Mathematica
Volumen 35, 2010, 179-195

# LAURENT SEPARATION, THE WIENER ALGEBRA AND RANDOM WALKS

## Gerd Jensen and Christian Pommerenke

Sensburger Allee 22 a, D--14055 Berlin, Germany; cg.jensen 'at' arcor.de

Technische Universität Berlin, Institut für Mathematik
D--10623 Berlin, Germany; pommeren 'at' math.tu-berlin.de

Abstract. Let \varphi, f0 belong to the algebra W of absolutely convergent complex Fourier series on T = {|z| = 1}. We define fn \in W by

(*) f1(z) = \varphi(z)f0(z) and fn+1(z) = \varphi(z)fn(z)+ for n \in N,

where (...)+ denotes the analytic part of the Laurent series. We derive a number of generating functions all of which contain

p(z,w) = exp([log(1 - w\varphi(z))]-) (|z| \ge 1, |w| < 1).

The 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 by

Sn+1 = S0 + X1 + ... + Xn,

where S0 is a random variable with generating function f0 specifying the initial distribution, the X\nu are i.i.d. with generating function \varphi and the random walk stops if it hits (-\infty,-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.

2000 Mathematics Subject Classification: Primary 30B10, 30H05, 60G50.

Key words: Generating functions, Laurent separation, Wiener algebra, Wiener-Hopf, random walks, ruin, minimum.

Reference to this article: G. Jensen and Ch. Pommerenke: Laurent separation, the Wiener algebra and random walks. Ann. Acad. Sci. Fenn. Math. 35 (2010), 179-195.

doi:10.5186/aasfm.2010.3510