Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 35, 2010, 179-195

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, *f*_{0} belong to the algebra *W* of absolutely
convergent complex Fourier series on **T** = {|*z*| = 1}. We define
*f*_{n} \in *W* by

(*) *f*_{1}(*z*) = \varphi(*z*)*f*_{0}(*z*)
and *f*_{n+1}(*z*) =
\varphi(*z*)*f*_{n}(*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

*S*_{n+1} = *S*_{0} + *X*_{1}
+ ... + *X*_{n},

where *S*_{0} is a random variable with generating function
*f*_{0} 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

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