Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

Korolev, V. Yu.; Zeifman, Alexander

doi:10.1016/j.spl.2020.108918

Cited by 7 publications

(1 citation statement)

References 28 publications

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“…Specifically, for a = 1, the fact that the limiting distribution of rv Y is Exp(1) represents a classic theorem due to Rényi [11], see also [12]. A generalization of Rényi's theorem to negative binomial distributions NB(r, p) of the second kind with arbitrary r > 0 was obtained by Korolev and Zeifman [13] based on an estimate of the Zolotarev distance [14] between the distributions of the normalized random sum U and E(r, r); for a review of relevant results and methodology, see the article by Korolev [9] and references therein. Although it is probably possible to prove Theorem 1 by reduction to the known limit theorems for the normalized random sum U, we here prefer, for greater insight and the reader's convenience, to give a direct, self-contained, and fairly elementary proof of Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

Hanin,

Pavlova

2023

Mathematics

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We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and a random term depending on the length distributions for the two classes. For long sequences selected from a large alphabet, computing misclassification error rates is not feasible either theoretically or computationally. To mitigate this problem, we computed limiting distributions for two versions of the normalized LLS applicable to long sequences whose class-specific length follows a translated negative binomial distribution (TNBD). The two limiting distributions turned out to be plain or transformed Erlang distributions. This allowed us to establish the asymptotic accuracy of the likelihood-based classification of random sequences with TNBD length distributions. Our limit theorem generalizes a classic theorem on geometric random sums due to Rényi and is closely related to the published results of V. Korolev and coworkers on negative binomial random sums. As an illustration, we applied our limit theorem to the classification of DNA sequences contained in the genome of the bacterium Bacillus subtilis into two classes: protein-coding genes and standard noncoding open reading frames. We found that TNBDs provide an excellent fit to the length distributions for both classes and that the limiting distributions capture essential features of the normalized empirical LLS fairly well.

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Section: Introductionmentioning

confidence: 99%

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

Hanin,

Pavlova

2023

Mathematics

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Convergence rates in the limit theorems for random sums of m-orthogonal random variables

Quang,

Huan,

Kien

2025

Statistics & Probability Letters

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Bounds for the Rate of Convergence in the Generalized Rényi Theorem

Korolev

2022

Mathematics

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In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. The main focus is on the upper bounds for the Zolotarev ζ-metric as the distance between the pre-limit and limit laws. New results are presented that extend existing estimates of the rate of convergence of geometric random sums (in the well-known Rényi theorem) to a considerably more general class of random indices whose distributions are mixed Poisson, including generalized negative binomial (e.g., Weibull-mixed Poisson), Pareto-type (Lomax)-mixed Poisson, exponential power-mixed Poisson, Mittag-Leffler-mixed Poisson, and one-sided Linnik-mixed Poisson distributions. A transfer theorem is proven that makes it possible to obtain upper bounds for the rate of convergence in the law of large numbers for mixed Poisson random sums with mixed exponential mixing distribution from those for geometric random sums (that is, from the convergence rate estimates in the Rényi theorem). Simple explicit bounds are obtained for ζ-metrics of the first and second orders. An estimate is obtained for the stability of representation of the Mittag-Leffler distribution as a geometric convolution (that is, as the distribution of a geometric random sum).

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Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

Cited by 7 publications

References 28 publications

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

Convergence rates in the limit theorems for random sums of m-orthogonal random variables

Bounds for the Rate of Convergence in the Generalized Rényi Theorem

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