Limit laws for Boolean convolutions

Jiun-Chau Wang

Vol. 237 (2008), No. 2, 349–371

DOI: 10.2140/pjm.2008.237.349

Abstract

We study the distributional behavior for products and sums of Boolean independent random variables in a general infinitesimal triangular array. We show that the limit laws of Boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to demonstrate several connections between the limiting distributional behavior of classical convolutions and that of Boolean convolutions. The proof of our results is based on the analytical apparatus developed by Bercovici and Wang for free convolutions.

Keywords

Boolean convolution, limit theorems, infinitesimal arrays

Mathematical Subject Classification 2000

Primary: 46L54, 46L53

Milestones

Received: 30 July 2007

Accepted: 4 June 2008

Published: 1 October 2008