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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 25, Fasc. 1,
pages 183 - 195
 

APPLICATION OF THE EXACT INVERSE OF THE TOEPLITZ MATRIX WITH SINGULAR RATIONAL SYMBOL RANDOM WALKS

Philippe Rambour
Jean-Marc Rinkel

Abstract: In the paper we study the random walks  sum n  X
  i=0  i  on the interval [0,N ] < Z , where X
  i  are i.i.d. random variables with characteristic function P = (l- cosh)| f|2. Here f is a rational function. We consider more precisely the case

             ---A-----
P = (1- cosh)|1 - aeih|2, 0 < a < 1.
where the distribution of the random variable Xi  is characterized. Using the results of previous works on the inverses of the Toeplitz matrices with singular symbol of rational regular part, we compute exact formulas for the expected number of visits and the hitting probabilities on the interval [0,N ]. From these exact expressions we deduce the formula for the asymptotic behavior of the quantities considered as N goes to infinity.

2000 AMS Mathematics Subject Classification: 47B39, 47BXX.

Key words and phrases: Toeplitz matrices, rational singular symbol, random walk on a finite interval, hitting probabilities.

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