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Random walks and periodic continued fractions

Published online by Cambridge University Press:  01 July 2016

Wolfgang Woess
Wolfgang Woess*
Affiliation:
Montanuniversität Leoben
*
∗Postal address: Institut für Mathematik und Angewandte Geometrie, Montanuniversität Leoben, A-8700 Leoben, Austria.
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Abstract

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

Keywords

LOCAL LIMIT THEOREMHARMONIC FUNCTIONSIMPLE RANDOM WALK ON A TREESPHERICAL FUNCTION
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 67 - 84
Copyright
Copyright © Applied Probability Trust 1985 

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References

Bender, E. A. (1974) Asymptotic methods in enumeration. SIAM Rev. 16, 485–515.Google Scholar
Figà-Talamanca, A. and Picardello, M. (1982) Spherical functions and harmonic analysis on free groups. J. Functional Anal. 47, 281–304.Google Scholar
Gerl, P. (1984) Continued fraction methods for random walks on ℕ and on trees. In Probability Measures on Groups, Proceedings, Oberwolfach 1983. Lecture Notes in Mathematics 1064, Springer-Verlag, Berlin, 131–146.Google Scholar
Gerl, P. and Woess, W. (1983) Simple random walks on trees.Google Scholar
Henrici, P. (1977) Applied and Computational Complex Analysis , Vol. 2. Wiley, New York.Google Scholar
Jones, W. B. and Thron, W. J. (1980) Continued Fractions. Encyclopedia of Math. and Appl. 11. Addison-Wesley, London.Google Scholar
Karlin, S. and Mcgregor, J. (1959) Random walks. Illinois J. Math. 3, 66–81.Google Scholar
Papangelou, F. (1967) Strong ratio limits, R-recurrence and mixing properties of discrete parameter Markov chains. Z. Wahrscheinlichkeitsth. 8, 259–297.Google Scholar
Perron, O. (1957) Die Lehre von den Kettenbrüchen , Band 2. Teubner, Stuttgart.Google Scholar
Sawyer, S. (1978) Isotropic random walks in a tree. Z. Wahrscheinlichkeitsth. 42, 279–292.CrossRefGoogle Scholar
Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. (Oxford) (II) 13, 7–28.CrossRefGoogle Scholar
Wall, H. S. (1948) Analytic Theory of Continued Fractions. Van Nostrand, Toronto.Google Scholar