Lacunary series and stable distributions

TL;DR

This paper establishes criteria for subsequences of random variables to converge to stable distributions, extending classical probability results on subsequences satisfying the CLT and LIL.

Contribution

It introduces new criteria for subsequences of random variables to converge to stable laws, generalizing well-known probabilistic theorems.

Findings

Provides conditions for convergence to stable distributions

Extends classical results on subsequences satisfying CLT and LIL

Offers a framework for analyzing weighted partial sums

Abstract

By well known results of probability theory, any sequence of random variables with bounded second moments has a subsequence satisfying the central limit theorem and the law of the iterated logarithm in a randomized form. In this paper we give criteria for a sequence $(X_n)$ of random variables to have a subsequence $(X_{n_k})$ whose weighted partial sums, suitably normalized, converge weakly to a stable distribution with parameter $0<\alpha<2$.