Baum-Katz type theorems with exact threshold

TL;DR

This paper establishes Baum-Katz type theorems under a minimal uniform moment condition for various dependent and independent sequences, leading to strong laws of large numbers with convergence rates.

Contribution

It introduces the weakest possible moment condition for Baum-Katz theorems applicable to dependent and independent sequences, extending classical results.

Findings

Proves Baum-Katz theorems under uniform moment bounds.

Shows the moment condition is optimal even for independent sequences.

Derives Marcinkiewicz-Zygmund strong laws with convergence rate estimates.

Abstract

Let $\{X_n\}_{n\geq 1}$ be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centered sequence with a suitable level of negative dependence. We prove Baum-Katz type theorems by only assuming that the variables $X_n$ satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centered, independent random variables. This leads to Marcinkiewicz-Zygmund type strong laws of large numbers with estimate for the rate of convergence.