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WROCŁAW UNIVERSITY
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TECHNOLOGY

Contents of PMS, Vol. 31, Fasc. 1,
pages 119 - 139
 

RANDOM WALKS ON THE NONNEGATIVE INTEGERS WITH A LEFT-BOUNDED GENERATOR

Charles Delorme
Jean-Marc Rinkel

Abstract: This paper studies the random walks S + ∑ X
 0      i  on the nonnegative integers, where the X
 i  ’s are independent identically distributed random variables with generating function of type Φ (z) = ∑    czi
         i≥ -s i  , s  a positive integer, with a convergence radius greater than 1 . We infer from a link between the number of zeros of z ↦→ 1- Φ (z) inside the unit disc and inf X
    i  a factorisation of the symbol f(θ) = 1 - Φ(eiθ) which allows a geometrical computation of the potentials associated with these random walks. Examples illustrate this theory.

2000 AMS Mathematics Subject Classification: Primary: 60G50; Secondary: 47B35, 15A09, 30Cxx.

Keywords and phrases: Potentials, random walks, Toeplitz matrices.

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