##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability Seminar

## Haosui Duanmu

#### UC Berkeley

## Nonstandard analysis and its application to Markov processes

##### Abstract:

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 useful 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 which has all the basic order logical properties of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with each other up to some multiplicative constants for discrete-time general-state-space reversible Markov processes satisfying certain condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and Metropolis-Hasting algorithms.

Host: Todd Kemp

### January 31, 2019

### 10:00 AM

### AP&M 6402

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