# The Asymptotic Behavior for Generalized Jiřina Process

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

## 2. Main Results

- (A1)
- $r:=\underset{x\to {0}^{+}}{lim}m\left(x\right),$ where $0<r<1$.
- (A2)
- There exits a function $\overline{m}\left(x\right)\ge m\left(x\right)$ for all $x\ge 0$ which satisfies that $\underset{x\ge 0}{inf}\overline{m}\left(x\right)\ge r$ and$$p\left(x\right):=|\overline{m}(1/x)-r|(=\overline{m}(1/x)-r)$$$${\int}_{1}^{+\infty}\frac{p\left(x\right)}{x}\mathrm{d}x<+\infty .$$
- (A3)
- For any $x\ge 0$, it satisfies $rx{\displaystyle {\int}_{0}^{+\infty}}{\mathrm{e}}^{-xF(x,s)}\mathrm{d}s\le 1$.
- (A4)
- $xp\left(\sqrt{x}\right)$ is non-decreasing, concave and $x{p}^{2}\left(\sqrt{x}\right)$ is concave.
- (A5)
- For any $x\ge 0$, it satisfies ${r}^{2}{x}^{2}{\displaystyle {\int}_{0}^{+\infty}}s{\mathrm{e}}^{-xF(x,s)}\mathrm{d}s\le 1$.

**Lemma**

**1.**

- (1)
- ${\int}_{1}^{+\infty}}\frac{h\left(x\right)}{x}\mathrm{d}x<+\infty ;$
- (2)
- $\sum _{n=1}^{\infty}}h\left(\u03f5{t}^{n}\right)<+\infty .$

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Lv, Y.; Huang, H.
The Asymptotic Behavior for Generalized Jiřina Process. *Axioms* **2023**, *12*, 13.
https://doi.org/10.3390/axioms12010013

**AMA Style**

Lv Y, Huang H.
The Asymptotic Behavior for Generalized Jiřina Process. *Axioms*. 2023; 12(1):13.
https://doi.org/10.3390/axioms12010013

**Chicago/Turabian Style**

Lv, You, and Huaping Huang.
2023. "The Asymptotic Behavior for Generalized Jiřina Process" *Axioms* 12, no. 1: 13.
https://doi.org/10.3390/axioms12010013