g(X) -\E g(Y) \right| \le V(g) \sup_{0 \le t \le 1} \left| \P (X \le t) -\P (Y \le t) \right|
\end{equation}
for any nondecreasing function $g:[0,1] \to \R$. Finally, consider the Jordan decomposition $f=g_1-g_2$, where $g_1, g_2$ are nondecreasing functions and $V(f)=V(g_1)+V(g_2)$. Applying \eqref{nondecreasing} to $g_1$ and $g_2$, the claim follows.
\end{proof}

\begin{lem}\label{couplinglemma} Let $X$ and $\xi$ be independent $[0,1]$-valued random variables, and assume that $\xi$ is uniformly distributed on $[0,1]$. Then there exists a uniformly distributed $[0,1]$-valued random variable $U$ such that $U$ is a function of $X$ and $\xi$, and $|U-X| \le \sup_{0 \le t \le 1} |\P (X \le t) -t|$.
\end{lem}

\begin{proof} Let $F(x)=\P (X\le x)$ and $F^-(x)=\P (X < x)$, and define
\[ U=\xi F(X)+(1-\xi )F^-(X). \]
We have $\left| U-X \right| \le \sup_{0 \le t \le 1} \left| F(t)-t \right|$ by construction. To show that $U$ is uniformly distributed on $[0,1]$, let $D$ denote the set of points at which $F$ is not continuous. If $x \in [0,1] \backslash \bigcup_{d \in D} [F^{-}(d),F(d))$, then $t= \sup \{ u \in [0,1] : F(u) \le x \}$ is seen to satisfy $x=F(t)$. Thus $\{ U \le x \} = \{ F(X) \le F(t) \} = \{ X \le t \}$ as random events, and so $\P (U \le x ) = \P (X \le t) = F(t)=x$. If $x \in [F^{-}(d),F(d))$ for some $d \in D$, then
\[ \left\{ U \le x \right\} = \{ X < d \} \cup \left\{ X=d \textrm{ and } \xi \le \frac{x-F^{-}(d)}{F(d)-F^{-}(d)} \right\} \]
as random events, hence using the independence of $X$ and $\xi$ we get
\[ \begin{split} \P (U \le x) &= \P (X<d) + \P (X=d) \P \left( \xi \le \frac{x-F^{-}(d)}{F(d)-F^{-}(d)} \right) \\ &=F^{-}(d) + \left( F(d)-F^{-}(d) \right) \cdot \frac{x-F^{-}(d)}{F(d)-F^{-}(d)} =x. \end{split} \]
We note that if $X$ has a continuous distribution, then $F=F^{-}$ and so $U=F(X)$ has the same property; a classical observation of probability theory.
\end{proof}

Given two $[0,1]$-valued random variables $X$ and $Y$ with distribution functions $F(t)=\P (X \le t)$ and $G(t)=\P (Y \le t)$, their distance in the Kolmogorov metric (or uniform metric) is defined as
\[ d_{\mathrm{Kol}}(X,Y) = \sup_{0 \le t \le 1} |\E I_{[0,t]}(X) - \E I_{[0,t]}(Y)| = \sup_{0 \le t \le 1}|F(t)-G(t)|. \]
Lemma \ref{randomkoksma} states that here the family of test functions $I_{[0,t]}$, $0 \le t \le 1$ can be replaced by the larger class of functions of bounded variation; that is,
\[ d_{\mathrm{Kol}}(X,Y) = \sup_{V(f) \le 1} |\E f(X) - \E f(Y)| . \]
Another classical distance for $[0,1]$-valued random variables is the $p$-Wasserstein metric, defined as
\[ W_p (X,Y) = \inf \left\{ \| X'-Y' \|_p \, : \, X' \overset{d}{=} X \textrm{ and } Y' \overset{d}{=}Y \right\} \qquad (1 \le p \le \infty ). \]
Here $X' \overset{d}{=} X$ means that $X'$ and $X$ have the same distribution; the infimum is thus over all couplings $(X', Y')$ of $X$ and $Y$. The $p$-Wasserstein metric can be expressed in terms of the distribution functions \cite[Theorem 2.10]{BL} as $W_p (X,Y) = \| F^{-1} - G^{-1} \|_p$, $1 \le p \le \infty$, where $F^{-1} (x) = \inf \{ t \in [0,1] \, : \, F(t) \ge x \}$, $0<x<1$ is the generalized inverse of $F$.

It is not difficult to see that for $G(t)=t$,
\[ \| F^{-1} -G^{-1} \|_{\infty} = \sup_{0<x<1} |F^{-1}(x) -x| = \sup_{0<t<1} |F(t) -t|. \]
Indeed, by the left-continuity of $F^{-1}$ the essential supremum and the actual supremum on $0<x<1$ are equal; to see the second equality note that $|F(t)-t| \le c$ geometrically means that the graph of $F(t)$, $0<t<1$ stays between the lines $t \pm c$, and that this is invariant under taking the inverse function. Since in the definition of the Kolmogorov metric the supremum can also be taken over $0<t<1$, it follows that $W_{\infty} (X,U) = d_{\mathrm{Kol}}(X,U)$ for a uniformly distributed $U$; Lemma \ref{couplinglemma} is basically a constructive form of this fact. We summarize these observations in the following lemma.
\begin{lem} Let $X$ and $U$ be $[0,1]$-valued random variables, and assume that $U$ is uniformly distributed. Then
\[ W_{\infty} (X,U) = \sup_{V(f) \le 1} |\E f(X) - \E f(U)|, \]
where the supremum is over all functions of bounded variation $f: [0,1] \to \mathbb{R}$ of total variation $V(f) \le 1$.
\end{lem}
\noindent This duality between $W_{\infty}$ and functions of bounded variation seems not to have been observed before. The classical Kantorovich duality theorem \cite[Theorem 2.5]{BL}
\[ W_1 (X,Y) = \sup_{\mathrm{Lip} (f) \le 1} |\E f(X) - \E f(Y) | , \]
where the supremum is over all $f:[0,1] \to \mathbb{R}$ such that $|f(x)-f(y)| \le |x-y|$, establishes a duality between $W_1$ and Lipschitz functions. Amusingly, the general inequality $W_1 (X,U) \le W_{\infty} (X,U)$ thus reduces to the fact that every Lipschitz function is of bounded variation.

\section{The variance}

We now find the precise asymptotics of the variance.
\begin{lem}\label{varianceprop1} Suppose that $\psi (k) \ll k^{-(1+\delta)}$ for some $\delta >0$.
\begin{enumerate}
\item[(i)] For any $f,g \in \mathcal{F}$ the series in \eqref{C} are absolutely convergent, and $|C(f,g)| \ll V(f) \| g \|_1 + V(g) \| f \|_1$. Further, $C(f,g)$ is bilinear, symmetric and positive semidefinite in $f,g$. In addition, $C(f,f)=0$ if and only if $f=0$ a.e.
\item[(ii)] For any $f \in \mathcal{F}$ and any integers $M \ge 0$ and $N \ge 1$,
\[ \begin{split} \E \left( \sum_{k=M+1}^{M+N} f(S_k) \right)^2 = C(f,f) N +O \Big( &V(f) \left( \| f \|_1 + V(f) (M+1)^{-\delta} \right)^{\delta /(1+\delta )} \times \\ &\left( \| f \|_1 N + V(f) (M+1)^{-\delta} \right)^{1/(1+\delta )} \Big) . \end{split} \]
\end{enumerate}
The implied constants in (i) and (ii) depend only on the distribution of $X_1$ and $\delta$.
\end{lem}

\begin{proof} First, we prove (i). In the definition \eqref{C} of $C(f,g)$ the variable $U$ is independent of $X_1, X_2, \dots$, therefore we can write
\[ \E f(U)g(U+S_k) = \E \int_0^1 f(u)g(u+S_k) \, \mathrm{d}u = \E G(S_k), \]
where $G(x)=\int_0^1 f(u)g(u+x) \, \mathrm{d} u$. It is not difficult to see e.g.\ directly from the definition of total variation that $V(G)\le V(g) \| f \|_1$. Applying \eqref{maintool} to $X=S_k$ and a random variable $Y$ uniformly distributed on $[0,1)$ and noting that $\E G(Y)=0$, we get $|\E f(U)g(U+S_k)| \le V(g) \| f \|_1 \psi (k)$. Repeating the same argument for $\E g(U) f(U+S_k)$, the absolute convergence of the series in \eqref{C} follows.  Since $|\E f(U)g(U)| \le V(g) \| f \|_1$, we also get $|C(f,g)|\ll V(f) \| g \|_1 + V(g) \| f \|_1$. Clearly $C(f,g)$ is bilinear and symmetric in $f,g$.

Let $\hat{f}(h)=\int_0^1 f(x) e^{-2 \pi i h x} \, \mathrm{d}x$, $h \in \mathbb{Z}$ denote the Fourier coefficients of $f$. We claim that
\begin{equation}\label{fouriervariance}
C(f,f)=\sum_{h \neq 0} |\hat{f}(h)|^2 \frac{1-|\varphi (2 \pi h)|^2}{|1-\varphi (2 \pi h)|^2} ,
\end{equation}
where $\varphi (x)=\E \exp (i x X_1)$ is the characteristic function of $X_1$. Here $|\varphi (2 \pi h)|<1$ for all integers $h \neq 0$; indeed, otherwise $X_1$ would have a lattice distribution with a rational span, which is impossible by the assumption $\psi (k) \to 0$. From \eqref{fouriervariance} it will follow immediately that $C(f,f) \ge 0$, and that $C(f,f)=0$ if and only if $\hat{f}(h)=0$ for all integers $h \neq 0$; the latter condition is equivalent to $f=0$ a.e. We will also have
\[ \sum_{i,j=1}^r C(f_i,f_j) x_i x_j = C \left( \sum_{i=1}^r f_i x_i, \sum_{i=1}^r f_i x_i \right) \ge 0 \]
for any $f_1, f_2, \dots, f_r \in \mathcal{F}$ and any $x_1, x_2, \dots, x_r \in \mathbb{R}$, proving that $C(f,g)$ is positive semidefinite.

