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Convergence of successive approximations to a stochastic fixed point

Published online by Cambridge University Press:  14 July 2016

Arun P. Sanghvi
Arun P. Sanghvi*
Affiliation:
ICF Incorporated
*
∗Postal address: ICF Incorporated, 1850 K St. NW, Suite 950, Washington D. C. 20006, U.S.A.
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Abstract

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).

Keywords

FIXED POINTMARKOV PROCESSESINVARIANT DISTRIBUTIONDOMINATED OPERATORRANDOM MAPPINGSQUASI-COMPACT OPERATORS
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 297 - 299
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research partially funded by NSF grant GK–38121.

References

Batt, J. and Berg, J. (1969) Linear bounded transformations on the space of continuous functions. J. Functional Analysis 4, 215–239.Google Scholar
Dinculeanu, N. (1966) Vector Measures. VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
Doeblin, W. and Fortet, R. (1937) Sur des chaînes à liaisons complète. Bull. Soc. Math. 65, 132–148.Google Scholar
Doob, J. L. (1952) Stochastic Processes. Wiley, New York.Google Scholar
Dubins, L. E. and Freedman, P.A. (1966) Invariant probabilities for certain Markov processes. Ann. Math. Statist. 37, 837–848.Google Scholar
Dunford, N. and Schwartz, J. T. (1957) Linear Operators, Part I. Wiley, New York.Google Scholar
Hanš, O. (1965) Random operator equations. 4th Berkeley Symp. Math. Statist. Prob. 2, 185–202.Google Scholar
Ionescu Tulcea, A. and Marinescu, G. (1950) Théorie ergodique pour des classes d'operations non-completement continues. Ann. Math. 52, 140–147.CrossRefGoogle Scholar
Isaac, R. (1962) Markov processes and unique stationary probability measures. Pacific J. Math. 12, 273–286.Google Scholar
Jamison, B. (1964) Asymptotic behavior of successive iterates of continuous functions under a Markov operator. J. Math. Anal. Appl. 9, 203–214.Google Scholar
Kakutani, S. (1962) Random ergodic theorems and Markoff processes with a stable distribution. 2nd Berkeley Symp. Math. Statist. Prob. Google Scholar
Karlin, S. (1953) Some random walks arising in learning models. Pacific J. Math. 3, 725–756.Google Scholar
Kennedy, M. (1957) A convergence theorem for a certain class of Markoff processes. Pacific J. Math. 7, 1107–1124.CrossRefGoogle Scholar
Norman, M. (1968) Some convergence theorems for stochastic learning models with distance diminishing operators. J. Math. Psych. 5, 61–101.Google Scholar
Ryll-Nardzewski, C. (1966) On fixed points of semi groups of endomorphisms of linear spaces. 5th Berkeley Symp. Math. Statist. Prob. 2, 55–61.Google Scholar
Yahav, J. S. (1975) On a fixed point theorem and its stochastic equivalent. J. Appl. Prob. 12, 605–611.CrossRefGoogle Scholar
Yosida, K. and Kakutani, S. (1939) Operator theoretical treatment of Markoff's process and mean ergodic theorem. Ann. Math. 42, 188–228.Google Scholar