Ergodicity of Markov Processes via Nonstandard Analysis

We introduce a class of hyperfinite Markov processes that behave like finite state space discrete-time Markov processes, allowing us to extend the Markov chain ergodic theorem to general state space continuous-time Markov processes under moderate conditions.

Format: Paperback / softback
Publication date: 30 November 2021
Publisher: American Mathematical Society

The understanding of the Markov chain ergodic theorem extends to discrete time-lines or state spaces. However, when it comes to general state space continuous-time Markov processes, there is a lack of a clear result. To address this, we introduce a class of hyperfinite Markov processes, known as general Markov processes, which exhibit behavior akin to finite state space discrete-time Markov processes. Through the application of methods from mathematical logic and nonstandard analysis, we demonstrate that, under certain moderate conditions, the transition probability of hyperfinite Markov processes aligns with the transition probability of standard Markov processes. Consequently, the Markov chain ergodic theorem for hyperfinite Markov processes implies the Markov chain ergodic theorem for general state space continuous-time Markov processes.

Introduction:
The Markov chain ergodic theorem establishes the convergence of a Markov chain to its stationary distribution, assuming certain conditions are met. In the context of discrete time-lines or state spaces, the theorem is well-understood. However, when it comes to general state space continuous-time Markov processes, the theorem lacks a clear result. This study aims to introduce a class of hyperfinite Markov processes, namely general Markov processes, which exhibit behavior similar to finite state space discrete-time Markov processes. By leveraging methods from mathematical logic and nonstandard analysis, we demonstrate that, under moderate conditions, the transition probability of hyperfinite Markov processes aligns with the transition probability of standard Markov processes. This result provides a pathway to understanding the ergodic behavior of general state space continuous-time Markov processes.

Hyperfinite Markov Processes:
A Markov chain is a sequence of random variables that satisfies certain transition probabilities. In the case of discrete time-lines or state spaces, the state space is finite, and the Markov chain can be represented by a transition matrix. However, in general state space continuous-time Markov processes, the state space is infinite, making it challenging to analyze directly. To overcome this limitation, we introduce the concept of hyperfinite Markov processes. Hyperfinite Markov processes are a generalization of finite state space Markov processes, where the state space is replaced by a countable infinite set. The transition probabilities are still defined using a transition matrix, but the matrix is now infinite-dimensional.

General Markov Processes:
We define general Markov processes as hyperfinite Markov processes with additional conditions. These conditions ensure that the general Markov processes behave like finite state space discrete-time Markov processes. For example, we require that the state space be compact, meaning that it is closed and bounded. We also impose a condition on the transition matrix, known as the ergodic theorem, which ensures that the Markov chain converges to its stationary distribution.

Alignment of Transition Probabilities:
Under moderate conditions, we show that the transition probability of a general Markov process aligns with the transition probability of a standard Markov process. This result provides a connection between hyperfinite Markov processes and standard Markov processes, which can be useful in various applications. For instance, it can be used to analyze the behavior of complex systems, such as financial markets, where continuous-time Markov processes are prevalent.

Implications of the Markov Chain Ergodic Theorem:
The Markov chain ergodic theorem for hyperfinite Markov processes implies the Markov chain ergodic theorem for general state space continuous-time Markov processes. This means that the convergence of a Markov chain to its stationary distribution holds for both hyperfinite and continuous-time Markov processes. This theorem has significant implications in the field of probability theory and stochastic processes, as it provides a framework for analyzing and understanding complex systems.

Conclusion:
In conclusion, we have introduced a class of hyperfinite Markov processes, namely general Markov processes, which behave like finite state space discrete-time Markov processes. We have demonstrated that, under moderate conditions, the transition probability of hyperfinite Markov processes aligns with the transition probability of standard Markov processes. This result provides a pathway.

Dimension: 254 x 178 (mm)
ISBN-13: 9781470450021