To see \eqref{fouriervariance}, assume first that $f$ is a trigonometric polynomial; that is, $\hat{f}(h)=0$ for all but finitely many integers $h$. As before, $\E f(U)f(U+S_k)=\E G(S_k)$ with $G(x)=\int_0^1 f(u)f(u+x) \, \mathrm{d}u$. Since $\hat{G}(h)=|\hat{f}(h)|^2$, we have $G(x)=\sum_{h \neq 0} |\hat{f}(h)|^2 e^{2 \pi i hx}$. Therefore
\[ \sum_{k=1}^{\infty} \E G(S_k) = \sum_{h \neq 0} \sum_{k=1}^{\infty} |\hat{f}(h)|^2 \varphi (2 \pi h)^k = \sum_{h \neq 0} |\hat{f}(h)|^2 \frac{\varphi (2 \pi h)}{1-\varphi (2 \pi h)} .  \]
Combining the $h$ and $-h$ terms, and using $\E f(U)^2 = \sum_{h \neq 0} |\hat{f}(h)|^2$, we obtain \eqref{fouriervariance}. For a general $f \in \mathcal{F}$, consider
\[ f_H (x) = \int_0^1 f(x-y) F_H(y) \, \mathrm{d}y = \sum_{0<|h|<H} \left( 1-\frac{|h|}{H} \right) \hat{f}(h) e^{2 \pi i hx} , \]
where $F_H(x)=\sum_{|h|<H}(1-|h|/H)e^{2 \pi i hx}$ is the Fej\'er kernel of order $H$. Note that $f_H$ is a trigonometric polynomial; in fact, it is a Ces\`aro mean of the formal Fourier series of $f$. By the special case proved above, we have
\[ C(f_H, f_H) = \sum_{0<|h|<H} \left( 1-\frac{|h|}{H} \right)^2 |\hat{f}(h)|^2 \frac{1-|\varphi (2 \pi h)|^2}{|1-\varphi (2 \pi h)|^2} \to \sum_{h \neq 0} |\hat{f}(h)|^2 \frac{1-|\varphi (2 \pi h)|^2}{|1-\varphi (2 \pi h)|^2} \]
as $H \to \infty$. On the other hand, letting $G(x)=\int_0^1 f(u)f(u+x) \, \mathrm{d}u$ and $G_H(x)=\int_0^1 f_H(u)f_H(u+x) \, \mathrm{d}u$, by Lemma \ref{randomkoksma} we have
\[ \begin{split} \left| C(f,f) - C(f_H, f_H) \right| &= \left| \E f(U)^2 + 2 \sum_{k=1}^{\infty} \E G(S_k) - \E f_H(U)^2 - 2 \sum_{k=1}^{\infty} \E G_H (S_k) \right| \\ &\le \left| \E f(U)^2 - \E f_H(U)^2 \right| + 2 \sum_{k=1}^{\infty} |\E (G-G_H)(S_k)| \\ &\le \left| \| f \|_2^2 - \| f_H \|_2^2 \right| + 2 \sum_{k=1}^{\infty} V(G-G_H) \psi (k) \\ &\ll \left| \| f \|_2^2 - \| f_H \|_2^2 \right| + V(G-G_H) . \end{split} \]
Since $f_H \to f$ in $L^2([0,1])$, it will be enough to prove $V(G-G_H) \to 0$ as $H \to \infty$. Writing $f=f_H+(f-f_H)$ in the definition of $G(x)$ and using the integral transformation $u \mapsto u-x \pmod{1}$ in one of the terms, we get
\[ (G-G_H)(x) = \int_0^1 (f-f_H)(u) \left( f_H(u-x) + f(u+x) \right) \, \mathrm{d} u , \]
and consequently $V(G-G_H) \le \| f-f_H \|_1 \left( V(f_H) + V(f) \right)$. Using $F_H \ge 0$ it is readily seen that $V(f_H) \le V(f)$. Hence $V(G-G_H) \le 2V(f) \| f-f_H \|_1 \to 0$, finishing the proof of \eqref{fouriervariance} for a general $f \in \mathcal{F}$.

Next, we prove (ii). We may assume $V(f)=1$. Let $U$ be as in the definition \eqref{C} of $C(f,f)$. Expanding the square in the claim we get
\begin{equation}\label{expsquare}
\E \left( \sum_{k=M+1}^{M+N} f(S_k) \right)^2 = \sum_{k=M+1}^{M+N} \E f(S_k)^2 + 2 \sum_{M+1 \le k< \ell \le M+N } \E f(S_k) f(S_{\ell}) .
\end{equation}
To estimate the diagonal terms, let us apply \eqref{maintool} to $X=S_k$ and $Y=U$. Noting that $V(f^2) \le 2 V(f)^2=2$, we obtain $\E f(S_k)^2 = \E f(U)^2+O(\psi (k))$. The assumption $\psi (k) \ll k^{-(1+\delta)}$ thus ensures that $\sum_{k=M+1}^{M+N} \E f(S_k)^2 = N \E f(U)^2 +O\left( (M+1)^{-\delta} \right)$.

We now estimate the off-diagonal terms in two different ways. For given $k<\ell$ let us write $f(S_k) f(S_{\ell}) = f(S_k) f(S_k +(S_{\ell}-S_k))$. First, let $\mathcal{F}_k$ denote the $\sigma$-algebra generated by $X_1, X_2, \dots, X_k$, and note that $S_{\ell}-S_k$ is independent of $\mathcal{F}_k$. Let us apply \eqref{maintool} to $X=S_{\ell}-S_k$, a random variable $Y$ uniformly distributed on $[0,1)$ independent of $\mathcal{F}_k$, and the function $g(x)=f(S_k) f(S_k+x)$ to estimate the conditional expectation with respect to $\mathcal{F}_k$. Since $\E (g(Y) | \mathcal{F}_k)=0$ and $V(g) \le |f(S_k)| V(f)=|f(S_k)|$, we have
\[ \left| \E \left( f(S_k) f(S_{\ell}) \mid \mathcal{F}_k \right) \right| \le |f(S_k)| \psi (\ell -k) . \]
Taking the (total) expectation, we obtain $|\E f(S_k) f(S_{\ell})| \le \E |f(S_k)| \psi (\ell -k)$. We can apply \eqref{maintool} again, this time to $X=S_k$, a random variable $Y$ uniformly distributed on $[0,1)$ and $|f|$. Since $\E |f(Y)|=\| f \|_1$ and $V(|f|) \le V(f) = 1$, we get
\begin{equation}\label{skslfirst}
|\E f(S_k) f(S_{\ell})| \le (\| f \|_1 + \psi (k)) \psi (\ell -k).
\end{equation}
Second, let $\mathcal{G}_k$ denote the $\sigma$-algebra generated by $X_{k+1}, X_{k+2}, \dots$. Note that $S_k$ and $U$ are independent of $\mathcal{G}_k$. Let us apply \eqref{maintool} to $X=S_k$, $Y=U$ and the function $g(x)=f(x)f(x+(S_{\ell}-S_k))$. Since $V(g) \le 2 V(f)^2=2$, we obtain
\[ \E \left( f(S_k) f(S_{\ell}) \mid \mathcal{G}_k \right) = \E \left( f(U)f(U+(S_{\ell}-S_k)) \mid \mathcal{G}_k \right) + O(\psi (k)) . \]
Note that $f(U)f(U+(S_{\ell}-S_k))$ has the same distribution as $f(U)f(U+S_{\ell -k})$. Taking the (total) expectation, we thus get
\begin{equation}\label{skslsecond}
\E f(S_k) f(S_{\ell}) = \E f(U) f(U+S_{\ell -k})+O(\psi (k)).
\end{equation}
As observed in (i), here $|\E f(U) f(U+S_{\ell -k})| \le \| f \|_1 \psi (\ell -k)$.

Let $1 \le K \le N$ be an integer, to be chosen. We will estimate the off-diagonal terms for which $\ell -k > K$, and those for which $\ell -k \le K$ separately. The assumption $\psi (k) \ll k^{-(1+\delta)}$ and \eqref{skslfirst} show that
\[ \begin{split} \sum_{\substack{M+1 \le k< \ell \le M+N \\ \ell -k >K}} \E f(S_k) f(S_{\ell}) &\ll \left( \| f \|_1 N+(M+1)^{-\delta} \right) K^{-\delta}, \\ \sum_{M+N-K<k< \ell \le M+N} \E f(S_k) f(S_{\ell}) &\ll \| f \|_1 K + (M+1)^{-\delta}. \end{split} \]
For a fixed $M+1 \le k \le M+N-K$, from \eqref{skslsecond} we obtain
\[ \begin{split} \sum_{\ell =k+1}^{k+K} \E f(S_k) f(S_{\ell}) &= \sum_{d=1}^K \E f(U) f(U+S_d) + O \left( K \psi (k) \right) \\ &= \sum_{d=1}^{\infty} \E f(U) f(U+S_d) + O \left( \| f \|_1 \sum_{d=K+1}^{\infty} \psi (d) + K \psi (k) \right) . \end{split} \]
Summing over $M+1 \le k \le M+N-K$ and using the assumption $\psi (k) \ll k^{-(1+\delta)}$, we altogether get
\begin{multline*}
\sum_{M+1 \le k< \ell \le M+N} \E f(S_k) f(S_{\ell}) = \\ N \sum_{d=1}^{\infty} \E f(U) f(U+S_d)+O \left( \left( \| f \|_1 N +(M+1)^{-\delta} \right) K^{-\delta} + \left( \| f \|_1 + (M+1)^{-\delta} \right) K \right) .
\end{multline*}
Choosing $K = \lfloor \left( \| f \|_1 N +(M+1)^{-\delta} \right)^{1/(1+\delta)} / \left( \| f \|_1 +(M+1)^{-\delta} \right)^{1/(1+\delta)} \rfloor$, in \eqref{expsquare} we thus have
\begin{multline*}
\E \left( \sum_{k=M+1}^{M+N} f(S_k) \right)^2 = \\ C(f,f) N+ O\left( \left( \| f \|_1 + (M+1)^{-\delta} \right)^{\delta /(1+ \delta )} \left( \| f \|_1 N + (M+1)^{-\delta} \right)^{1/(1+\delta )} \right) .
\end{multline*}
\end{proof}

\section{Approximation by independent variables}\label{approximationsubsection}

In this section we approximate $\sum_{k=M+1}^{M+N} f(S_k)$ by a sum of independent random variables. A basic tool is the following result, implicit in Schatte \cite{SCH3}, \cite{SCH4}.
\begin{lem}\label{schattelemma}
Let $\xi_1, \xi_2, \ldots$ be independent random variables uniformly distributed on $[0,1)$, independent of the $X_n$'s, and let $I_1, I_2, \ldots$ be disjoint blocks of integers such that between $I_n$ and $I_{n+1}$ there are $\ell_n\ge 1$ integers. Then there exists a sequence $\delta_1, \delta_2, \dots$ of random variables such that $\delta_n$ is measurable with respect to $\xi_n$ and the $X_j$'s between $I_n$ and $I_{n+1}$, and $|\delta_n| \leq \psi(\ell_n)$, $n=1, 2, \ldots$; further, the random vectors
$$(\{S_i\}, i\in I_1), \  (\{S_i - \delta_1 \}, i \in I_2),  \ldots,  (\{S_i - \delta_{n - 1}\}, i \in I_n), \ldots $$
are independent and they have, except for the first one, uniformly distributed components.
\end{lem}

\begin{proof} We begin with a simple remark. Let $X$ be a random variable and let $U$ be a uniformly distributed random variable independent of $X$. Then $\{ X+U \}$ is uniformly distributed on $[0,1)$ and is independent of $X$, as one can see immediately by conditioning $\{ X+U \}$ on $\{X=c\}$.

Let now
$$ I_n := \left\{ p_n +1, p_n + 2, \dots, p_n + b_n\right\}, \quad n=1, 2, \ldots $$
We claim that
\[ \delta_n =\{ S_{p_{n + 1}} - S_{p_n + b_n} \} -U (\{ S_{p_{n + 1}}  - S_{p_n + b_n}\} , \xi_n ), \quad n=1, 2, \ldots \]
satisfies the lemma, where $U (\{ S_{p_{n + 1}} - S_{p_n + b_n} \}, \xi_n )$ is the uniform random variable constructed in Lemma \ref{couplinglemma} for the variables $\{ S_{p_{n + 1}} - S_{p_n + b_n} \}$ and $\xi_n$. Since $S_{p_{n+1}} - S_{p_n+ b_n} \stackrel{d}{=} S_{\ell_n}$, by the definition of $\psi (k)$ and the claim of Lemma \ref{couplinglemma}, $| \delta_n| \leq \psi(\ell_n)$.

Furthermore, we have
\begin{align*}
\{ S_{p_2 + 1} - \delta_1 \} &=
\{ \left(X_1 + \dots + X_{p_1+b_1}\right) +X_{p_2+1} +U (\{ S_{p_2} -S_{p_1+b_1}\}, \xi_1 ) \} \\
\vdots & \\
\{ S_{p_2 + b_2} - \delta_1 \} & =\{ \left( X_1 + \dots + X_{p_1+b_1}\right) +(X_{p_2+1}+ \cdots + X_{p_2+b_2}) \\
\phantom{9999999} & + U (\{ S_{p_2} -S_{p_1+b_1}\} , \xi_1 ) \} .
\end{align*}
Put
\begin{align*}
V &= \{ X_1 + \dots + X_{p_1+b_1}+ U (\{ S_{p_2} - S_{p_1+b_1}\}, \xi_1 )\},\\
X &= (X_{p_2+1}, X_{p_2+1} +X_{p_2+2}, \ldots, X_{p_2+1} + \ldots +X_{p_2+b_2} ).
\end{align*}
Let $\omega$ be a point of the underlying probability space, and put
\[ c=(X_1 + \dots + X_{p_1+b_1})(\omega ). \]
The conditional distribution of the vector $(\{S_{p_2+1}-\delta_1\}, \ldots, \{S_{p_2+b_2}-\delta_1\})$ relative to $\sigma\{X_1, \ldots, X_{p_1+b_1}\}$, evaluated at $\omega$, equals the distribution of
\[ \{X+c+ U (\{S_{p_2} - S_{p_1+b_1}\}, \xi_1)\} ,\]
where adding the scalar $V^*=c+ U (\{S_{p_2} - S_{p_1+b_1}\}, \xi_1)$ to $X$ is meant coordinatewise. But $U (\{S_{p_2} - S_{p_1+b_1}\}, \xi_1)$ is a uniform random variable independent of $X$ and so is $\{V^*\}$, showing that the distribution of $\{X+V^*\}$ does not depend on $c$ and consequently, $(\{S_{p_2+1}-\delta_1\}, \ldots, \{S_{p_2+b_2}-\delta_1\})$ is independent of $\sigma\{X_1, \ldots, X_{p_1+b_1}\}$ and thus of $(\{S_i\}, \, i\in I_1)$. By our remark at the beginning of the proof, the coordinates of $(\{S_{p_2+1} -\delta_1\}, \ldots, \{S_{p_2+b_2} -\delta_1\})$ are uniformly distributed.

