Convergence of series of moments on general exponential inequality

Abstract

For an array $\{X_{n,j},1⩽j⩽k_n,n⩾1\}$ of random variables and a sequence $\left\{c_n\right\}$ of positive numbers, sufficient conditions are given under which, for all $ϵ>0$, $\sum _{n=1}^{\mathrm{\infty }}{c}_n\mathbb{E}\left[\underset{1⩽i⩽k_n}{max}\right|\sum _{j=1}^i(X_{n,j}-\mathbb{E}{X}_{n,j}{I}_{\left\{\right|X_{n,j}|⩽\delta \}})|-ϵ{]}_{+}^p<\mathrm{\infty },$ where $x_{+}$ denotes the positive part of x and $p⩾1$, $\delta >0$. 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 rth mean of cumulative sum type estimators of the change in the mean of dependent observations.

Keywords:

• Maximum of partial sums
• convergence of series of moments
• dependent random variables
• CUSUM-type estimators

2010 Mathematics Subject Classifications:

• 60F15
• 62F12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is funded by national funds through the Fundação para a Ciência e a Tecnologia (FCT), I.P., under the scope of projects UIDB/04035/2020 (GeoBioTec) and UIDB/00297/2020 (Center for Mathematics and Applications).