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Asymptotic Properties for Branching Random Walks with Immigration in Random Environments

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Abstract

We focus on an \({\mathbb {R}}^d\)-valued discrete time branching random walk with immigration in a random environment, where the environment \(\xi =(\xi _n)\) is a stationary and ergodic sequence indexed by time \(n\in {\mathbb {N}}\). For the process, the partition function is the moment generating function of the counting measure describing the dispersion of individuals at time n. Let \(Z_n(t)\) be the partition function of the process with immigration, and \({\bar{Z}}_n(t)\) be that of the process without immigration. For \(t\in {\mathbb {R}}^d\) fixed, we are interested in the relationships and differences between the moments of \(Z_n(t)\) and those of \({\mathbb {E}}[{\bar{Z}}_n(t)|\xi ]\). We show asymptotic properties of annealed moments and quenched moments of all orders of \(Z_n(t)\) and discover the differences from the corresponding moments of \({\mathbb {E}}[{\bar{Z}}_n(t)|\xi ]\). Then, with the help of the moments of \(Z_n(t)\), we establish large and moderate deviation results associated with the free energy \(\frac{1}{n}\log Z_n(t)\).

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Acknowledgements

The authors would like to express their heartfelt thanks to referees for their valuable suggestions and comments. This work was supported by Shandong Provincial Natural Science Foundation (No. ZR2021MA085).

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, China

    Chunmao Huang

  2. School of General Education, Wenzhou Business College, Wenzhou, 325035, China

    Xin Wang

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  1. Chunmao Huang
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  2. Xin Wang
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Correspondence to Xin Wang.

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Communicated by See Keong Lee.

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Appendix

Appendix

Lemma 1.24

Fix \(t\in {\mathbb {R}}^d\) and denote \(f(x)=\mathbb Em_0(xt)^{p/x}\) \((x>0)\). Let \(b>a>0\). Assume that f(x) is well-defined on [a, b]. Then, we have

$$\begin{aligned} \sup _{a\le x\le b}f(x)=\max \{f(a),f(b)\}. \end{aligned}$$

Proof

By the continuity of f(x), there exists a \(x_0\in [a,b]\) such that \(f(x_0)=\sup _{a\le x\le b}f(x)\). We shall prove that \(f(x_0)=\max \{f(a),f(b)\}\). It is clear that \(f(x_0)\ge \max \{f(a),f(b)\}\). Now we suppose that \(f(x_0)>\max \{f(a),f(b)\}\), which means that \(x_0\in (a,b)\). Then, there exists \(\theta \in (0,1)\) such that \(x_0=\theta a+(1-\theta )b\). By Hölder’s inequality, we have

$$\begin{aligned} f(a)< & {} f(\theta a+(1-\theta )b)\\\le & {} \mathbb Em_0(at)^{\frac{p\theta }{x_0}}m_0(bt)^{\frac{p(1-\theta )}{x_0}}\\\le & {} \left[ \mathbb Em_0(at)^{\frac{p}{a}}\right] ^{\frac{a\theta }{x_0}}\left[ \mathbb Em_0(bt)^{\frac{p}{b}}\right] ^{\frac{b(1-\theta )}{x_0}}\\= & {} f(a)^{\frac{a\theta }{x_0}}f(b)^{\frac{b(1-\theta )}{x_0}}, \end{aligned}$$

which means that \(f(a)<f(b)\). Similarly, we can also deduce that \(f(b)<f(a)\), which leads to the contradiction. \(\square \)

Lemma 1.25

[ [20], Lemma A2] Let f(s) be a convex function defined on \([a,b]\subset (0,\infty )\). Then,

$$\begin{aligned} \sup _{a\le s\le b}\left\{ \frac{1}{s}f(s)\right\} =\max \left\{ \frac{1}{a}f(a),\frac{1}{b}f(b)\right\} . \end{aligned}$$

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Huang, C., Wang, X. Asymptotic Properties for Branching Random Walks with Immigration in Random Environments. Bull. Malays. Math. Sci. Soc. 46, 179 (2023). https://doi.org/10.1007/s40840-023-01573-4

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  • DOI: https://doi.org/10.1007/s40840-023-01573-4

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