Some Conditional Limiting Theorems For Markov Processes

Limit theorems constitute a core part in the theory of Markov processes. In manysituations one may find Markov models with certain loss of probability. They are notergodic in the standard sense and yet with clear (and properly trivial) limit trends, e.g.,they will eventually be absorbed by certain states. For such processes, conditional limittheorems are usually more interesting than the trivial one. In this thesis, two types of con-ditional limit distributions—-quasi-stationary distribution and quasi-ergodic distributionare investigated from various viewpoints, reflecting diferent conditional stability. Themain contents of the thesis are divided into the following three parts:First, under certain hypothesis we prove a quasi-mixing theorem for Markov pro-cesses on general state spaces. From this theorem, not only the existence of quasi-stationary distributions and quasi-ergodic distributions follow, but also a certain condi-tional independence of the processes is showed, a phase transition concerning quasi-stationary distribution and quasi-ergodic distribution is exhibited. We then show thatany fractional Yaglom limit is quasi-ergodic, which corresponds to conditional Law ofLarge Numbers. We also demonstrate that the quasi-ergodic distribution is the standardergodic distribution of the conditional limiting process obtained by certain “exponentialh-transformation” of the original process;Next, we focus on continuous-time Markov chains. We characterize the quasi-ergodic distribution and the decay parameter in terms of Donsker-Varadhan’s I-function.We show that the decay parameter is the infimum of the I-function, and this infimum istypically attained at the quasi-ergodic distribution. A necessary and sufcient conditionfor the infimum to be attained is provided. As an application, we derive a special newdual form of minmax formula for the decay parameter;Finally, we study birth-death processes, for which we first show that the uniquenessof quasi-stationary distribution guarantees that the quasi-ergodic distribution is the onlyquasi-average limit, which attracts all initial distributions. Then we study the problem offinite dimensional approximation for decay parameter, quasi-stationary distribution andquasi-ergodic distribution. We show that under certain conditions, the rate of approx-imation for the decay parameter is exponentially fast. We also show that assumption (H1) guarantees the convergence of the finite approximation for both the quasi-stationarydistribution and the quasi-ergodic distribution.