A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums

Abstract

Let be a sequence of independent and identically distributed (i.i.d.) random variables and denote , . In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences , we have for a nondegenerate distribution , and is a bounded Lipschitz 1 function, then almost surely, where stands for the standard normal distribution function, ,and , .