Convergence of series of moments on general exponential inequality

TL;DR

This paper establishes conditions for the convergence of series involving moments of maxima of partial sums of dependent random variables, with applications to change point estimation in dependent data.

Contribution

It provides general sufficient conditions for convergence of moment series in dependent structures, extending classical results and applying to change point estimators.

Findings

Convergence criteria for series of moments of maxima of dependent sums.

Application to consistency of change point estimators.

Extension of exponential inequalities to dependent data.

Abstract

For an array $\left\{X_{n,j}, \, 1 \leqslant j \leqslant k_{n}, n \geqslant 1 \right\}$ of random variables and a sequence $\{c_{n} \}$ of positive numbers, sufficient conditions are given under which, for all $\varepsilon > 0$, $\sum_{n=1}^{\infty} c_{n} \mathbb{E} \bigg[{\displaystyle \max_{1 \leqslant i \leqslant k_{n}}} \Big\lvert\sum_{j=1}^{i} (X_{n,j} - \mathbb{E} \, X_{n,j} I_{\left\{\lvert X_{n,j} \rvert \leqslant \delta \right\}}) \Big\rvert - \varepsilon \bigg]_{+}^{p} < \infty,$ where $x_{+}$ denotes the positive part of $x$ and $p \geqslant 1$, $\delta > 0$. Our statements are announced in a general setting allowing to conclude the previous convergence for well-known dependent structures. As an application, we study complete consistency and consistency in the $r$th mean of cumulative sum type estimators of the change in the mean of dependent observations.