Title: Convergence of series of moments on general exponential inequality
Authors: João Lita da Silva, Vanda Lourenço
(Submitted on 2 Mar 2021 (v1), last revised 24 Jun 2021 (this version, v2))
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.
MSC classes: 60F15, 62F12
From: João Lita da Silva [view email]
[v1] Tue, 2 Mar 2021 19:33:58 GMT (8kb)
[v2] Thu, 24 Jun 2021 11:15:45 GMT (123kb)