Geometric ergodicity in a weighted sobolev space
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Peer-reviewed
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Abstract
For a discrete-time Markov chain
- The transition kernel
has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space of functions with norm, $$ |f|{v,1} = \sup{x \in \Re^\ell} \frac{1}{v(x)} \max {|f(x)|, |\partial_1 f(x)|,\ldots,|\partial_\ell f(x)|}, $$ where is a Lyapunov function and . - The Markov chain is geometrically ergodic in
: There is a unique invariant probability measure and constants and such that, for each , any initial condition , and all : $$\Big| \text{E}_x[f(X(t))] - \pi(f)\Big| \le Be^{-\delta t}v(x),\quad |\nabla \text{E}_x[f(X(t))] |_2 \le Be^{-\delta t} v(x), $$ where . - For any function
there is a function solving Poisson's equation: [ h-Ph = f-\pi(f). ] Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents.
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Keywords
Markov chain, stochastic Lyapunov function, discrete spectrum, sensitivity process, weighted Sobolev space, Lyapunov exponent
Journal Title
Annals of Probability
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Journal ISSN
0091-1798
2168-894X
2168-894X
Volume Title
48
Publisher
Institute of Mathematical Statistics