Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits

TL;DR

This paper investigates the convergence in distribution of sums of exchangeable random variables, highlighting the role of mixtures of stable laws as potential limit distributions and providing conditions for such convergence.

Contribution

It establishes necessary and sufficient conditions for convergence to mixtures of stable laws in exchangeable sums, advancing understanding of their limit behavior.

Findings

Mixtures of stable laws serve as limits for sums of exchangeable variables.

Necessary and sufficient conditions for convergence are derived.

Sufficient conditions for convergence within a single row of exchangeable arrays are provided.

Abstract

The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws - as limits in law of normed sums in different rows of the array - is emphasized. Necessary and sufficient conditions for convergence to a specific form in the above class of measures are then given. Moreover, sufficient conditions for convergence of sums in a single row are proved. Finally, a potentially useful variant of the formulation of the results just summarized is briefly sketched, a more complete study of it being deferred to a future work.