Asymptotics of running maxima for $\varphi$-subgaussian random double arrays

TL;DR

This paper investigates the asymptotic behavior of running maxima in double arrays of -subgaussian random variables, providing detailed results for various tail distribution scenarios and specific classes.

Contribution

It offers new asymptotic results for maxima of -subgaussian double arrays, extending understanding of their tail behaviors and maxima distributions.

Findings

Asymptotic formulas for maxima of -subgaussian arrays derived.

Results specified for different tail distribution classes.

Main theorems cover various -subgaussian scenarios.

Abstract

The article studies the running maxima $Y_{m,j}=\max_{1 \le k \le m, 1 \le n \le j} X_{k,n} - a_{m,j}$ where $\{X_{k,n}, k \ge 1, n \ge 1\}$ is a double array of $\varphi$-subgaussian random variables and $\{a_{m,j}, m\ge 1, j\ge 1\}$ is a double array of constants. Asymptotics of the maxima of the double arrays of positive and negative parts of $\{Y_{m,j}, m \ge 1, j \ge 1\}$ are studied, when $\{X_{k,n}, k \ge 1, n \ge 1\}$ have suitable "exponential-type" tail distributions. The main results are specified for various important particular scenarios and classes of $\varphi$-subgaussian random variables.