The limit law of maximum of discrete partial-sums distribution

Let X₁,X₂, . . .,X_N, N ∈ \({\mathbb{N}}\) $\mathbb{N}$ , be independent but not necessarily identically distributed discrete and integervalued random variables. Assume that X₁ ⩾ m₁, X₂ ⩾ m₂, . . . , X_N ⩾ m_N almost surely, where m₁,m₂, . . . , m_N are some integer numbers such that m₁ + m₂ + ⋯ +m_N < 0, and X_k \(\overset{d}{=}\) $\stackrel{d}{=}$ X_k+N for all k ∈ \({\mathbb{N}}\) $\mathbb{N}$ 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 ∈ \({\mathbb{Z}}\) $\mathbb{Z}$ , via (a) inclusion–exclusion principle-based sum-product of the roots of G_N(s) = 1, where G_N(s) is the probability generating function of S_N = X₁+X₂+ ⋯ +X_N, (b) the probability mass function of S_N, and (c) the expectation ES_N.