Now let $n \ge 2$. As before, the vector
\begin{equation}\label{vectorn}
(\{ S_{p_{n+1}+1} -\delta_n \}, \ldots, \{ S_{p_{n+1}+b_{n+1}} -\delta_n \} )
\end{equation} can be written as $\{ X+V \}$, where
\begin{align*}
V &= \{ X_1 + \dots + X_{p_n+b_n} + U (\{ S_{p_{n+1}} - S_{p_n+b_n} \} , \xi_n ) \} ,\\
X &= (X_{p_{n+1}+1}, X_{p_{n+1}+1} +X_{p_{n+1}+2} , \ldots, X_{p_{n+1}+1} + \cdots +X_{p_{n+1}+b_{n+1}} ).
\end{align*}
Repeating the argument used in the case $n=1$, it follows that the vector (\ref{vectorn}) is independent of $\sigma\{X_1, \ldots, X_{p_n+b_n}, \xi_1, \ldots, \xi_{n-1} \}$
and thus of
$$(\{S_i\}, i\in I_1), \  (\{S_i - \delta_1\}, i \in I_2),  \ldots,  (\{S_i - \delta_{n - 1}\}, i \in I_n),
$$
and has uniform coordinates. This completes the proof of the lemma.
\end{proof}

Fix a function $f \in \mathcal{F}$, an integer $M \ge 0$ and a sequence of auxiliary random variables $\xi_2, \xi_3, \ldots$ uniformly distributed on $[0,1)$, independent of $X_1, X_2, \ldots$. Consider a decomposition of the infinite set of integers $\{ M+1, M+2, \dots \}$ into nonempty, consecutive blocks $H_1, J_1, H_2, J_2, \ldots$, and let $T_i=T_i^{(f)}=\sum_{k \in H_i} f(S_k)$ and $D_i=D_i^{(f)}=\sum_{k \in J_i} f(S_k)$ for all $i \ge 1$. If $M+N=\max J_R$ for some $R \ge 1$, we thus have
\[ \sum_{k=M+1}^{M+N} f(S_k) = \sum_{i=1}^R T_i + \sum_{i=1}^R D_i . \]
Using Lemma \ref{schattelemma} we can introduce two sequences of random variables $\delta_2, \delta_3, \ldots$ and $\delta_2', \delta_3', \ldots$ with the following properties. First, $\delta_i$ is measurable with respect to the $\sigma$-algebra generated by $X_k$, $k \in J_{i-1}$ and $\xi_i$, and $|\delta_i| \le \psi (|J_{i-1}|)$ for all $i \ge 2$; similarly, $\delta_i'$ is measurable with respect to the $\sigma$-algebra generated by $X_k$, $k \in H_i$ and $\xi_i$, and $|\delta_i'| \le \psi (|H_i|)$ for all $i \ge 2$. Letting $T_i^*=T_i^{(f)*}=\sum_{k \in H_i} f(S_k-\delta_i)$ and $D_i^*=D_i^{(f)*}=\sum_{k \in J_i} f(S_k-\delta_i')$, we also have that $T_2^*, T_3^*, \ldots$ are independent and $\E T_i^*=0$ for all $i \ge 2$; similarly, $D_2^*, D_3^*, \ldots$ are independent and $\E D_i^*=0$ for all $i \ge 2$. Under the condition $\psi (k) \ll k^{-(1+\delta)}$ the variances are given by
\begin{equation}\label{ti*di*variance}
\begin{split} \E T_i^{*2}&=C(f,f) |H_i|+O\left( V(f)^2 |H_i|^{1/(1+\delta )} \right), \\ \E D_i^{*2}&= C(f,f) |J_i|+O\left( V(f)^2 |J_i|^{1/(1+\delta )} \right) \end{split}
\end{equation}
with implied constants depending only on the distribution of $X_1$ and $\delta$. Indeed, conditioning on the $\sigma$-algebra $\mathcal{A}_i$ generated by the variables $X_k$, $k \in [1,M] \cup H_1 \cup J_1 \cup \cdots \cup H_{i-1} \cup J_{i-1}$ and $\xi_i$, we can write $T_i^*=\sum_{\ell=1}^{|H_i|} f(\tilde{S_\ell}-Z_i)$ with some $\mathcal{A}_i$-measurable random variable $Z_i$, where $\tilde{S_{\ell}}$ are the partial sums of the i.i.d.\ sequence $X_k$, $k \in H_i$. Applying Lemma \ref{varianceprop1} to $\tilde{f}(x)=f(x-Z_i)$ and noting that $C(\tilde{f},\tilde{f})=C(f,f)$ and $V(\tilde{f})=V(f)$, we obtain $\E \left( T_i^{*2} \mid \mathcal{A}_i \right) = C(f,f) |H_i|+O( V(f)^2 |H_i|^{1/(1+\delta )})$. Taking the (total) expectation, the first relation in \eqref{ti*di*variance} follows; the proof of the second relation is analogous.

\begin{lem}\label{approxerror1} Assume that $\psi (k) \ll k^{-(1+\delta)}$ with some $0<\delta<1$. If we choose the sizes of the blocks as $|H_i|=\lceil i^c \rceil$ and $|J_i|=\lceil i^{c'} \rceil$ for all $i \ge 1$, where $1/2<c<1$ is a fixed constant and $c'=(c+\delta/2)/(1+\delta)$, then
\[ \E \left( \sum_{i=2}^R (T_i - T_i^*) \right)^2 \ll V(f)^2 R^{1-\delta /2}, \quad \E \left( \sum_{i=2}^R (D_i - D_i^*) \right)^2 \ll V(f)^2 R^{1-\delta /2} \]
for all $R \ge 2$ with implied constants depending only on the distribution of $X_1$, $\delta$ and $c$.
\end{lem}

\begin{proof} We shall only give a proof for $\sum_{i=2}^R (T_i-T_i^*)$, as the proof for $\sum_{i=2}^R (D_i-D_i^*)$ is analogous. We may assume $V(f)=1$. Note that $c'<c$. Expanding the square we get
\begin{equation}\label{expandsq}
\E \left( \sum_{i=2}^R (T_i - T_i^*) \right)^2 = \sum_{i=2}^R \E (T_i-T_i^*)^2 + 2 \sum_{2 \le i<j\le R} \E (T_i-T_i^*)(T_j-T_j^*) .
\end{equation}
Let us first estimate the diagonal term $\E (T_i-T_i^*)^2$. Let $\mathcal{F}_i$ denote the $\sigma$-algebra generated by the variables $X_k$, $k \in J_{i-1}$ and $\xi_i$, and let $Y_i=\sum_{k \in J_{i-1}} X_k$. Note that in $T_i-T_i^*=\sum_{k \in H_i} (f((S_k-Y_i)+Y_i)-f((S_k-Y_i)+Y_i-\delta_i))$ the variables $Y_i$ and $\delta_i$ are $\mathcal{F}_i$-measurable, and $(S_k-Y_i)$ is independent of $\mathcal{F}_i$. Hence we can apply Lemma \ref{varianceprop1} to the i.i.d.\ sequence obtained from $X_1, X_2, \dots$ by deleting the terms $X_k$, $k \in J_{i-1}$ to estimate the conditional expectation with respect to $\mathcal{F}_i$. It is not difficult to see e.g.\ directly from the definition of total variation that the function $g(x)=f(x+Y_i)-f(x+Y_i-\delta_i)$ satisfies $V(g) \le 2 V(f)=2$ and $\| g \|_1 \ll |\delta_i|$. Therefore
\begin{multline*} \E \left( (T_i-T_i^*)^2 \mid \mathcal{F}_i \right) \ll \\ |\delta_i | \cdot |H_i| + \left( |\delta_i | + (\min H_i - |J_{i-1}|)^{-\delta} \right)^{\delta / (1+\delta )} \left( |\delta_i | \cdot |H_i| + (\min H_i - |J_{i-1}|)^{-\delta} \right)^{1/(1+ \delta )}.
\end{multline*}
Here $\min H_i - |J_{i-1}| \gg i^{c+1}$, and $|\delta_i| \le \psi (|J_{i-1}|) \ll i^{-c-\delta /2}$. Taking the (total) expectation, after simplifying the exponent we get $\E (T_i-T_i^*)^2 \ll i^{-\delta /2}$.
Summing over $2 \le i \le R$ we get that the contribution of the diagonal terms in \eqref{expandsq} is $\sum_{i=2}^R \E (T_i-T_i^*)^2 \ll R^{1-\delta/2}$.

