An Anscombe-type theorem

TL;DR

This paper extends the Anscombe theorem by establishing new conditions under which sequences of random variables evaluated at random indices converge stably, with examples illustrating the applicability of these conditions beyond existing ones.

Contribution

It introduces novel conditions for stable convergence of random variables at random indices, expanding the applicability of the Anscombe theorem.

Findings

New conditions for stable convergence are established.

Examples demonstrate cases where existing conditions fail but new ones succeed.

Theoretical framework applicable to exchangeability and random sums.

Abstract

Let (X_n) be a sequence of random variables (with values in a separable metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to converge stably (in particular, in distribution) are provided. Some examples, where such conditions work but those already existing fail, are given as well. Key words and phrases: Anscombe theorem, Exchangeability, Random indices, Random sums, Stable convergence