Tuesday, September 18th @ 11:00-12:30 PM
Nonstandard Analysis and its Application to Markov Processes
Haosui (Kevin) Duanmu, UC Berkeley
Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in probability theory as well as stochastic processes. Nonstandard analysis allows construction of a single object - a hyperfinite probability space - which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings. In this talk, for every general-state-space continuous-time Markov process satisfying appropriate conditions, we construct a hyperfinite Markov process to represent it. Hyperfinite Markov processes have all the first order logical properties of a finite Markov process. We establish ergodicity of a large class of general-state-space continuous-time Markov processes via studying their hyperfinite counterpart. We also establish the asymptotical equivalence between mixing times, hitting times and average mixing times for discrete-time general-state-space Markov processes satisfying moderate condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and a large class of Metropolis-Hasting algorithms.