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On absorption probabilities for a random walk between two different barriers. (English) Zbl 0765.60071
For the simple random walk on \(\{0,1,\dots,N\}\) having only steps \(\pm 1\) and 0, an absorbing state at \(N\) and a partially reflecting boundary at 0 formulas for the absorption probabilities and the first two moments of the absorption time are given.

MSC:
60G50 Sums of independent random variables; random walks
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[1] Blasi ( A. ) .- On a random walk between a reflecting and absorbing barrier , Ann. Probability , 4 ( 1976 ) pp. 695 - 696 . Article | MR 407996 | Zbl 0342.60050 · Zbl 0342.60050
[2] Cox ( D.R. ) and Miller ( H.D. ) .- The theory of stochastic processes , Methuen , London ( 1965 ). MR 192521 | Zbl 0149.12902 · Zbl 0149.12902
[3] Feller ( W. ) .- An Introduction to probability theory and its applications , ( third edition) Vol. 1 , John Wiley , New York ( 1968 ). MR 228020 | Zbl 0155.23101 · Zbl 0155.23101
[4] Hardin ( J.C. ) and Sweet ( A.L. ) .- A note on absorption probabilities for a random walk between a reflecting and an absorbing barrier , J. Appl. Prob. , 6 ( 1969 ) pp. 224 - 226 . MR 243622 | Zbl 0176.47801 · Zbl 0176.47801
[5] Kennedy ( D.P. ) .- Some martingales related to cumulative sum tests and single-server queues , Stoch. Proc. Appl. , 4 ( 1976 ) pp. 261 - 269 . MR 420834 | Zbl 0338.60030 · Zbl 0338.60030
[6] Khan ( R.A. ) .- On cumulative sum procedures and the SPRT applications , J. R. Statist. Soc. , B 64 ( 1984 ) pp. 79 - 85 . MR 745219 | Zbl 0543.62058 · Zbl 0543.62058
[7] Munford ( A.G. ) .- A first passage problem in a random walk with a quality control application , J. R. Statist. Soc. , B 43 ( 1981 ) pp. 142 - 146 . MR 626758 | Zbl 0477.60082 · Zbl 0477.60082
[8] Murthy ( K.P.N. ) and Kehr ( K.W. ) .- Mean first-passage time of random walks on a random lattice , Phys. Rev. , A 40 , N 4 ( 1989 ) pp. 2082 - 2087 .
[9] Neuts ( M.F. ) .- Absorption probabilities for a random walk between a reflecting and an absorbing barrier , Bull. Soc. Math. Belgique , 15 ( 1963 ) pp. 253 - 258 . MR 160275 | Zbl 0118.33501 · Zbl 0118.33501
[10] Stadje ( W. ) .- Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues , J. Appl. Prob. , 24 ( 1987 ) pp. 200 - 214 . MR 876181 | Zbl 0611.60093 · Zbl 0611.60093
[11] Weesakul ( B. ) .- The random walk between a reflecting and an absorbing barrier , Ann. Math. Statist. , 32 ( 1961 ) pp. 765 - 769 . Article | MR 125641 | Zbl 0109.10903 · Zbl 0109.10903
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