Spectral theory and limit theorems for geometrically ergodic Markov processes

Abstract

Consider the partial sums {S t } of a real-valued functional F (Φ(t)) of a Markov chain {Φ(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained:Spectral theory: Well-behaved solutionsf can be constructed for the "multiplicative Poisson equation" (e αF P )f = λf , where P is the transition kernel of the Markov chain, and α ∈ C is a constant. The functionf is an eigenfunction, with corresponding eigenvalue λ, for the kernel (e αF P ) = e αF (x) P (x, dy).A "multiplicative" mean ergodic theorem: For all complex α in a neighborhood of the origin, the normalized mean of exp(αS t ) (and not the logarithm of the mean) converges tof exponentially fast, wheref is a solution of the multiplicative Poisson equation.Edgeworth Expansions: Rates are obtained for the convergence of the distribution function of the normalized partial sums S t to the standard Gaussian distribution. The first term in this expansion is of order (1/ √ t), and it depends on the initial condition of the Markov chain through the solution F of the associated Poisson equation (and not the solutionf of the multiplicative Poisson equation).Large Deviations: The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line.Exact Large Deviations Asymptotics: Rates of convergence are obtained for the large deviations estimates above. The polynomial pre-exponent is of order (1/ √ t), and its coefficient depends on the initial condition of the Markov chain through the solutionf of the multiplicative Poisson equation.Extensions of these results to continuous-time Markov processes are also given.

Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

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The ODE Method and Spectral Theory of Markov Operators

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Self-normalized Large Deviations for Markov Chains

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