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As a result, E ( α$_{i}$ ( δ$_{n}$, 0) α$_{j}$ ( δ$_{n}$, 0)) = 0. Moreover, 

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As a result, 

Lemma S1.3 yields E( M ∗ $_{n}$( t )) = 0 and 

where $^{F}$⌊ nt ⌋ denotes the empirical distribution function of ˜ ψ ( X$_{i}$ ), i ≥ 1, i.e. 

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$^{Note that sup$_{0}$}$≤ x ≤ 1 ( $^{F}$⌊ nt ⌋ ( δ$_{n}$ + x ) − $^{F}$⌊ nt ⌋ ( x ) ) converges to 0 almost surely due to the Glivenko-Cantelli theorem for stationary, ergodic sequences and since F$_{˜}$ ψ ( X 1 ) is a con- tinuous distribution function. It follows that its expected value converges to 0, as well. Therefore, the right-hand side of (S1.12) converges to 0. Thus, M ∗ $_{n}$( t ) converges to 0 in probability. This implies fidi-convergence of M ∗ n as a process with values in D [0 , 1]. Again, 

