Summary
In this paper, we continue the study undertaken in our earlier paper [M1]. One of the main results here can be described as follows. LetX 0,X 1, ... be a sequence of iid random affine maps from (R +)d into itself. Let us write:W n ≡X n X n −1...X 0 andZ n ≡X 0 X 1...X n , where “composition” of maps is the rule of multiplication. By the attractorA(u),u∈(R +)d, we mean the setA u={y∈(R+)d:P(Wn u∈N i.o.) > 0 for every openN containingy}. It is shown that the attractorA(u), under mild conditions, is the support of a stationary probability measure, when the random walk (Z n ) has at least one recurrent state.
References
[AC] Arnold, L., Crauel, H.: Iterated function systems and multiplicative ergodic theory. In: Pinsky, M., Wihstutz, V. (eds.) Diffusion proc. and rel. problems in analysis, vol. II, pp. 283–305. Boston Basel Stuttgart: Birkhäuser 1992
[BD] Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. R. Soc. Lond., Ser. A399, 243–275 (1985)
[G] Grintsevichyus, A.K.: On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theor. Probab. Appl.19, 163–168 (1974)
[L] Ljapin, E.S.: Semigroups. Providence, R.I.: Am. Math. Soc. 1963
[M1] Mukherjaa, A.: Recurrent random walks in nonnegative matrices: attractors of certain iterated function systems. Probab. Theory Relat. Fields91, 297–306 (1992)
[M2] Mukherjea, A.: Convergence in distribution of products of random matrices: a semigroup approach. Trans. Am. Math. Soc.303 (no. 1), 395–411 (1987)
[M3] Mukherjea, A.: Limit theorems Stochastic matrices, ergodic Markov chains and measures on semigroups. In: Bharucha-Reid, A.T. (ed.) Prob. analysis and related topics, vol. 2, pp. 143–203. New York London: Academic Press 1979
[MT] Mukherjea, A., Tserpes, N.A.: Measures on topological semigroups (Lect. Notes Math., vol. 547) Berlin Heidelberg New York: Springer 1976
[V] Vervaat, W.: On a stochastic difference equation and a representation of positive infinitely divisible random variables Report 7625. Nijmegen, The Netherlands: Math. Inst. Katholieke Universiteit 1976
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mukhrjea, A. Recurrent random walks in nonnegative matrices II. Probab. Th. Rel. Fields 96, 415–434 (1993). https://doi.org/10.1007/BF01200204
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01200204
Mathematics Subject Classifications (1991)
- 60 B 10
- 60 J 15