The Discrete Wiener-Hopf Equation Whose Kernel is a Probability Distribution with Positive Drift

Abstract

We consider the discrete Wiener-Hopf equation with inhomogeneous term $$g = \{ {g_j}\} _{j = 0}^\infty \in {l_\infty }$$; the kernel of the equation is an arithmetic probability distribution generating a random walk drifting to +∞. We prove that the previously obtained formula for the Wiener-Hopf equation with general arithmetic kernel for g ∈ l1 is a solution to the equation for g ∈ l∞ and that successive approximations converge to the solution. The asymptotics of the solution is established in the following cases with account taken of their peculiarities: (1) g ∈ l1; (2) g ∈ l∞; (3) gj → const as j → ∞; (4) g ∉ l1 and gj ↓ 0 as j → ∞.