Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

By Ioannis Kontoyiannis, Sean P. Meyn. 2005

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In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a "maximal," well-behaved solution to the "multiplicative Poisson equation," defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

Subfields: Finance, Management Information Systems, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2831469234c7633aefd5adc9ca80f5cfa7c5b05c

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Citations in the corpus

Geometric ergodicity and the spectral gap of non-reversible Markov chains (2009, unknown): “, [16, 22, 21, 8, 5, 12, 11, 13, 23, 19, 3, 2, 17, 14, 7, 6], and the relevant references therein.”
Bernstein type's concentration inequalities for symmetric Markov processes (2010, unknown): “(1.1)”
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].”
Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions (2014, background): “Given the work of [26] and the number of exponential moments that we need to treat, it seems that multiplicative drift conditions (see [17]) could be adopted.”
Spectral theory and limit theorems for geometrically ergodic Markov processes (2002, background): “Both M+p and B+p are positive cones, and they are closed under addition.”
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] .”
Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, background): “Note that since we focus on an L2 framework, the natural normalization is the square root of the variance and we can only expect Gaussian fluctuations.”
Control variates for estimation based on reversible Markov chain Monte Carlo samplers (2012, result): “…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).”
Geometric ergodicity of the Bouncy Particle Sampler (2018, background): “It is also possible to consider moderate deviation [21, 12] or large deviation principle [46, 26]”