msc;60e10;characteristic functions; other transforms
Journal of Applied Probability (31)
Advances in Applied Probability (8)
Journal of the Australian Mathematical Society (3)
Bulletin of the Australian Mathematical Society (1)
LMS Journal of Computation and Mathematics (1)
Applied Probability Trust (39)
WEAK SUBORDINATION OF MULTIVARIATE LÉVY PROCESSES
MSC 2010: Stochastic processes
MSC 2010: Parametric inference
MSC 2010: Applications
MSC 2010: Markov processes
MSC 2010: Distribution theory - Probability
KEVIN W. LU
Journal: Bulletin of the Australian Mathematical Society , First View
Published online by Cambridge University Press: 09 July 2019, pp. 1-2
Asymptotic expansions and saddlepoint approximations using the analytic continuation of moment generating functions
MSC 2010: Approximations and expansions
Ronald W. Butler
Journal: Journal of Applied Probability / Volume 56 / Issue 1 / March 2019
Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$ , have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks' likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.
On the double Laplace transform of the truncated variation of a Brownian motion with drift
Rafał Marcin Łochowski
Journal: LMS Journal of Computation and Mathematics / Volume 19 / Issue 1 / 2016
The aim of this paper is to find a formula for the double Laplace transform of the truncated variation of a Brownian motion with drift. In order to find the double Laplace transform, we also prove some identities for the Brownian motion with drift, which may be of independent interest.
Approximating the Laplace transform of the sum of dependent lognormals
MSC 2010: Probabilistic methods, simulation and stochastic differential equations
MSC 2010: Integral transforms, operational calculus
Patrick J. Laub, Søren Asmussen, Jens L. Jensen, Leonardo Rojas-Nandayapa
Journal: Advances in Applied Probability / Volume 48 / Issue A / July 2016
Let (X 1,...,X n ) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and let S n =e X 1 +⋯+e X n . The Laplace transform ℒ(θ)=𝔼e-θS n ∝∫exp{-h θ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form and I(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacing h θ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙* is presented, and it is shown that I(θ)→ 1 as θ→∞. A variety of numerical methods for evaluating I(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density of S n ) are also given.
Runs in coin tossing: a general approach for deriving distributions for functionals
MSC 2010: Limit theorems
Lars Holst, Takis Konstantopoulos
Journal: Journal of Applied Probability / Volume 52 / Issue 3 / September 2015
Published online by Cambridge University Press: 30 March 2016, pp. 752-770
We take a fresh look at the classical problem of runs in a sequence of independent and identically distributed coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types. This generalizes and unifies already existing approaches. We give several examples, derive asymptotics, and pose some further questions.
Finite-Dimensional Distributions of a Square-Root Diffusion
Michael B. Gordy
Journal: Journal of Applied Probability / Volume 51 / Issue 4 / December 2014
Print publication: December 2014
We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.
On Multiply Monotone Distributions, Continuous or Discrete, with Applications
MSC 2010: Distribution theory
Claude Lefèvre, Stéphane Loisel
This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.
Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations
MSC 2010: Numerical methods in Fourier analysis
Ankush Agarwal, Santanu Dey, Sandeep Juneja
We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed, and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.
The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion
Adam Metzler
In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.
On a Class of Distributions Stable Under Random Summation
L. B. Klebanov, A. V. Kakosyan, S. T. Rachev, G. Temnov
Journal: Journal of Applied Probability / Volume 49 / Issue 2 / June 2012
Published online by Cambridge University Press: 04 February 2016, pp. 303-318
We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.
A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps
Hermine Biermé, Agnès Desolneux
We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.
Exact Lower Bounds on the Exponential Moments of Truncated Random Variables
Iosif Pinelis
Exact lower bounds on the exponential moments of min(y, X) and X 1{X < y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y, X) over the truncation X 1{X < y} are demonstrated. An application to option pricing is given.
Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions
A. Kuznetsov
Published online by Cambridge University Press: 14 July 2016, pp. 1023-1033
In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.
On zero-truncating and mixing Poisson distributions
J. Valero, M. Pérez-Casany, J. Ginebra
Journal: Advances in Applied Probability / Volume 42 / Issue 4 / December 2010
The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.
An alternative characterization for matrix exponential distributions
Mark Fackrell
A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.
Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent
Valeri T. Stefanov, Frank Ball
Journal: Advances in Applied Probability / Volume 41 / Issue 2 / June 2009
Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.
On Prolific Individuals in a Supercritical Continuous-State Branching Process
Jean Bertoin, Joaquin Fontbona, Servet Martínez
We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.
Exact distributions for reward functions on semi-Markov and Markov additive processes
MSC 2010: Special processes
Valeri T. Stefanov
The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.
Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution
M. Möhle
We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.
B-stability
Colin Mallows, Larry Shepp
A random variable Y is branching stable (B-stable) for a nonnegative integer-valued random variable J with E(J)>1 if Y *J ∿cY for some scalar c, where Y *J is the sum of J independent copies of Y. We explore some aspects of this notion of stability and show that, for any Y 0 with finite nonzero mean, if we define Y n+1=Y n *J /E(J) then the sequence Y n converges in law to a random variable Y ∞ that is B-stable for J. Also Y ∞ is the unique B-stable law with mean E(Y 0). We also present results relating to random variables Y 0 with zero means and infinite means. The notion of B-stability arose in a scheme for cataloguing a large network of computers.