Asymptotics of Running Maxima for φ-Subgaussian Random Double Arrays

The article studies the running maxima Ym,j=max1≤k≤m,1≤n≤jXk,n−am,j where {Xk,n,k ≥ 1,n ≥ 1} is a double array of φ-subgaussian random variables and {am,j,m ≥ 1,j ≥ 1} is a double array of constants. Asymptotics of the maxima of the double arrays of positive and negative parts of {Ym,j,m ≥ 1,j ≥ 1} are studied, when {Xk,n,k ≥ 1,n ≥ 1} have suitable “exponential-type” tail distributions. The main results are specified for various important particular scenarios and classes of φ-subgaussian random variables.