Laurent separation, the Wiener algebra and random walks

Abstract

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.

A Proof of Yamashita’s Conjecture on Area Integral

“…Since A 1/2 (r ) = πr 2 , Theorem 2 follows and hence, another proof of Yamashita's conjecture. Before we proceed to prove Theorem 3, it is worth mentioning certain basic properties of the functional given by A β (r ) in (5), where…”

“…Moreover, many useful probabilities can be expressed in terms of the Taylor coefficients of z/ f (z), its derivatives, or their combinations and as a consequence of it, certain known inequalities for such combinations allow us to find explicit estimates for probabilities. For example, the authors in [5][6][7]14,15] recently have found many interesting applications in the theory of the analytic fixed point function and even in questions in probability.…”