Comparison of Sums of independent Identically Distributed Random Variables

TL;DR

This paper compares the tail distributions of partial sums of i.i.d. Banach space-valued random variables, establishing inequalities that relate the probabilities of large deviations for different partial sums.

Contribution

It introduces a universal inequality linking tail probabilities of partial sums at different indices for Banach space-valued i.i.d. variables.

Findings

Established a universal constant c for tail probability inequalities.

Derived tail distribution maximal inequalities for partial sums.

Compared tail behaviors of partial sums in Banach spaces.

Abstract

Let S_k be the k-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of ||S_k|| with that of ||S_j||, and deduce some tail distribution maximal inequalities. Theorem: There is universal constant c such that for j < k Pr(||S_j|| > t) <= c Pr(||S_k|| > t/c).