# On a New Characterization of Harris Recurrence for Markov Chains and Processes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Definition**

**1.**

- (i)
- ${\tilde{P}}_{x}(({\tilde{X}}_{0},\cdots ,{\tilde{X}}_{n})\in \xb7)={P}_{x}(({X}_{0},\cdots ,{X}_{n})\in \xb7);$
- (ii)
- $({U}_{i}:i\ge 0)$ is a sequence of independent and identically distributed (iid) random variables (rv’s) under ${\tilde{P}}_{x}.$

**Definition**

**2.**

- (i)
- ${R}_{n}$ is independent of $({\tilde{X}}_{{R}_{n}+k}:k\ge 0)$ under ${\tilde{P}}_{x}$;
- (ii)
- ${\tilde{P}}_{x}(({\tilde{X}}_{{R}_{n}+k}:k\ge 0)\in \xb7)={\int}_{S}\lambda (dy){P}_{y}(({X}_{k}:k\ge 0)\in \xb7).$

**Theorem**

**1.**

**Definition**

**3.**

**Theorem**

**2.**

**Definition**

**4.**

**Theorem**

**3.**

**Proof.**

**Definition**

**5.**

- (i)
- ${\tilde{P}}_{x}((\tilde{X}(t):t\ge 0)\in \xb7)={P}_{x}((X(t):t\ge 0)\in \xb7);$
- (ii)
- $({U}_{i}:i\ge 0)$ is a sequence of iid rv’s on $[0,1]$ under ${\tilde{P}}_{x}.$

**Definition**

**6.**

**Theorem**

**4.**

**Definition**

**7.**

**Theorem**

**5.**

**Proof.**

## 3. A Further Generalization

**Definition**

**8.**

**Theorem**

**6.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Glynn, P.; Qu, Y.
On a New Characterization of Harris Recurrence for Markov Chains and Processes. *Mathematics* **2023**, *11*, 2165.
https://doi.org/10.3390/math11092165

**AMA Style**

Glynn P, Qu Y.
On a New Characterization of Harris Recurrence for Markov Chains and Processes. *Mathematics*. 2023; 11(9):2165.
https://doi.org/10.3390/math11092165

**Chicago/Turabian Style**

Glynn, Peter, and Yanlin Qu.
2023. "On a New Characterization of Harris Recurrence for Markov Chains and Processes" *Mathematics* 11, no. 9: 2165.
https://doi.org/10.3390/math11092165