Bounds for the Rate of Convergence in the Generalized Rényi Theorem

Abstract

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).

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

“…The first is the distribution of the "waiting time" to r-th "success" in a sequence of Bernoulli trials with the success probability p including the first r successes, while the second is the distribution of the number of "failures" preceding the r-th success. The latter distribution has a natural extension, sometimes called the Pólia distribution, for any real number r > 0 [9]. For compelling biological reasons associated with the structure of genes and elucidated in Section 5, see also [10], we will be modeling the length of DNA segments using negative binomial distributions NB(r, p) of the first kind with integer r ≥ 1.…”

“…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.…”

“…On the mathematical side, our limit theorem would be more practical if augmented with a tight estimate of the Zolotarev metric [9,14] or another suitable distance [19] between the empirical distribution of the normalized log-likelihood score and its theoretical limiting counterpart.…”