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On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

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  • Gerold Alsmeyer

Abstract

A result by Elton(6) states that an iterated function system $$M_n = F_n (M_{n - 1} ),{\text{ }}n \geqslant 1,$$ of i.i.d. random Lipschitz maps F 1,F 2,... on a locally compact, complete separable metric space $$(\mathbb{X},d)$$ converges weakly to its unique stationary distribution π if the pertinent Liapunov exponent is a.s. negative and $$\mathbb{E}\log ^ + d(F_1 (x_0 ),x_0 ) 0, where L 1 denotes the Lipschitz constant of F 1. The same and also polynomial rates have been recently obtained in Alsmeyer and Fuh(1) by different methods. In this article, necessary and sufficient conditions are given for the positive Harris recurrence of (M n ) n≥0 on some absorbing subset $$\mathbb{H}{\text{ of }}\mathbb{X}$$ . If $$\mathbb{H} = \mathbb{X}$$ and the support of π has nonempty interior, we further show that the same respective moment conditions ensuring the weak convergence rate results mentioned above now lead to polynomial, respectively geometric rate results for the convergence to π in total variation ∥⋅∥ or f-norm ∥⋅∥ f , f(x)=1+d(x,x 0) η for some η∈(0,p]. The results are applied to various examples that have been discussed in the literature, including the Beta walk, multivariate ARMA models and matrix recursions.

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  • Gerold Alsmeyer, 2003. "On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results," Journal of Theoretical Probability, Springer, vol. 16(1), pages 217-247, January.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022290807360
    DOI: 10.1023/A:1022290807360
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    References listed on IDEAS

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    1. Niemi, S. & Nummelin, E., 1986. "On non-singular renewal kernels with an application to a semigroup of transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 22(2), pages 177-202, July.
    2. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    3. Alsmeyer, Gerold & Fuh, Cheng-Der, 2001. "Limit theorems for iterated random functions by regenerative methods," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 123-142, November.
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