Limit theorems for forward and backward processes of numbers of non-empty urns in infinite urn schemes

TL;DR

This paper investigates the joint asymptotic behavior of forward and backward processes in an infinite urn scheme, establishing Gaussian process convergence and developing statistical tests for scheme homogeneity.

Contribution

It introduces a new analysis of joint forward-backward processes in infinite urn schemes, including Gaussian process limits and parameter estimation methods.

Findings

Weak convergence to a two-dimensional Gaussian process.

Covariance depends only on the regular decrease exponent.

Parameter estimates with normal asymptotics for joint distribution.

Abstract

We study the joint asymptotics of forward and backward processes of numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. We obtain parameter estimates that have a normal asymototics for its joint distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn scheme on the number of thrown balls.