On the asymptotic behavior of the sequence and series of running maxima from a real random sequence

For a sequence {Xn,n≥1} of random variables, set Yn=max1≤k≤nXk−an, where {an,n≥1} is a sequence of constants to be specified. We obtain the limiting behavior of the sequences of positive and negative parts of {Yn,n≥1} when the tail distribution of {Xn,n≥1} satisfies suitable “exponential-type” conditions. Next, we consider the rate convergence of the positive part to zero (results similar to complete convergence).