Approximations for sums of three-valued 1-dependent symmetric random variables

Gabija Liaudanskaitė,Vydas Čekanavičius

doi:10.15388/namc.2020.25.16843

Gabija Liaudanskaitė, Vydas Čekanavičius

Open Access

https://doi.org/10.15388/namc.2020.25.16843

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Abstract

The sum of symmetric three-point 1-dependent nonidentically distributed random variables is approximated by a compound Poisson distribution. The accuracy of approximation is estimated in the local and total variation norms. For distributions uniformly bounded from zero,the accuracy of approximation is of the order O(n–1). In the general case of triangular arrays of identically distributed summands, the accuracy is at least of the order O(n–1/2). Nonuniform estimates are obtained for distribution functions and probabilities. The characteristic functionmethod is used.

Highlights

We consider the direct extension of 2-runs statistic to a symmetric case
In the papers devoted to discrete approximations of weakly dependent random variables, it is typical to assume their nonnegativeness, for example, see [4, 14, 19, 20]
As far as we know, so far there was no attempt to apply compound Poisson approximation to the sums of 1dependent rvs taking positive and negative values. For independent rvs, it is well-known that symmetry can considerably improve the accuracy of approximation

Summary

Introduction

We consider the direct extension of 2-runs statistic to a symmetric case. Note that, due to its explicit structure, k-runs (and, especially, 2-runs) statistic is arguably the best investigated case of m-dependent rvs, see [2, 11, 20]. In the papers devoted to discrete approximations of weakly dependent random variables (rvs), it is typical to assume their nonnegativeness, for example, see [4, 14, 19, 20]. As far as we know, so far there was no attempt to apply compound Poisson approximation to the sums of 1dependent rvs taking positive and negative values. For independent rvs, it is well-known that symmetry can considerably improve the accuracy of approximation. We retain a similar structure, replacing Bernoulli variables by symmetric three-point rvs. We use the difference of two independent Poisson variables with the same mean for approximation. The section will provide the already known results related to the approximations of m-dependent random variables.

Known results

Setting

Results

Inversion formulas

Heinrich’s lemma

Proofs of the main results