On convergence of the distributions of statistics with random sample sizes to normal variance-mean mixtures

TL;DR

This paper establishes a general transfer theorem for multivariate random sequences with random sample sizes, demonstrating their convergence to normal variance-mean mixtures under natural conditions.

Contribution

It introduces a transfer theorem for multivariate random sequences with random indexes and characterizes their convergence to normal variance-mean mixtures.

Findings

Proves a transfer theorem for multivariate random sequences with random indexes.

Provides necessary and sufficient conditions for convergence of randomly indexed sequences.

Shows convergence of statistics from samples with random sizes to normal variance-mean mixtures.

Abstract

We prove a general transfer theorem for multivariate random sequences with independent random indexes in the double array limit setting. We also prove its partial inverse providing 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 statistics constructed from samples with random sizes. Under rather natural conditions we prove the theorem on convergence of the distributions of such statistics to normal variance-mean mixtures.