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Let  Open image in new window  be a sequence of independent and identically distributed (i.i.d.) random variables and denote  Open image in new window ,  Open image in new window . 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  Open image in new window ,  Open image in new window  we have  Open image in new window  for a nondegenerate distribution  Open image in new window , and  Open image in new window  is a bounded Lipschitz 1 function, then  Open image in new window  almost surely, where  Open image in new window  stands for the standard normal distribution function,  Open image in new window  ,and  Open image in new window ,  Open image in new window .

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