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A central limit theorem for iterated random functions

  • Wei Biao Wu (a1) and Michael Woodroofe (a1)

Abstract

A central limit theorem is established for additive functions of a Markov chain that can be constructed as an iterated random function. The result goes beyond earlier work by relaxing the continuity conditions imposed on the additive function, and by relaxing moment conditions related to the random function. It is illustrated by an application to a Markov chain related to fractals.

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Corresponding author

Postal address: Department of Statistics, The University of Michigan, 4062 Frieze Building, 105 South State St, Ann Arbor, MI 48109-1285, USA
∗∗ Email address: wbwu@umich.edu

References

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Benda, M. (1998). A central limit theorem for contractive stochastic dynamical systems. J. Appl. Prob. 35, 200205.
Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 4576.
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Durrett, R., and Resnick, S. (1978). Functional limit theorems for dependent variables. Ann. Prob. 6, 829846.
Falconer, K. (1990). Fractal Geometry. John Wiley, New York.
Gordin, M. I., and Lifsic, B. (1978). The central limit theorem for stationary Markov processes. Doklady 19, 392394.
Hutchinson, J. (1981). Fractals and self similarity. Indiana Univ. Math. J. 30, 713747.

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