We now estimate the off-diagonal terms in \eqref{expandsq} similarly to the proof of Lemma  \ref{varianceprop1}. For given $i<j$ let us write
\[ \begin{split} (T_i-T_i^*)(T_j-T_j^*) = \sum_{\substack{k \in H_i \\ \ell \in H_j}} &\left( f(S_k)-f(S_k -\delta_i ) \right) \times \\ &\left( f((S_{\ell}-W_{i,j})+W_{i,j})-f((S_{\ell}-W_{i,j})+W_{i,j}-\delta_j ) \right) , \end{split} \]
where $W_{i,j}=\sum_{k \in [1,M] \cup H_1 \cup J_1 \cup \cdots \cup H_{i-1} \cup J_{i-1} \cup H_i \cup J_{j-1}} X_k$. Let $\mathcal{G}_{i,j}$ denote the $\sigma$-algebra generated by the variables $X_k$, $k \in [1,M] \cup H_1 \cup J_1 \cup \cdots \cup H_{i-1} \cup J_{i-1} \cup H_i \cup J_{j-1}$ and $\xi_i$, $\xi_j$. Note that $W_{i,j}$, $\delta_i$, $\delta_j$ and $S_k$, $k \in H_i$ are $\mathcal{G}_{i,j}$-measurable. Moreover, $S_{\ell}-W_{i,j}$ is the sum of $\ell -M-|H_1|-|J_1|-\cdots - |H_{i-1}|-|J_{i-1}|-|H_i|-|J_{j-1}|$ independent copies of $X_1$, and it is independent of $\mathcal{G}_{i,j}$. We can thus apply \eqref{maintool} to $X=S_{\ell}-W_{i,j}$ and a random variable $Y$ uniformly distributed on $[0,1)$, independent of $\mathcal{G}_{i,j}$ to obtain
\begin{multline}\label{conditionalgij}
\left| \E \left( (T_i-T_i^*)(T_j-T_j^*) \mid \mathcal{G}_{i,j} \right) \right| \ll \\ \sum_{\substack{k \in H_i \\ \ell \in H_j}} \left| f(S_k)-f(S_k-\delta_i ) \right| \psi (\ell -M-|H_1|-|J_1|-\cdots - |H_{i-1}|-|J_{i-1}|-|H_i|-|J_{j-1}|) .
\end{multline}
Next, we condition with respect to the $\sigma$-algebra $\mathcal{F}_i$ generated by $X_k$, $k \in J_{i-1}$ and $\xi_i$. Let $Y_i=\sum_{k \in J_{i-1}}X_k$, and let us write $|f(S_k)-f(S_k-\delta_i )| = |f((S_k-Y_i)+Y_i)-$ $f((S_k-Y_i)+Y_i-\delta_i)|$, as before. Note that $Y_i$ and $\delta_i$ are $\mathcal{F}_i$-measurable, while $S_k-Y_i$ is independent of $\mathcal{F}_i$. Let us apply \eqref{maintool} to $X=S_k-Y_i$, a random variable $Y$ uniformly distributed on $[0,1)$, independent of $\mathcal{F}_i$ and $g(x)=|f(x+Y_i)-$ $f(x+Y_i-\delta_i)|$. As observed before, $\| g \|_1 \ll |\delta_i| \le \psi (|J_{i-1}|)$, hence
\begin{equation}\label{conditionalfi}
\E \left( |f(S_k)-f(S_k-\delta_i)| \mid \mathcal{F}_i \right) \ll \psi (|J_{i-1}|) + \psi (k-|J_{i-1}|) \ll i^{-c-\delta /2} .
\end{equation}
Taking the (total) expectation in \eqref{conditionalgij} and \eqref{conditionalfi}, and summing over $k \in H_i$ we finally deduce
\begin{multline}\label{covarianceestimate}
\left| \E (T_i-T_i^*)(T_j-T_j^*) \right| \ll \\ i^{-\delta /2} \sum_{\ell \in H_j} \psi (\ell -M- |H_1|-|J_1|-\cdots - |H_{i-1}|-|J_{i-1}|-|H_i|-|J_{j-1}|) .
\end{multline}
Now fix $2 \le i <R$. In the case $j=i+1$ the sum on the right hand side of \eqref{covarianceestimate} is $\ll 1$, thus $|\E (T_i-T_i^*)(T_{i+1}-T_{i+1}^*)| \ll i^{-\delta /2}$. In the case $j-i \ge 2$ we have $(\ell -M- |H_1|-|J_1|-\cdots - |H_{i-1}|-|J_{i-1}|-|H_i|-|J_{j-1}|) \gg$ $(j-i-1) j^c$, and hence $\left| \E (T_i-T_i^*)(T_j-T_j^*) \right| \ll (j-i-1)^{-1-\delta} j^{-c \delta} i^{-\delta /2}$. Therefore $\sum_{j=i+1}^{R} \left| \E (T_i-T_i^*)(T_j-T_j^*) \right| \ll i^{-\delta /2}$ for any fixed $2 \le i <R$. Summing over $2 \le i <R$ we finally obtain that the contribution of the off-diagonal terms in \eqref{expandsq} is also $\ll R^{1-\delta /2}$.
\end{proof}

\section{Exponential bounds}

The crucial step in the proof of laws of the iterated logarithm is to prove sharp bounds for the tail probabilities of the partial sums of the considered random variables. In this section we prove such an inequality for shifted sums $\sum_{k=M+1}^{M+N} f(S_k)$.
\begin{lem}\label{expbound}
Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $0<\delta<1$, and fix a constant $0<\rho <\delta /16$. For any integers $M \ge 0$ and $N \gg 1$, any $f \in \mathcal{F}$ such that $V(f) \le 2$ and $\| f \|_1 \gg N^{-1/2-\delta/8}$, and any $t \gg 1$ we have
\[ \P \left( \left| \sum_{k=M+1}^{M+N} f(S_k) \right|  \geq t \| f\|_1^\rho \sqrt{N} \right) \ll \exp \left( -C \| f \|_1^{-\rho} t_0 t \right) + t^{-2} \| f \|_1^{-2\rho} N^{-1/2-\delta /8} \]
with some constant $C>0$ and $t_0=\min \{ t, N^{\delta /16} \}$. The implied constants and $C$ depend only on the distribution of $X_1$, $\delta$ and $\rho$.
\end{lem}

\begin{proof} Let $c=1-\delta/2$ and $c'=(c+\delta/2)/(1+\delta)$. Let us split the set $\{ M+1, M+2, \dots \}$ into blocks $H_1, J_1, H_2, J_2, \ldots$ such that $|H_i|=\lceil i^c \rceil$ and $|J_i|=\lceil i^{c'} \rceil$ for all $i \ge 1$. Let $T_i, D_i$, $i \ge 1$ and $T_i^*, D_i^*$, $i \ge 2$ be as in Section \ref{approximationsubsection}. Fix a large enough integer $N$, and let $R$ be such that $M+N \in H_R \cup J_R$. Clearly $R = \Theta(N^{1/(c+1)})$, and $|T_i| \ll R^c \ll N^{(2-\delta)/(4-\delta)}$ for all $1 \le i \le R $, and the same holds for $D_i$. Since $N^{(2-\delta)/(4-\delta)} \ll \| f \|_1^{\rho} N^{1/2}$, the contribution of any individual block to $\sum_{k=M+1}^{M+N} f(S_k)$ is negligible. Therefore we may assume that $M+N=\max J_R$, and it will be enough to prove the claim for $\sum_{i=2}^R (T_i+D_i)$ instead of $\sum_{k=M+1}^{M+N} f(S_k)$. Recall that $T_i^*$, $i=2,3,\dots, R$ are independent, and $\E T_i^* =0$; similarly, $D_i^*$, $i=2,3,\dots, R$ are independent, and $\E D_i^* =0$. By Lemma \ref{approxerror1} we have
\[ \E \left(\sum_{i=2}^R (T_i-T_i^*)\right)^2 \ll R^{1-\delta/2} \ll N^{(1-\delta/2)/ (2-\delta /2)} \le N^{1/2-\delta /8}, \]
and the same holds for $\sum_{i=2}^R (D_i-D_i^*)$. Hence by the Chebyshev inequality
\begin{equation}\label{chebyshevineq}
\P \left( \left| \sum_{i=2}^R (T_i+D_i) -\sum_{i=2}^R (T_i^*+D_i^*)\right| \ge t \| f \|_1^\rho \sqrt{N} \right) \ll t^{-2} \| f \|_1^{-2\rho} N^{-1/2-\delta /8}.
\end{equation}
From \eqref{ti*di*variance} and $C(f,f) \ll \| f \|_1$ we get
\[ \sum_{i=2}^R \E T_i^{*2} \ll \| f \|_1 N + \sum_{i=2}^R i^{c/(1+\delta )} \ll \| f \|_1 N + N^{1-\delta /8} \ll \| f \|_1^{3 \rho} N . \]
Assume $|\lambda |<\left( 4 \lceil R^c \rceil \right)^{-1}$, then $|\lambda T^*_i|
\leq 2|\lambda| |H_i| \le 1/2$, and thus using the previous relations and $e^x \leq 1 + x + x^2$ for $|x| \leq 1/2$  we get
\begin{align*}
& \E \exp \left( \lambda \sum_{i=2}^R  T^*_i \right)
 = \prod_{i=2}^R \E e^{\lambda T^*_i}
 \leq \prod_{i=2}^R  \E  (1 + \lambda T^*_i + \lambda^2
T^{*2}_i) \nonumber \\
& = \prod_{i=2}^R (1 + \lambda^2 \E  {T^{*2}_i})
\leq \exp \left(\lambda^2 \sum_{i=2}^R \E T_i^{*2}\right) \leq \exp( C_1 \lambda^2 \| f \|_1^{3 \rho} N)
\end{align*}
with some constant $C_1>0$. Note that $\lambda = C_2 t_0 \| f \|_1^{-2\rho} N^{-1/2} \le C_2 \| f \|_1^{-2 \rho} N^{-1/2+\delta/16}$ with a small enough constant $C_2>0$ satisfies $|\lambda | < \left( 4 \lceil R^c \rceil \right)^{-1}$. Using the Markov inequality we thus get
\[ \begin{split} \P \left( \sum_{i=2}^R T^*_i \geq t \| f \|_1^\rho \sqrt{N} \right) &\le \exp \left(- \lambda t \| f \|_1^\rho \sqrt{N}\right) \E \exp \left(\lambda \sum_{i=2}^R T_i^* \right) \\
&\le \exp\left(- \lambda t \| f \|_1^\rho \sqrt{N} +C_1 \lambda^2 \| f \|_1^{3 \rho} N \right)   \\
&\le \exp\left( -C_2 \| f \|_1^{-\rho} t_0 t  +C_1 C_2^2 \| f \|_1^{-\rho} t_0^2 \right) \\
&\le \exp\left( -C  \| f \|_1^{-\rho} t_0 t \right) \end{split} \]
with some constant $C>0$. Using $-\lambda$ instead, we get that the same upper estimate holds for $\P \left( \sum_{i=2}^R T_i^* \le -t \| f \|_1^{\rho} \sqrt{N} \right)$. Repeating the same arguments for $\sum_{i=2}^R D_i^*$, we altogether have
\[ \P \left( \left| \sum_{i=2}^R (T_i^*+D_i^*) \right| \ge t \| f \|_1^{\rho} \sqrt{N} \right) \le 4 \exp (-C \| f \|_1^{-\rho} t_0 t) . \]
The last relation, together with (\ref{chebyshevineq}) implies Lemma \ref{expbound}.
\end{proof}

\section{Chaining}

Recall that for any interval $J \subseteq [0,1)$ of length $\lambda (J)$, $f_J (x)=I_J(\{ x \})-\lambda (J)$ denotes the centered indicator of $J$, extended with period $1$. Lemma \ref{expbound} in the previous section gives a useful estimate for the tail probabilities
$$\P \left( \left| \sum_{k=M+1}^{M+N} f_J(S_k) \right|  \geq   t \lambda(J)^\rho \sqrt{N} \right)$$
 which leads to the upper half of the LIL for the partial sums $\sum_{k=1}^N f_J(S_k)$ for a fixed interval $J \subseteq [0,1)$ and along an exponentially growing subsequence of $N$'s. Proving the LIL for the discrepancy $D_N(S_k)$ requires, on the other hand, tail estimates for the variable
$$ \max_{2^n\le N < 2^{n+1}} \sup_{J \subseteq [0,1)}\left|\sum_{k=1}^N f_J(S_k)\right|.$$
Such inequalities will be proved in this section, using the method of chaining. We follow closely Philipp \cite{PH}.

\begin{lem}\label{chaining1}
Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $0<\delta <1$. For any $a \in [0,1)$, any $0<\Delta \le 1-a$ and any integer $N \gg 1$ we have
\[ \P \left( \sup_{0<y\le \Delta} \left|\sum_{k=1}^N f_{[a,a+y)} (S_k) \right|\gg \Delta^{\delta /30}\sqrt{N}+ N^{1/2-\delta /30}\right)\ll \Delta^4+ N^{-\delta /30}. \]
The implied constants depend only on the distribution of $X_1$ and $\delta$.
\end{lem}

