Application of the exact inverse of the Toeplitz matrix with singular rational symbol to random walks. (English) Zbl 1102.47062

Rambour, Philippe; Rinkel, Jean-Marc

Probab. Math. Stat. 25, No. 1, 183-195 (2005).

Using the results of previous works on the inverses of the Toeplitz matrices with singular symbol of rational regular part [P. Rambour, J.–M. Rinkel and A. Seghier, C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 11, 857–860 (2000; Zbl 0965.15002); “Exact inverse of the Toeplitz matrix with singular rational symbol” (Prépublication d’Orsay 2004-52) (2004)], the authors compute exact formulars for the expected number of visits and the hitting probabilities on the interval \([0, N]\). From these exact expressions, they deduce the formula for the asymptotic behavior of the quantities considered as \(N\) goes to infinity.

Reviewer: Kun Soo Chang (Seoul)

MSC:

47N30	Applications of operator theory in probability theory and statistics
47B39	Linear difference operators
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
60G50	Sums of independent random variables; random walks

Keywords:

Toeplitz matrices; rational singular symbol; random walk on a finite interval; hitting probabilities