On convergence of the distributions of random sequences with independent random indexes to variance-mean mixtures

TL;DR

This paper establishes conditions under which distributions of random sequences with independent random indexes converge to variance-mean mixtures, extending transfer theorems and analyzing sums of independent variables.

Contribution

It introduces a generalized transfer theorem for randomly indexed sequences and provides necessary and sufficient conditions for their distributional convergence.

Findings

Proves convergence of sums to normal variance-mean mixtures under moment conditions.

Provides a partial inverse theorem characterizing convergence conditions.

Extends transfer theorems to more general random index settings.

Abstract

We prove a version of a general transfer theorem for random sequences with independent random indexes in the double array limit setting under relaxed conditions. We also prove its partial inverse providing the necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed random variables. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance-mean mixtures.