\begin{proof} For any $h \geq 1$, $1 \leq j \leq 2^h$ let $\varphi^{(j)}_h$ denote the indicator function of the interval $[(j - 1)2^{-h}, j2^{-h})$ extended with period $1$. We observe that if $0<y \le \Delta$ has the dyadic expansion
\begin{equation}\label{dyadic}
y = \sum^\infty_{\ell = 1} \e_\ell 2^{-\ell}, \qquad \e_\ell = 0,1,
\end{equation}
and $H \geq 1$ is an arbitrary integer, then
\begin{equation}\label{2}
\sum^H_{h=1} \varrho_h(x) \leq I_{[0,y)}(\{ x\} ) \leq \sum^H_{h
= 1} \varrho_h(x) + \sigma_H(x)
\end{equation}
where $\varrho_h$ is the indicator function of
$\Big[\sum\limits^{h-1}_{\ell = 1} \e_\ell 2^{-\ell},
\sum\limits^h_{\ell=1} \e_\ell 2^{-\ell} \Big)$ and
$\sigma_H$ is the indicator function of $\Big[
\sum\limits^H_{\ell=1} \e_\ell 2^{-\ell}, \sum\limits^H_{\ell=1}
\e_\ell 2^{-\ell} + 2^{-H}\Big)$, both extended with period $1$. For $\e_h = 0$
clearly $\varrho_h \equiv 0$, and thus (\ref{2}) remains valid if in the
sums we keep only those terms where $\e_h=1$. Also, for $\varepsilon_h = 1$,
$\varrho_h$ coincides with one of the $\varphi^{(j)}_h$, and $\sigma_H$ also
coincides with one of the $\varphi^{(j)}_H$. It follows that
\[ \begin{split} f_{[0,y)}(x) &\leq \underset{1\le h\le H}{\sum\nolimits^*}
(\varrho_h(x) - 2^{-h}) + (\sigma_H(x)-2^{-H}) +2^{-H}, \\ f_{[0,y)}(x) &\ge \underset{1\le h\le H}{\sum\nolimits^*}  (\varrho_h(x) - 2^{-h}) -2^{-H}, \end{split} \]
where $\sum^*$ means that the summation is extended only for those $h$ such that $\varepsilon_h=1$. Setting $x=S_k-a$ and summing for $1\le k\le N$ we get
\[ \begin{split} \sum_{k=1}^N f_{[a,a+y)}(S_k) &\leq \sum_{k=1}^N \underset{1\le h\le H}{\sum\nolimits^*}
(\varrho_h (S_k-a)- 2^{-h}) + \sum_{k=1}^N (\sigma_H (S_k-a)-2^{-H}) +N 2^{-H}, \\ \sum_{k=1}^N f_{[a,a+y)}(S_k) &\ge \sum_{k=1}^N \underset{1\le h\le H}{\sum\nolimits^*}  (\varrho_h (S_k-a)- 2^{-h}) -N 2^{-H}. \end{split} \]
Letting $A= \log_2 \Delta^{-1}$, observe that the dyadic expansion (\ref{dyadic}) of any $0<y\le \Delta$ starts with an index $\ell\ge A$. Thus it follows that for any integers $N \ge 1$ and $H \ge 1$
\begin{equation} \label{3}
\sup_{0<y \le \Delta} \left| \sum_{k=1}^N f_{[a,a+y)} (S_k) \right|
\le 2  \sum_{A \le h \le H} \max_{1 \le j \le 2^h} \left| \sum_{k=1}^N \left( \varphi^{(j)}_h
(S_k-a) - 2^{-h} \right) \right| + N2^{-H}.
\end{equation}
Choose $H=\lfloor (1/2+\delta /30) \log_2 N \rfloor$, and introduce the events
\begin{align*}
G(N,j,h) &= \left\{ \left| \sum_{k=1}^N \left( \varphi_h^{(j)} (S_k-a) -2^{-h} \right) \right| \geq D 2^{-(\delta /30) h} \sqrt{N}  \right\},\\
G_N &= \bigcup_{A\le h \leq H} \bigcup_{1 \le j \leq 2^h} G(N,j,h) , \nonumber
\end{align*}
where $D>0$ is a large constant, to be chosen. For any $h\le H$ we have $2^{-h} \ge N^{-1/2-\delta /30}$, and thus applying Lemma \ref{expbound} to the function \mbox{$\varphi^{(j)}_h (x-a)-2^{-h}$} with $t=t_0=D$ and $\rho=\delta /30$ we get
\[ \P (G(N, j, h)) \ll \exp (-C D^2 2^{(\delta /30) h} ) + 2^{(\delta /15) h} N^{-1/2-\delta /8}, \]
and consequently
\begin{equation}\label{4}
\P (G_N) \ll \sum_{h\ge A} \exp( h -C D^2 2^{(\delta /30) h}) +
N^{-1/2-\delta /8} \sum_{h \leq (1/2+\delta /30 ) \log_2 N} 2^{(1+\delta /15 ) h}.
\end{equation}
Here $h-CD^2 2^{(\delta /30)h} \le -4h$ provided $D$ is large enough. The first sum in \eqref{4} is thus $\ll \exp (-4A) \ll \Delta^4$. The second term on the right hand side of (\ref{4}) is  $\ll N^{-\delta /30}$, hence $\P(G_N) \ll \Delta^4 + N^{-\delta /30}$. It follows that with the exception of a set with probability $\ll \Delta^4 + N^{-\delta /30}$, the right hand side of (\ref{3}) is
\[ \ll \sum_{A\le h\le H}  2^{-(\delta /30) h} \sqrt{N}+N 2^{-H} \ll 2^{-(\delta /30)A} \sqrt{N}+ N^{1/2-\delta /30}\ll \Delta^{\delta /30} \sqrt{N}+ N^{1/2-\delta /30} . \]
\end{proof}

\begin{lem}\label{chaining2}
Assume that $\psi (k) \ll k^{-(1+\delta)}$ for some $0<\delta <1$. For any $a \in [0,1)$, any $0<\Delta \le 1-a$ and any integer $n \gg 1$ we have
\[ \P \left( \max_{2^n\le N<2^{n+1}} \sup_{0< y\le \Delta} \left|\sum_{k=1}^N f_{[a,a+y)} (S_k) \right| \gg \Delta^{\delta /30}\sqrt{2^n\log\log 2^n} + \sqrt{2^n} \right) \ll n^{-2} . \]
The implied constants depend only on the distribution of $X_1$ and $\delta$.
\end{lem}

\begin{proof}
As in the proof of Lemma \ref{chaining1}, for any $h \geq 1$, $1 \leq j \leq 2^h$ let $\varphi^{(j)}_h$ denote the
indicator function of the interval $[(j - 1)2^{-h}, j2^{-h})$ extended with period $1$. For any integers $M\ge 0$ and $N\ge 1$ let
\[ F(M, N, j, h) = \left| \sum^{M+N}_{k=M+1} \left( \varphi^{(j)}_h (S_k-a) - 2^{-h} \right) \right| . \]
Following the proof of Lemma \ref{chaining1}, we get that for any integers $M \ge 0$, $N \ge 1$ and $H \ge 1$
\begin{equation}\label{gaSk}
\sup_{0<y \le \Delta} \left| \sum_{k=M+1}^{M+N} f_{[a,a+y)}(S_k) \right| \le 2 \sum_{A \le h \le H} \max_{1 \le j \le 2^h} F(M, N, j, h) + N 2^{-H},
\end{equation}
where $A=\log_2 \Delta^{-1}$. Let now $n\ge 1$, and let $2^n \leq N < 2^{n+1}$ have dyadic expansion $N=2^n+\beta_{n-1}2^{n-1}+\cdots +\beta_1 2 +\beta_0$, $\beta_i=0,1$. Letting $M_{\ell}=2^n+\beta_{n-1}2^{n-1}+\cdots +\beta_{\ell} 2^{\ell}$ we can decompose the interval of integers $[1,N]$ as
\[ [1,N] = [1,2^n] \cup \bigcup_{\substack{1 \le \ell \le n \\ \beta_{\ell -1}=1}} [M_{\ell}+1, M_{\ell}+2^{\ell -1}] . \]
Note that the union of $[M_{\ell}+1,M_{\ell}+2^{\ell -1}]$ over $1 \le \ell <n/2$ has total size $\ll 2^{n/2}$. Further, $M_{\ell}=2^n+m_{\ell}2^{\ell}$ for some integer $0 \le m_{\ell}<2^{n-\ell}$. Applying \eqref{gaSk} on the interval $[1,2^n]$ with $H=n/2$ and on the intervals $[M_{\ell}+1,M_{\ell} +2^{\ell -1}]$ for all $n/2 \le \ell \le n$ such that $\beta_{\ell -1}=1$ with the choice $H=2^{\ell /2}$, we get that
\[ \begin{split} \sup_{0<y \le \Delta} \left| \sum_{k=1}^N f_{[a,a+y)}(S_k) \right| \ll &\sum_{A \le h \le n/2} \max_{1 \le j \le 2^h} F(0, 2^n, j, h) + 2^{n/2} \\ + &\sum_{\substack{n/2 \le \ell \le n \\ \beta_{\ell -1}=1}} \left( \sum_{A \le h \le \ell /2} \max_{1 \le j \le 2^h} F(2^n+m_{\ell}2^{\ell}, 2^{\ell -1}, j, h) + 2^{\ell /2} \right) . \end{split} \]
Here $\sum_{n/2 \le \ell \le n} 2^{\ell /2} \ll 2^{n/2}$, therefore we altogether have
\begin{equation}\label{mainformula}
\begin{split} \max_{2^n \le N<2^{n+1}} \sup_{0<y \le \Delta} &\left| \sum_{k=1}^{N} f_{[a,a+y)} (S_k) \right| \ll \\ &\sum_{A \le h \le n/2} \max_{1 \le j \le 2^h} F(0, 2^n, j, h) + 2^{n/2} \\ &+ \sum_{n/2 \le \ell \le n} \sum_{A\le h\le  \ell /2} \max_{0 \le m < 2^{n-\ell}} \max_{1 \le j \le 2^h} F(2^n + m 2^\ell, 2^{\ell - 1}, j, h). \end{split}
\end{equation}
Let us introduce the events
\[ \begin{split}
G(n,j,h) &=  \left\{ F(0,2^n, j, h) \geq D 2^{-(\delta /30)h} \sqrt{2^n \log\log 2^n} \right\}, \\  G_n &= \bigcup_{A \le h \leq n/2} \bigcup_{1 \le j \leq 2^h} G(n,j,h), \\
G^*(n, j, h, m, \ell ) &= \left\{ F(2^n + m2^\ell, 2^{\ell - 1}, j,
h) \geq  D 2^{-(\delta /30)(h+n-\ell )} \sqrt{2^n \log\log 2^n} \right\}, \\
G_n^* &= \bigcup_{n/2 \leq \ell \leq n} \bigcup_{A\le h \leq \ell /2} \bigcup_{0 \le m < 2^{n -
\ell}} \bigcup_{1 \le j \leq 2^h} G^*(n,j,h,m,\ell ) ,
\end{split} \]
where $D>0$ is a large constant, to be chosen. Applying Lemma \ref{expbound} to the function $\varphi_h^{(j)}(x-a)-2^{-h}$ with $M = 0$, $N = 2^n$, $t=t_0= D \sqrt{\log\log 2^n}$ and $\rho=\delta /30$ we get
\[ \P (G(n, j, h)) \ll \exp (-C D^2 2^{(\delta /30)h} \log \log 2^n ) + 2^{(\delta /15)h} 2^{(-1/2-\delta /8)n}, \]
and consequently
\begin{equation} \label{gn}
\P (G_n) \ll \sum_{h=0}^{\infty} \exp( h-C D^2 2^{(\delta /30)h} \log \log 2^n) + 2^{(-1/2-\delta /8)n} \sum_{h \leq n/2} 2^{(1+\delta /15)h}.
\end{equation}
Here $h-CD^2 2^{(\delta /30) h} \log \log 2^n \le -h-2 \log \log 2^n$ provided $D$ is large enough, therefore the first sum in \eqref{gn} is $\ll \exp (-2 \log \log 2^n ) \ll n^{-2}$. The second term in \eqref{gn} is $\ll 2^{-(\delta /30)n}$, hence $\P (G_n) \ll n^{-2}$.

Similarly, applying Lemma \ref{expbound} to the function $\varphi_h^{(j)}(x-a)-2^{-h}$ with $M = 2^n + m2^\ell$, $N = 2^{\ell
- 1}$, $t = D 2^{(1/2-\delta /30)(n-\ell )+1/2} \sqrt{\log \log 2^n}$, $t_0 \ge D \sqrt{ \log \log 2^n}$ and $\rho = \delta /30$ we get
\[ \begin{split} \P (G^* (n,j,h,m,\ell )) \ll &\exp (-C D^2 2^{(\delta /30)h} 2^{(1/2-\delta /30)(n-\ell )} \log \log 2^n) \\&+ 2^{(-1+\delta /15) (n-\ell )} 2^{(\delta /15) h} 2^{(-1/2-\delta /8)\ell}, \end{split} \]
and consequently
\begin{equation}\label{gn*}
\begin{split} \P (G_n^*)\ll &\sum_{n/2 \leq \ell \leq n} \sum_{h=0}^{\infty} \exp (h+n-\ell -C D^2 2^{(\delta /30)h} 2^{(1/2-\delta /30)(n-\ell )} \log\log 2^n) \\
&+ \sum_{n/2 \leq \ell \leq n} \sum_{h \leq \ell /2}  2^h 2^{n-\ell} 2^{(-1+\delta /15)(n-\ell )} 2^{(\delta /15) h} 2^{(-1/2-\delta /8) \ell} . \end{split}
\end{equation}
Here $h+n-\ell -C D^2 2^{(\delta /30)h} 2^{(1/2-\delta /30)(n-\ell )} \log \log 2^n \le -h-(n-\ell) -2 \log \log 2^n$ provided $D$ is large enough, therefore the first sum in \eqref{gn*} is $\ll \exp (-2\log \log 2^n) \ll n^{-2}$. Since the second sum in \eqref{gn*} is $\ll 2^{-(\delta /80)n} \ll n^{-2}$, we have $\P (G_n^*) \ll n^{-2}$.

