Abstract
A result by Elton(6) states that an iterated function system
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 \(\mathbb{E}\log ^ + d(F_1 (x_0 ),x_0 ) < \infty \) for some \(x_0 \in \mathbb{X}\). Diaconis and Freedman(5) showed the convergence rate be geometric in the Prokhorov metric if \(\mathbb{E}L_1^p < \infty {\text{ and }}\mathbb{E}d(F_1 (x_0 ),x_0 )^p < \infty \) for some p>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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REFERENCES
Alsmeyer, G., and Fuh, C. D. (2001). Limit theorems for iterated random functions by regenerative methods. Stoch. Proc. Appl. 96, 123–142.
Alsmeyer, G., and Fuh, C. D. (2001). Stoch. Proc. Appl.Corrigendum in 97, 341–345.
Bougerol, P., and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20, 1714–1730.
Brandt, A. (1986). The stochastic equation Yn+1=AnYn+Bn with stationary coefficients. Adv. Appl. Probab. 18, 211–220.
Chan, K. S., and Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. Appl. Probab. 17, 666–678.
Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76.
Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stoch. Proc. Appl. 34, 39–47.
Halmos, P., and Savage, L. J. (1948). Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Statist. 20, 225–241.
Meyn, S. P., and Tweedie, R. L. (1992). Stability of Markovian processes, I. Adv. Appl. Probab. 24, 542–574.
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability, Springer, London.
Niemi, S., and Nummelin, E. (1986). On non-singular renewal kernels with an application to a semigroup of transition kernels. Stoch. Proc. Appl. 22, 177–202.
Petrucelli, J. D., and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Probab. 21, 270–286.
Tong, H. (1990). Non-linear Time Series. A Dynamical Systems Approach, Oxford University Press, Oxford.
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Probab. 11, 750–783.
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Alsmeyer, G. On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results. Journal of Theoretical Probability 16, 217–247 (2003). https://doi.org/10.1023/A:1022290807360
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DOI: https://doi.org/10.1023/A:1022290807360