Almost sure central limit theorem for self-normalized partial sums and maxima

Abstract

Let \(X, X_1, X_2,\ldots \) be a sequence of independent and identically distributed random variables with zero mean and finite second moment. A universal result in almost sure central limit theorem for the self-normalized partial sums \(S_n/V_n\) and maxima \(M_n\) is established, where \(S_n=\sum _{i=1}^nX_i, V^2_n=\sum _{i=1}^nX^2_i,\) and \(M_n=\max _{1\le i\le n}X_i.\)