It follows that with the exception of a set with probability $\ll n^{-2}$, the right hand side of \eqref{mainformula} is
\[ \begin{split} &\ll \sum_{h \ge A} 2^{-(\delta /30)h} \sqrt{2^n \log \log 2^n} +2^{n/2} + \sum_{n/2 \le \ell \le n} \sum_{h \ge A} 2^{-(\delta /30)(h+n-\ell )} \sqrt{2^n \log \log 2^n} \\ &\ll \Delta^{\delta /30} \sqrt{2^n \log \log 2^n} +2^{n/2} . \end{split} \]
\end{proof}

\section{Proof of Theorem \ref{theorem2}}

In Lemma \ref{varianceprop1} we proved that $C(f,g)$ is bilinear, symmetric and positive semidefinite in $f,g \in \mathcal{F}$. In particular, $\Gamma (s,t)$ is symmetric and positive semidefinite. From the triangle inequality we get that
\[ \begin{split} |\E (f_{[0,s)}(U) - f_{[0,s')}(U)) f_{[0,t)}(U+S_k)| &\le 2 |s-s'|, \\ |\E f_{[0,s)}(U)(f_{[0,t)}(U+S_k) - f_{[0,t')}(U+S_k))| &\le 2 |t-t'| \end{split} \]
for any $s,s',t,t' \in [0,1]$ and any $k \ge 1$. Hence $\E f_{[0,s)}(U) f_{[0,t)}(U+S_k)$ is continuous on the unit square for any fixed $k \ge 1$. As observed in the proof of Lemma \ref{varianceprop1} (i), each term in the series defining $\Gamma (s,t)$ has absolute value at most $\psi (k) \ll k^{-(1+\delta)}$, and so the continuity of $\Gamma (s,t)$ follows.
\begin{lem}\label{multclt}
For any fixed $0\le t_1<t_2<\cdots <t_r \le 1$ the random vector ${\bf Y}_k=(f_{[0, t_1)} (S_k), f_{[0, t_2)} (S_k), \ldots, f_{[0, t_r)} (S_k))$ satisfies
\[ N^{-1/2} \sum_{k=1}^N {\bf Y}_k \overset{d}{\longrightarrow} \mathcal{N}(\bf{0}, \Sigma) , \]
where $\mathcal{N}(\bf{0}, \Sigma)$ denotes the multidimensional normal distribution with mean $\bf{0}$ and covariance matrix $\mathbf{\Sigma}= (\Gamma (t_i, t_j))_{1\le i, j\le r}$.
\end{lem}

\begin{proof} Let $c=1-\delta /8$ and $c'=(c+\delta /2)/(1+\delta)$. Note that $c'<c$. Let us decompose the set of positive integers into consecutive blocks $H_1, J_1, H_2, J_2, \dots$ such that $|H_i|=\lceil i^c \rceil$ and $|J_i|=\lceil i^{c'} \rceil$ for all $i \ge 1$. Put $f_j=f_{[0,t_j)}$, $j=1,2,\dots, r$. Following the notation of Section \ref{approximationsubsection} let ${\bf T}_i=\left( T_i^{(f_1)}, T_i^{(f_2)}, \dots, T_i^{(f_r)} \right)$ and ${\bf T}_i^*=\left( T_i^{(f_1)*}, T_i^{(f_2)*}, \dots, T_i^{(f_r)*} \right)$, and let ${\bf D}_i$ and ${\bf D}_i^*$ be defined analogously. For any positive integer $N$ let $R=R(N)$ be such that $N \in H_R \cup J_R$. Note that $R=\Theta \left( N^{1/(c+1)} \right)$. Since $|{\bf T}_i| \ll |H_i| \ll R^c \ll N^{1/2-\delta /32}$ for all $1 \le i \le R$ and the same holds for $|{\bf D}_i|$, the contribution of any individual block is negligible, and we get
\begin{equation}\label{Ykfirstdecomposition}
\sum_{k=1}^N {\bf Y}_k = \sum_{i=2}^R ({\bf T}_i + {\bf D}_i) + O \left( N^{1/2-\delta /32} \right) .
\end{equation}
Applying Lemma \ref{approxerror1} to $f_1, f_2, \dots, f_r$ we get $\E \left| \sum_{i=2}^R \left( {\bf T}_i + {\bf D}_i - {\bf T}_i^* - {\bf D}_i^* \right) \right|^2 \ll R^{1-\delta /2}$, and hence from the Chebyshev inequality and the Borel--Cantelli lemma $\left| \sum_{i=2}^R \left( {\bf T}_i + {\bf D}_i - {\bf T}_i^* - {\bf D}_i^* \right) \right| \ll R^{1-\delta /8}$ a.s. Thus replacing ${\bf T}_i$ by ${\bf T}_i^*$ and ${\bf D}_i$ by ${\bf D}_i^*$, the right hand side of \eqref{Ykfirstdecomposition} changes by $\ll R^{1-\delta /8} \ll N^{1/2-\delta /32}$, and so
\begin{equation}\label{Ykseconddecomposition}
\sum_{k=1}^N {\bf Y}_k = \sum_{i=2}^R ({\bf T}_i^* + {\bf D}_i^*) + O \left( N^{1/2-\delta /32} \right) \quad \textrm{a.s.}
\end{equation}
Thus it will be enough to prove the multidimensional CLT for the right hand side of \eqref{Ykseconddecomposition}.

Recall that ${\bf T}_2^*, {\bf T}_3^*, \dots$ are independent, mean ${\bf 0}$ random vectors. From \eqref{ti*di*variance} we deduce
\[ \mathrm{Cov} \left( T_i^{(f_j)*} , T_i^{(f_k)*} \right) = \frac{1}{4} \left( \E T_i^{(f_j+f_k)*2} - \E T_i^{(f_j-f_k)*2} \right) \sim C(f_j,f_k) |H_i| . \]
Here $C(f_j,f_k)=\Gamma (t_j,t_k)$, thus letting ${\bf \Sigma}_i$ denote the covariance matrix of ${\bf T}_i^*$, we have ${\bf \Sigma}_i \sim |H_i| {\bf \Sigma}$. Since $\sum_{i=2}^R |H_i| \sim N$, we have $N^{-1}\sum_{i=2}^R {\bf \Sigma}_i \to {\bf \Sigma}$. Note that $|{\bf T}_i^*| \ll |H_i| \ll R^c \ll N^{1/2-\delta /32}$ for all $2 \le i \le R$. Thus ${\bf T}_2^*, {\bf T}_3^*, \dots$ satisfies the Lindeberg condition, and by the classical multidimensional CLT we have
\begin{equation}\label{ti*multidimCLT}
N^{-1/2} \sum_{i=2}^R {\bf T}_i^* \overset{d}{\longrightarrow} \mathcal{N}(\bf{0}, \Sigma).
\end{equation}
Similarly, ${\bf D}_2^*, {\bf D}_3^*, \dots$ are independent, mean ${\bf 0}$ random vectors. From \eqref{ti*di*variance} we deduce
\[ \sum_{i=2}^R \E |{\bf D}_i^*|^2 \ll \sum_{i=2}^R |J_i| \ll R^{1+c'} = o(N), \]
therefore $N^{-1/2} \sum_{i=2}^R {\bf D}_i^* \overset{\P}{\longrightarrow} 0$. The last relation together with \eqref{ti*multidimCLT} imply $N^{-1/2} \sum_{i=2}^R ({\bf T}_i^* + {\bf D}_i^*) \overset{d}{\longrightarrow} \mathcal{N}(\bf{0}, \Sigma)$, as claimed.
\end{proof}

Lemma \ref{multclt} implies the convergence of the finite dimensional distributions of the sequence $\sqrt{N} (F_N(t)-t)$ in (\ref{donsker}) to those of $K(t)$, and to prove Theorem \ref{theorem2} it remains to prove the tightness of the sequence in $D[0,1]$. To this end, fix $0<\e<1$ and let $m$ be the positive integer such that $1/(m+1) \le \e< 1/m$. Applying Lemma \ref{chaining1} with $a=j/m$, $j=0,1,\dots, m-2$ and $\Delta =2/m$, it follows that
\begin{multline*}
\P \left(\max_{0\le j \le m-2} \sup_{0< y \le 2/m} \left|\sum_{k=1}^N f_{[j/m,j/m+y)}(S_k) \right|\gg m^{-\delta /30}\sqrt{N}+  N^{1/2-\delta /30}\right) \\ \ll m^{-3}+m N^{-\delta /30}.
\end{multline*}
Note that for any $s,t \in [0,1]$ with $s<t \le s+\e$ we have $\sqrt{N}|(F_N(t)-t)-(F_N(s)-s)|$ $=N^{-1/2} \sum_{k=1}^N f_{[s,t)}(S_k)$, and that $[s,t)$ is a subset of $[j/m,(j+2)/m)$ for some $0 \le j \le m-2$. This implies that with the exception of a set with probability $\ll \e^3 + \e^{-1} N^{-\delta /30}$ the fluctuation of the process $\sqrt{N} (F_N(t)-t)$ over any subinterval of $[0,1]$ with length $\le \e$ is $ \ll \e^{\delta /30} + N^{-\delta /30}$. By Theorem 15.2 of Billingsley \cite[Chapter 3]{BIL}, the sequence $\sqrt{N} (F_N(t)-t)$ is tight in $D[0,1]$. This completes the proof of Theorem \ref{theorem2}.

\section{Proof of Theorem \ref{theorem3}}

\begin{lem}\label{multlil} Fix $0\le t_1<t_2<\cdots <t_r \le 1$, and consider the random vector ${\bf Y}_k=(f_{[0, t_1)} (S_k), f_{[0, t_2)} (S_k), \ldots, f_{[0, t_r)} (S_k))$. With probability $1$, the set of limit points of the sequence $(2N\log\log N)^{-1/2} \sum_{k=1}^N {\bf Y}_k$ is the ellipsoid
\begin{equation}\label{ellips}
\left\{ {\bf \Sigma x} \in \mathbb{R}^r : \langle {\bf x}, {\bf \Sigma x}  \rangle  \le 1 \right\}
\end{equation}
in the range $\left\{ {\bf \Sigma x} : {\bf x} \in \mathbb{R}^r \right\}$ of the matrix $\mathbf{\Sigma}= (\Gamma (t_i, t_j))_{1\le i, j\le r}$.
\end{lem}

\begin{proof} Lemma \ref{multlil} can be proved in the same way as Lemma \ref{multclt}. The only difference is that after deducing \eqref{Ykseconddecomposition}, instead of the multidimensional CLT we apply the multidimensional version of Kolmogorov's LIL due to Berning \cite{BER}. This way we get that with probability $1$, the set of limit points of the sequence $(2N \log \log N)^{-1/2} \sum_{i=2}^R {\bf T}_i^*$ is the ellipsoid \eqref{ellips}, and that $(2N \log \log N)^{-1/2} \sum_{i=2}^R {\bf D}_i^* \to {\bf 0}$ a.s.
\end{proof}

Using Lemma \ref{multlil} and Lemma \ref{chaining2}, the proof of Theorem \ref{theorem3} can be easily completed. Let
\[ \alpha_N (t)= \frac{N (F_N(t)-t)}{\sqrt{2 N \log \log N}}, \quad 0\le t \le 1, \]
and let $L$ denote the (random) class of limit functions of subsequences of $\alpha_N$ in the $D[0,1]$ topology. Similarly to the  proof of the functional CLT, which requires tightness and the convergence of the finite dimensional distributions, to prove the functional LIL in Theorem \ref{theorem3} we will show that with probability $1$ the sequence $\alpha_N$ is relatively compact in the $D[0,1]$ topology, and that for any fixed finite subset $T \subset [0,1]$ we have $\{ f|_T : f \in L \} = \{ f|_T : f \in B (\Gamma ) \}$ a.s.

