The limit theorem for maximum of partial sums of exchangeable random variables

TL;DR

This paper extends classical limit theorems to exchangeable random variables, showing that under certain conditions, the maximum of their partial sums converges to the classical distribution, with more general cases involving conditional negative drift.

Contribution

It provides a limit theorem for the maximum of partial sums of exchangeable variables, generalizing the Erd"os-Kac result to broader conditions.

Findings

Limit distribution matches classical results under CLT conditions.

Conditional negative drift influences the limiting distribution.

Results apply to exchangeable variables with zero mean and variance one.

Abstract

We obtain the analogue of the classical result by Erd\"os and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central limit theorem of Blum et al. hold, the limit coincides with the classical one. Under more general assumptions, the probability of the random variables having conditional negative drift appears in the limiting distribution.