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 on the interval ,
where are i.i.d. random variables with characteristic function Here
is a rational function. We consider more precisely the case
where the distribution of the random variable
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
From these exact expressions we deduce the formula
for the asymptotic behavior of the quantities considered as
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.