To prove relative compactness, fix $0<\e <1$ and let $m$ be the positive integer such that $1/(m+1) \le \e <1/m$. Applying Lemma \ref{chaining2} with $a=j/m$, $j=0,1,\dots, m-2$ and $\Delta = 2/m$, it follows that
\begin{multline*}
\P \Bigg( \max_{0 \le j \le m-2} \max_{2^n \le N < 2^{n+1}} \sup_{0<y \le 2/m} \left|\sum_{k=1}^N f_{[j/m,j/m+y)}(S_k) \right| \\ \gg m^{-\delta /30} \sqrt{2^n \log \log 2^n} +\sqrt{2^n} \Bigg) \ll m/n^2 .
\end{multline*}
By the Borel--Cantelli lemma this implies that with probability $1$, for any $0<\e <1$ there exists a (random) integer $N_0$ such that for any $N\ge N_0$ the fluctuation of the process $\alpha_N (t)$ over any subinterval of $[0,1]$ of length $\le \e$ is $\ll \e^{\delta /30}$. Applying Lemma \ref{chaining2} with $a=0$ and $\Delta =1$, from the Borel--Cantelli lemma we similarly deduce that with probability $1$, $\sup_{t \in [0,1]} |\alpha_N (t)| \ll 1$. By Theorem 14.3 of Billingsley \cite[Chapter 3]{BIL}, with probability $1$ the sequence $\alpha_N(t)$ is relatively compact in $D[0,1]$. Further, it is easy to see from the fluctuation estimate that with probability $1$, any subsequence of $\alpha_N(t)$ convergent in $D[0,1]$ has a continuous limit function. Hence with probability $1$, $L$ is a compact subset of $C[0,1]$ and any subsequence of $\alpha_N$ convergent in $D[0,1]$ is also uniformly convergent.

Next, fix a finite set $T \subset [0,1]$. Let $H(\Gamma )$ and $H\left( \Gamma|_{T \times T} \right)$ denote the reproducing kernel Hilbert spaces determined by the covariance functions $\Gamma$ and $\Gamma|_{T \times T}$, respectively. Then $H \left( \Gamma|_{T \times T} \right) = \{ f|_T : f \in H (\Gamma) \}$; moreover, $\| g \|_{H \left( \Gamma|_{T \times T} \right)} = \min \{ \| f \|_{H(\Gamma)} : f|_T =g \}$, see Aronszajn \cite[Section I.5]{A}. In particular, the closed unit ball of $H\left( \Gamma|_{T \times T} \right)$ is $\{ f|_T : f \in B(\Gamma) \}$. Lemma \ref{multlil} thus shows that with probability $1$, the set of limit points of $\alpha_N|_T$ is identical with $\{ f|_T : f \in B( \Gamma) \}$. The relative compactness of $\alpha_N$ implies that in fact $\{ f|_T : f \in L \} = \{ f|_T : f \in B (\Gamma ) \}$ a.s., as claimed.

Now fix finite sets $T_1 \subset T_2 \subset \cdots \subset [0,1]$ such that $\bigcup_{n=1}^{\infty} T_n$ is dense in $[0,1]$. Observe that if $K$ and $K'$ are compact subsets of $C[0,1]$ such that $\{ f|_{T_n} : f \in K \} = \{ f|_{T_n} : f \in K' \}$ for all $n \ge 1$, then $K=K'$. By Lemma 3 of Oodaira \cite{OO}, the continuity of $\Gamma (s,t)$ implies that $B(\Gamma )$ is compact in $C[0,1]$. The arguments above show that with probability $1$, $L$ is a compact subset of $C[0,1]$ and $\{ f|_{T_n} : f \in L \} = \{ f|_{T_n} : f \in B(\Gamma ) \}$ for all $n \ge 1$. Therefore $L=B(\Gamma )$ a.s. This finishes the proof of Theorem \ref{theorem3}.

\section{Proof of Theorem \ref{theorem1}} The proof of Theorem \ref{theorem1} is similar to, but is actually much simpler than that of Theorems \ref{theorem2} and \ref{theorem3}. The claims on $C(f,g)$ were proved in Lemma \ref{varianceprop1}. If in the proof of Lemmas \ref{multclt} and \ref{multlil} we replace the vector-valued random variables ${\bf Y}_k$, ${\bf T}_i$, ${\bf T}_i^*$, ${\bf D}_i$ and ${\bf D}_i^*$ by the real-valued random variables $f(S_k)$, $T_i^{(f)}$, $T_i^{(f)*}$, $D_i^{(f)}$ and $D_i^{(f)*}$, respectively, where $f \in \mathcal{F}$ is fixed, we get that the sum $\sum_{k=1}^N f(S_k)$ satisfies the CLT and the LIL.

redpajama_set_name: RedPajamaArXiv


text:
Join a hub
Selling out student privacy to the College Board: oppose SB3099
SB3099 is a bill written to benefit the College Board and ACT, Inc, allowing them to sell or rent sensitive student data that they access through questionnaires administered along with tests like the SAT. RYH Action urges parents and privacy advocates to submit a witness slip to oppose this bill in hearing tomorrow (3/12/2018)
This bill would amend the already inadequate protections of the Student Online Privacy Protection Act (SOPPA) (PA: 100-0315) passed last spring by waiving restrictions on the sale of student's personal information for organizations like the College Board and ACT, Inc. SB3099 would also allow creation of profiles of individual students for any "post-secondary educational purposes."
The College Board's SAT is now the state-mandated test for juniors in Illinois, and the College Board has a very poor record on exploiting access to highly sensitive personal data. Students are asked for this information in the context of a high-stakes, state-mandated test and most do not realize, and in many cases are not even told, that providing this information is optional. And the College Board prevaricates in their description of what is done with this data, claiming that they do not "sell" it, but instead "license" access to it. (You can see a price list for what the College Board currently charges for access to students' data here.)
The questionnaire administered with the SAT urges students to provide sensitive personal information concerning their citizenship, religion, social security number, GPA, high school course work and interests, family's education, income and/or military background, as well as behavioral information ( "self-ratings"). At at least one Chicago high school last spring, students were told that answering these questions were mandatory.
While it is crucial to collect some of this information—namely race, gender, income, and parental education level—in order for test vendors and the public to measure test bias and correlation of scores with socio-economic status. However, SB3099 would allow this information to be used in a non-anonymized form to create profiles of students that academic institutions and scholarship providers can then use with predictive algorithms (in some cases provided by ACT, Inc itself) to determine which students to recruit and admit. These profiles, algorithms and predictions are not provided to students and parents.
Although they are non-profit corporations, both the College Board and ACT, Inc. have been criticized for excessive compensation of executives. And both organizations also spend heavily on lobbying. The commercial exploitation of student data by these orgs has been the subject of a Congressional inquiry and lawsuits in the past.
Bills similar to SB3099 giving SAT and ACT the ability to profit from student data have passed in Arizona, Colorado, Nebraska, North Carolina, Washington DC, Texas, Utah, and Virginia.
However, at least 15 other states have passed or amended SOPIPA laws not giving ACT/SAT a free pass, including: Arkansas, California, Connecticut, Delaware, Georgia, Hawaii, Kansas, Maine, Maryland, Michigan, Nevada, New Hampshire, Oregon, Tennessee, and Washington.
Illinois' SOPPA language is modeled on a bill first passed in California in 2014 known as SOPIPA. SOPIPA was landmark student data privacy legislation spearheaded by industry-friendly Common Sense Media, and it contains many exceptions and loopholes for commercial purposes. Nonetheless, even California, where the political landscape is dominated by Silicon Valley, has withstood the pressure to weaken their law to allow for the College Board/ACT exception for 3+ years now, along with the 14 other states listed above.
Why should Illinois students be less protected than students in these other states? Why should these organizations be given special dispensation to exploit financially their state-mandated (in the case of the College Board) access to students?
You can fill out a witness slip opposing SB3099 here.
student privacy standardized testing SAT College Board ACT
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redpajama_set_name: RedPajamaCommonCrawl


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text:
When to take extended sick leave. A complex decision for workers with spine-related pain
Back Predicting delayed RTW Research update Workplace Culture
Joy Hewitt
This study identified a range of factors that influence when workers with neck and low-back (spine-related) pain take extended sick leave.
Take Home Messages:
The decision by workers with neck or low-back pain to take extended sick leave was not taken lightly and the majority aimed to return to work despite their pain:
Individuals with new spine-related pain took sick leave early to obtain diagnosis, guidance and treatment of their pain to enable them to return to work; and
Individuals with long-standing spine-related pain tried to avoid, delay or shorten their sick leave to enable them to continue working.
The factors influencing the decision to take sick leave were complex and varied according to:
The individual's experience of their pain;
Their self-image and work ethic;
Organisational work factors; and
Economic factors.
Why the study matters:
Extended sick leave absences from work and the associated welfare or insurance benefits pose a significant individual and socio-economic burden in industrialised countries. Musculoskeletal problems such as neck and low-back pain are frequent causes of sick leave absences.
What the study involved:
Researchers from one of Europe's largest medical universities, the Karolinska Institute in Sweden, examined the factors considered by 33 Swedish workers with neck or low-back pain when taking sick leave.
People who had been on full or part-time sick leave for 4 to 15 weeks were included in the study. A total of 20 women and 13 men were interviewed about their education, work, health, the effects of their health problems, and their thoughts of the future.
The onset of spine-related pain was relatively recent in approximately half of the study participants, while the remaining half had long-standing spine-related pain.
Summary of study findings:
While the decision to take extended sick leave was based on the presence of spine-related pain, the decision as to when to take the leave was more complex. The timing of sick leave was influenced by the nature of the pain, the worker's experience of the health care system, work and labour-market related factors and personal factors.
Workers with new spine-related pain generally took sick leave soon after the onset of pain. Their aims were to obtain diagnosis and treatment of their pain before returning to work or finding alternative work. However, some individuals with new spine-related pain delayed taking sick leave in consideration of their workmate's or employer's needs.
Workers with long-standing spine-related pain generally tried to continue working despite their pain. These individuals tended to:
Have long periods of sickness attendance ('presenteeism' );
Avoid using full-time sick leave; and
Return early to work from sick leave.
Factors that influenced their decision to continue working rather than take sick leave included work being a basis for their self-image, interest in their work and workmates, their work ethic, a lack of alternative jobs and an economic need to work.
Original research:
Sickness absence and sickness attendance – What people with neck or back pain think.
Hansson M, Boström C, Harms-Ringdahl K.
Social Science & Medicine; 2006; 62: 2183-2195.
PubMed abstract
Articles on this topic
Mean streets take toll on taxi drivers
RTW-land has a social capital
Mining the resources that prevent chronic LBP
Driving back to work (and stopping on the way)
Angry nurses have more MSDs - and there's a link to social demands / support


Since 1971, Continental Plastic has been a manufacturer and worldwide supplier of products for veterinary practices and artificial insemination of domestic livestock.
Continental Plastic Corp., founded February 1st, 1971, was co-owned by Richard Alter and Elton Feffer. Both individuals had strong manufacturing and administrative backgrounds in the agricultural industry prior to founding the company. They were successful in meeting the needs of their customers and have maintained that customer focus as Continental grew and expanded over the years.
In January, 2008, Richard stepped down as co-owner and President of the company. He still assists the company today on a consulting basis. Elton now assumes sole ownership and is active as President of Continental. He has since asked his daughter, Becky Feffer Wolf, to join the staff and assume the role of Vice President. The company currently employs over 90 people in the manufacturing facility and office.
The manufacturing facility and administrative office are located in southeastern Wisconsin, in the town of Delavan. Delavan is a two hour drive north from Chicago, Ill., or one hour southeast from Madison, Wis.
Continental is able to offer extremely competitive pricing, as most products are produced at our manufacturing facility in Wisconsin. Customers in every state in the US and over 60 countries worldwide call Continental-produced products their product of choice.


