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Register a new userModerate deviations of density-dependent Markov chains
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Xiaofeng Xue
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The density-dependent Markov chain (DDMC) introduced in cite{Kurtz1978} is a continuous time Markov process applied in fields such as epidemics, chemical reactions and so on. In this paper, we give moderate deviation principles of paths of DDMC under some generally satisfied assumptions. The proofs for the lower and upper bounds of our main result utilize an exponential martingale and a generalized version of Girsanovs theorem. The exponential martingale is defined according to the generator of DDMC.
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Mingzhou Xu
, Kun Cheng (School of Information Engineering
, n Jingdezhen Ceramic Institute
2020
We prove that moderate deviations for empirical measures for countable nonhomogeneous Markov chains hold under the assumption of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chains in Ces`aro sense.
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Ces`aro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gartner-Ellis theorem and exponential equivalent method.
Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure frac{1}{n}sum_1^ndelta_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean frac{1}{n}sum_1^npsi(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.
The Poisson--Dirichlet distribution arises in many different areas. The parameter $theta$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $theta$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $theta$ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson--Dirichlet distribution for large $theta$, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results.
We recover the Donsker-Varadhan large deviations principle (LDP) for the empirical measure of a continuous time Markov chain on a countable (finite or infinite) state space from the joint LDP for the empirical measure and the empirical flow proved in [2].
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