On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables

Irina Shevtsova,Mikhail Tselishchev

doi:10.3390/math8111917

Abstract

Using the generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with a finite non-zero first moment, we prove moment-type error-bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent random variables with positive finite expectations, in particular, to geometric random sums, generalizing the previous results to alternating and dependent random summands. We also extend the notions of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) distributions to alternating random variables in terms of the corresponding distribution functions and provide a criteria in terms of conditional expectations similar to the classical one. As corollary, we provide simplified error-bounds in the case of NBUE/NWUE conditional distributions of random summands.

Highlights

According to the generalized Rényi theorem, a geometric random sum of independent identically distributed (i.i.d.) nonnegative random variables (r.v.’s), normalized by its mean, converges in distribution to the exponential law when the expectation of the geometric number of summands tends to infinity
Peköz and Röllin [2] developed Stein’s method for the exponential distribution and obtained moment-type estimates for the exponential approximation to geometric and non-geometric random sums with non-negative summands completing Kalashnikov’s bounds in the Kantorovich distance. Their method was substantially based on the equilibrium transform of non-negative random variables, yielding the technical restriction on the support of the random summands under consideration
Using the above notation and techniques, we prove moment-type error bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent r.v.’s with positive finite expectations (Theorem 1), which generalize the results of [2] to alternating random summands and results of [3] to dependent random summands

Summary

Introduction

According to the generalized Rényi theorem, a geometric random sum of independent identically distributed (i.i.d.) nonnegative random variables (r.v.’s), normalized by its mean, converges in distribution to the exponential law when the expectation of the geometric number of summands tends to infinity. As described in [3], it is defined in terms of d.f.’s in the same way as for probability measures in (1) and allows an alternative representation as an area between d.f.’s of its arguments (similar to the last expression in (2)) This generalization retains the property of the homogeneity of order 1 (see ([3], Lemma 1)). Using the above notation and techniques, we prove moment-type error bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent r.v.’s with positive finite expectations (Theorem 1), which generalize the results of [2] to alternating random summands and results of [3] to dependent random summands. We provide simplified error-bounds in cases of NBUE/NWUE conditional distributions of random summands, generalizing those obtained in [2]

Main Results

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