Convergence of series of moments on general exponential inequality

2021/03/02 by João Lita da Silva, da Silva, João Lita, Vanda Lourenço +1
Decision Sciences · Mathematics · #60F15 #62F12 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2103.01996

openalex publication_date 2021/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an array \X_n,j, 1 \leqslant j \leqslant k_n, n \geqslant 1 \ of random variables and a sequence \c_n \ of positive numbers, sufficient conditions are given under which, for all ε > 0, ∑_n=1^∞ c_n 𝔼 [ max_1 \leqslant i \leqslant k_n |∑_j=1ⁱ (X_n,j - 𝔼 X_n,j I_\| X_n,j | \leqslant δ\) | - ε ]₊^p < ∞, where x₊ denotes the positive part of x and p \geqslant 1, δ> 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.

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