Self-Normalized Concentration Inequalities of Marginal Mean with Sample Variance Only

TL;DR

This paper introduces a new family of self-normalized concentration inequalities for marginal means applicable to a wide range of dependent processes, using only observable variance measures without requiring strong mixing assumptions.

Contribution

It develops explicit, data-driven concentration bounds for marginal means under martingale and mixing conditions, without needing predictable variance proxies or decay assumptions.

Findings

Provides bounds that are explicit and easy to apply.

Applicable to processes with complex dependence structures.

No need for summability or decay conditions on mixing coefficients.

Abstract

(This is the third version of a working paper.) We develop a family of self-normalized concentration inequalities for marginal mean under martingale-difference structure and $\phi/\tilde{\phi}$-mixing conditions, where the latter includes many processes that are not strongly mixing. The variance term is fully data-observable: naive sample variance in the martingale case and an empirical block long-run variance under mixing conditions. Thus, no predictable variance proxy is required. No specific assumption on the decay of the mixing coefficients (e.g. summability) is needed for the validity. The constants are explicit and the bounds are ready to use.