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

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