Abstract. Let ϕ, f 0 belong to the algebra W of absolutely convergent complex Fourier series on T = {|z| = 1}. We define f n ∈ W bywhere (. . .) + denotes the analytic part of the Laurent series. We derive a number of generating functions all of which containThe Laurent separation is a discrete equivalent to the Wiener-Hopf factorization of probability theory and allows us to obtain rather concrete results.The recursion ( * ) comes from the study of the random walk on Z defined bywhere S 0 is a random variable with generating function f 0 specifying the initial distribution, the X ν are i.i.d. with generating function ϕ and the random walk stops if it hits (−∞, −1], which is a version of the ruin problem. We also consider the technical problems which arise if X is replaced by −X. The results will also be applied to the minimum problem for random walks.
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