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Let X1,X2,...,XN, N ∈ N , be independent but not necessarily identically distributed discrete and integervalued random variables. Assume that X1 ⩾ m1, X2 ⩾ m2,... , XN ⩾ mN almost surely, where m1,m2,... , mN are some integer numbers such that m1 + m2 + ⋯ +mN < 0, and Xk = d Xk+N for all k ∈ N in the sequence X1,X2,.... In this paper, we make use of some known results to provide a closed-form expression of the limit distribution function P(max{X1, X1+X2,...} ⩽ x) = P(X1 ⩽ x, X1 + X2 ⩽ x,...), x ∈ Z , via (a) inclusion–exclusion principle-based sum-product of the roots of GN(s) = 1, where GN(s) is the probability generating function of SN = X1+X2+ ⋯ +XN, (b) the probability mass function of SN, and (c) the expectation ESN. [ABSTRACT FROM AUTHOR]

Limit distributions of maxima of dependent Gaussian sequence are different according to the convergence rate of their correlations. For three different conditions on convergence rate of the correlations, in this paper, we establish the Piterbarg theorem for maxima of stationary Gaussian sequences. [ABSTRACT FROM AUTHOR]

For a fixed integer n ≥ 2, let X ,..., X be independent random variables (r.v.s) with distributions F,..., F, respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F, i = 1 ,..., n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X + ⋯ + X and Y. By taking different G's, we obtain general forms of some existing results. [ABSTRACT FROM AUTHOR]