Spectral theory and limit theorems for geometrically ergodic Markov processes

By Ioannis Kontoyiannis, Sean P. Meyn. The Annals of Applied Probability, 2002

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f ,w hereP is the transition kernel of the Markov chain and α ∈ C is a constant. The function ˇ f 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(αSt ) (and not the logarithm of the mean) converges to ˇ f exponentially fast, where ˇ f 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 St 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 solution ˇ f 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 preexponent is of order (1/ √ t) and its coefficient depends on the initial condition of the Markov chain through the solution ˇ f of the multiplicative Poisson equation. Extensions of these results to continuous-time Markov processes are also given.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/594e60f1a431c19a3c1ad8199f3252f1dd1b1c0e

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Citations in the corpus

A non‐conservative Harris ergodic theorem (2019, background): “These semigroups appear in particular for the study of branching processes [16, 27, 71] and large deviations [47, 63, 64].”
Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, methodology): “We also explain how the behavior out of equilibrium (i.e. X0 6∼ µ) may be recovered from the behavior at equilibrium (i.e. X0 ∼ µ) by using propagation of chaos (decorrelation), for instance via Lyapunov criteria ensuring a quick convergence in law of Xt to µ as t → ∞.”
Geometric ergodicity and the spectral gap of non-reversible Markov chains (2009, background): “Therefore, the chain is geometrically ergodic in the sense of [12], where it is also shown that it must admit a central gap in L∞.”
Ergodic Markov processes and Poisson equations (lecture notes) (2016, background): “A nontrivial issue of distinguishing the cases σ(2) > 0 and σ(2) = 0 for stationary Markov chains is under discussion in [3] for finite MC where a criterion has been established for σ(2) = 0; this criterion was extended to more general cases in [24].”
Perturbation analysis for denumerable Markov chains with application to queueing models (2004, methodology): “In the present work, we use the concepts of Lyapunov functions and ν-geometric ergodicity ([9], [ 16 ], [21], [27], [28]) that allow us to treat the cases that satisfy neither Doeblin conditions nor the strong ergodicity condition.”
Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes (2005, background): “32 4.2 Representations of the Univariate Convex Dual . . . . . . . . . . . . . . . . . . 35 4.3 Characterization of the Bivariate Convex Dual . . . . . . . . . . . . . . . . . . 38”
Control variates for estimation based on reversible Markov chain Monte Carlo samplers (2012, background): “…behaviour of estimators similar to those considered above can be found in Meyn (2006) and Meyn (2007), chapter 11, and finer asymptotics (including large deviations bounds and Edgeworth expansions) can be derived under stronger assumptions from the results in Kontoyiannis and Meyn (2003, 2005).”
Risk-sensitive control of continuous time Markov chains (2014, background): “The existence of the above limit follows from the multiplicative ergodic theorems proved in [18] and [19] .”
The Markov chain Monte Carlo revolution (2008, background): “These include central limit theorems and large deviations bounds for averages 1/N ∑ f(Xi) [56].”