On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures. Issue 3 (2nd July 2016)

• Record Type: Journal Article
• Authors: Korolev, V. Yu.; Zeifman, A. I.
• Is Part Of: Stochastic models. Volume 32:Issue 3(2016)
• Page Start: 414
• Page End: 432
• Publication Date: 2016-07-02
• Subjects: Normal variance–mean mixture -- random index -- random Lindeberg condition -- random sequence -- random sum -- transfer theorem; 60E07 -- 60F99; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/15326349.2016.1155464
• Languages: English
• ISSNs: 1532-6349

Abstract

ABSTRACT: We prove a 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.