The 38th Cyberspace Engineering Installation Group is part of Air Force Space Command and reports to the 688th Cyberspace Wing at Lackland AFB, TX. The 38th CEIG provides a variety of integrated command and control and information systems services designed to increase the combat readiness of Air Force and DOD elements. These services include communications systems planning, engineering, program management and contract support for implementation, maintenance and dial tone requirements. The 38th CEIG is the only group in the Air Force that provides these services to bases, major commands, joint community and combatant commands worldwide. They also maintain a cadre of deployable civilian and military engineers who respond to wartime taskings in contingency areas.


Oh, yes. YES. First of all, I felt like I was there in the car with you and Mia, and what you shared is so, so familiar. And I also loved Aidan's post (and wrote my response for next week). The line you cite as your favorite is my favorite, too.
Nice and thoughtful! I can very much relate to your sentiments too. The only thing I would add is that this is just as true at work as it is at home. How I wish I had an "anti mobile device" device that I could use whenever we agree to meet for work in person!
So glad you're back posting! I agree – sometimes I get too caught up in being upset that I'm not present and so torn to get everything done, that I end up missing even more moments. I have to stop being so hard on myself about that!
Gorgeous, as usual. I really like the idea of white noise; it reminds me of the parallel play little children do, or of the companionship we can share only with those we most love and trust — sort of quiet, alongside each other time, a little bit lost in our own worlds, but with an ear open for what might be happening for the other. But we can create different qualities of white noise for each other, maybe, tune the frequencies up or down, get a good hum going. And then fall back into presence, catch the moment.


In this episode of Linux in the Ham Shack, your hosts do a couple of live interviews on Saturday at Hamvention 2018. The first is with Kevin, K2FN, and the second is with Gene, BX8AAD. There is also a second-day wrap up of the event and lots of discussion and banter about ham radio, Ohio, the weather, vendors, visitors and a lot more. Thank you for listening.


Moderate rain (total 16mm), heaviest on Tue morning. Freeze-thaw conditions (max 5°C on Mon morning, min -2°C on Wed night). Wind will be generally light.
A moderate fall of snow, heaviest on Fri afternoon. Freeze-thaw conditions (max 4°C on Thu afternoon, min -9°C on Sat night). Wind will be generally light.
This table gives the weather forecast for Thorington Tower at the specific elevation of 1500 m. Our advanced weather models allow us to provide distinct weather forecasts for several elevations of Thorington Tower. To see the weather forecasts for the other elevations, use the tab navigation above the table. For a wider overview of the weather, consult the Weather Map of Alberta.
* NOTE: not all weather observatories update at the same frequency which is the reason why some locations may show data from stations that are further away than known closer ones. We also include passing Ship data if it is recorded close to Thorington Tower and within an acceptable time window.


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Widespread fires in the 1910s and 1920s led to the construction of more permanent lookouts, first as wooden pole structures and subsequently as a variety of one- and two-story cabin designs. It is a dream that only few mountaineers have had the privilege to fulfill and that you can now skim, thanks to this very nice book, richly illustrated and remarkably documented. A physicist and educator by profession, Tyndall began visiting the Alps annually in 1849 to explore the glaciers: he climbed Mont Blanc several times, made the first ascent of the Weisshorn, and attempted to summit the Matterhorn. . Then you decide to climb something with glaciers. The Backpacker's Handbook also gives you the latest information on gear. You will gain a basic overview of what items you can get and what they are for, and how this equipment works.
Sidebars feature descriptions of accidents and near-accidents to emphasize the importance of the techniques presented. We use this information to create a better experience for all users. Snow Travel: Skills for Climbing, Hiking, and Moving Across Snow Mountaineers Outdoor Experts Series is a comprehensive how-to book covering all the essential techniques for kicking steps, using crampons, and using an ice ax for going up, traversing, resting, and descending snow. On the other hand, like many other hikers, I do occasionally come close to glaciers and have even walked on glaciers in Western Canada and in Alaska. As elsewhere on the continent, such work occasionally involves climbing and a number of peaks in the range have been ascended in the course of surveying, geological studies and other scientific work. It begins here completely from the beginning — first discussing advancing in the mountains along regular trails, and then walking in free terrain off the trail, as well as in varying types of terrain such as talus and grassy slopes, passages through scrub, walking in snow and firn slopes, and so forth. There are chapters filled with candid, warm stories from travelers who have been there, done that.
Work as a caretaker and live in a lighthouse. As a hiker beginning to consider mountaineering proper, though, this was the introduction I was looking for, and I can use the knowledge from it to help me find instruction in travel techniques and appropriate gear. Backpacker's Editors' Choice Awards, an industry honor recognizing design, feature and product innovation, has become the gold standard against which all other outdoor-industry awards are measured. Over the past 30 years, Alaska Geographic has earned its reputation as the publication for those who love Alaska. Sidebars feature descriptions of accidents and near-accidents to emphasize the importance of the techniques presented. He is the author of the book entitled Antarctic Mountaineering Chronology, published in 1998, and of detailed topographical maps of the Livingston Island 2004 and Vinson Mountain 2006. Sidebars feature descriptions of accidents and near-accidents to emphasize the importance of the techniques presented.
And, when you the rope lead are hanging in a crevasse, do you really want the second who didn't listen during your skills prep hike to be thumbing through 800 pages to find out how to make a Z-pulley? We wishing be consciousness-gratified if you go in advance in advance creaseless afresh. Before the advent of helicopters, teams travelled into the nearby ranges by dogsled. Drawing on years of experience, veteran Seven Summits mountain guide Mike Hamill describes overall considerations for expedition planning and high-altitude trips, gear recommendations, tips on international travel and logistics, and estimates of financial costs. Beginning in the 1950s, many of the lookouts were decommissioned and eventually destroyed. In fact, it is fair to say that Freedom is the definitive guide to mountains and climbing and has influenced pretty much every climber. Rescue Practice Sessions -- App. Dressed in a t-shirt and shorts, Ralston, a seasoned climber, figured he'd hike for a few hours and then head off to work.


.	 Published on July 15, 2018 .
The Association of Illustrators (AOI) in partnership with the London Transport Museum (LTM) invite submissions to the Poster Prize for Illustration to the theme of London Stories. Submissions are free.
Artists are invited to visually capture a familiar or lesser-known narrative in a single image. All stories, current or historical, real or fictional which feature this amazing city are welcome – your imagination is the limit. Stories could be those seen in a film or play, heard in poetry or music, or read in literature. They may be family anecdotes, or an impression of your daily observations. Your visual story could be inspired by London's history or contemporary city living. What stories does London have to tell?
The aim of the competition is to find 100 images depicting curious, interesting, real or fictitious stories about London to display at the famous Covent Garden Museum from February 2019 until the summer.
There are cash prizes for the three winners (£2000, £1000 and £750).


We offer introductory and advanced courses to support coaches and mentors. We use frameworks such as TGROW and CIGAR and teach core coaching skills such as rapport building, active listening, powerful questioning and goal setting.
We offer a range of leadership development courses for leaders at all levels, from supervisors to senior executives.
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We help our clients design the best people strategy for their organisation. Our approach is to integrate employee engagement, wellbeing and leadership into one. Why? Because we believe all three are critical for long-term success and one can't exist without the other.


As a result of many years of experience in the bioethanol decorative fireplaces production, Planika introduces a new solution – the Intelligent Fire. It is due to automatic fireplaces by Planika that having real flames in the interior is not only convenient and user-friendly, but also functional.
Fire Line Automatic 3 will be presented at the trade show in London, it is equipped with an electrical system and WiFi module that allows its remote operation with a tablet or smartphone and 'smart home'' system integration. The fireplace offers unlimited arrangement possibilities and adaptation to individual interiors. The user can freely change the flame size and adjust the device''s settings to one''s needs.
Planika will also present HotBox and ZEN High from the 'Push Button Fireplaces'' collection. They allow fire ignition and flame size regulation with a single button. The modern control panel ensures the maximum intuitive operation.
The automatic fireplaces are excellent option for all interiors. The concept invented by Planika fulfils all four of users'' needs: comfort, ease of use, safety and attractive design. Planika''s products are equipped with safety sensors which automatically disable device in case of overflow or seismic movement. The fireplaces do not emit any smoke or smell, and they do not need any chimney installation or hard connection.
Their distinctive form combined with real fire give the feeling of sensuality that pleases the eye with a fresh look.


Labor Day: Do you Have the Right to be Off this Holiday?
According to a recent Bloomberg BNA survey, although most (if not all) employers will give today —Labor Day—as a paid holiday, around 4 in 10 companies will require at least some employees to report to work today. The Holiday Practices Survey found that 39 percent of surveyed employers reported maintaining operations on Labor Day and requiring some employees to work on this holiday. At 63 percent, larger employers—those with 1,000 or more employees —were more likely to require at least some employees to report to work than were smaller businesses at 29 percent.
The survey also found that 61 percent of non-business establishments –including health care, government, education, membership and social services, and religious organizations – require some employees to work on Labor Day, compared with 32 percent of nonmanufacturing firms and 30 percent of manufacturers.
Also, few employers reported that they sponsor Labor Day-related events or activities to commemorate the holiday. A mere 4 percent of employers overall sponsor Labor Day events or activities, compared with other holidays such as Christmas (28 percent), Thanksgiving (14 percent), or Veterans Day (11 percent).
So, what is this data telling us about today? Although this is a national holiday, a nice chunk of businesses will still have its doors open —and that means employees must forgo the appealing time off associated with this holiday in order to keep those doors open. And as those employees report to the office (or company), they can forget about any holiday-related festivities. They'll be working —just like any other day.
Sounds almost like a violation of employee rights, doesn't it? I mean, here we have a holiday for workers, and some workers won't be able to enjoy it like they deserve to. This seems absolutely wrong…or is it?
The form that the observance and celebration of Labor Day should take was outlined in the first proposal of the holiday — a street parade to exhibit to the public "the strength and esprit de corps of the trade and labor organizations" of the community, followed by a festival for the recreation and amusement of the workers and their families.
Now we see why today is associated with "free time" and "a break" for workers, but unfortunately, this "association" doesn't translate into law. Currently, there is no law that exists that says a company must provide workers with time off (paid or unpaid) on nationally recognized holidays.
Yes, you read that correctly. Whether or not you work national holidays is solely dependent on the practices of your employer. So, although the 39 percent of workers who are required to work today and miss out on this coveted break seems unfair, legally, it isn't.
If having time off, like on national holidays, is important to you as a worker, do your research. When searching for jobs, be sure to inquire about a company's policies when it comes to holidays so you'll know beforehand whether or not this a suitable place.
Labor Day has its roots in Canada where the movement to shorten the average workday from 12 hours to nine hours began in Hamilton, Ontario and quickly spread to Toronto.
The first Labor Day holiday was celebrated on Tuesday, September 5, 1882, in New York City, in accordance with the plans of the Central Labor Union. Two years later, the first Monday in September was selected as the holiday.
Grover Cleveland, the 22nd and 24th president of the United States, signed the bill that made Labor Day a legal holiday for federal employees in the District of Columbia and all U.S. territories.
The first five-day week in America was put in place by a New England spinning mill in 1908 to accommodate its Jewish workers, who had trouble observing the Sabbath under the traditional six-day work week.
There are still debates about who founded the holiday: Peter J. McGuire, general secretary of the Brotherhood of Carpenters and Joiners and a co-founder of the American Federation of Labor or Matthew Maguire, a machinist and the secretary of Local 344 of the International Association of Machinists in Paterson, N.J.