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

Abstract

Let {Xn;n≥1} be a sequence of independent and identically distributed (i.i.d.) random variables and denote Sn=∑k=1nXk, Mn=max⁡1≤k≤n⁡Xk. 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 (an>0), (bn) we have (Mn-bn)/an→𝒟G for a nondegenerate distribution G, and f(x,y) is a bounded Lipschitz 1 function, then lim⁡n→∞⁡(1/Dn)∑k=1ndkf(Sk/k,(Mk-bk)/ak)=∬-∞∞f(x,y)Φ(dx)G(dy) almost surely, where Φ(x) stands for the standard normal distribution function, Dn=∑k=1ndk ,and dk=(exp⁡((log⁡k)α))/k, 0≤α<1/2.