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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References
Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36(2), 544–581 (2004)
Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Springer, New York (1998)
Grama, I., Liu, Q., Miqueu, E.: Harmonic moments and large deviations for a supercritical branching process in a random environment. Electron. J. Probab. 22(99), 1–23 (2017)
Gao, Z., Liu, Q.: Exact convergence rates in central limit theorems for a branching random walk with a random environment in time. Stoch. Proc. Appl. 126(9), 2634–2664 (2016)
Gao, Z., Liu, Q.: Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time. Bernoulli. 24(1), 772–800 (2018)
Gao, Z., Liu, Q., Wang, H.: Central limit theorems for a branching random walk with a random environment in time. Acta. Math. Sci. 34B(2), 501–512 (2014)
Huang,C., Liu,Q.: Branching random walk with a random environment in time. Available at arXiv: 1407.7623
Huang, C., Liu, Q.: Moments, moderate and large deviations for a branching process in a random environment. Stoch. Proc. Appl. 122, 522–545 (2012)
Huang, C., Liang, X., Liu, Q.: Branching random walks with random environments in time. Front. Math. China. 9(4), 835–842 (2014)
Huang, C., Wang, X., Wang, X.: Large and moderate deviations for a \({\mathbb{R} }^d\)-valued branching random walk with a random environment in time. Stochastics. 92(6), 944–968 (2020)
Huang, C., Wang, C., Wang, X.: Moments and large deviations for supercritical branching processes with immigration in random environments. Acta. Math. Sci. 42B(1), 49–72 (2022)
Li, D., Zhang, M.: Harmonic moments and large deviations for a critical Galton-Watson process with immigration. Sci. China. Math. 64, 1885–1904 (2021)
Li, Y., Liu, Q., Peng, X.: Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment. Stat. Probabil. Lett. 147, 57–65 (2019)
Mallein, B., Miloś, P.: Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. Stoch. Proc. Appl. 129(9), 3239–3260 (2019)
Nakashima, M.: Branching random walks in random environment and super-Brownian motion in random environment. Ann. Inst. Henri. Poincaré. Probab. Stat. 51(4), 1251–1289 (2015)
Ney, P.E., Vidyashankar, A.N.: Harmonic moments and large deviation rates for supercritical branching process. Ann. Appl. Probab. 13, 475–489 (2003)
Sun, Q., Zhang, M.: Harmonic moments and large deviations for supercritical branching processes with immigration. Front. Math. China. 12, 1201–1220 (2017)
Wang, X., Huang, C.: Convergence of martingale and moderate deviations for a branching random walk with a random environment in time. J. Theor. Probab. 30, 961–995 (2017)
Wang, X., Huang, C.: Convergence of complex martingale for a branching random walk in a time random environment. Electron. Commun. Probab. 24(41), 1–14 (2019)
Wang, X., Liang, X., Huang, C.: Convergence of complex martingale for a branching random walk in an independent and identically distributed environment. Front. Math. China. 16(1), 187–209 (2021)
Wang,X., Li,M., Huang,C.: Limit theorems for a branching random walk with immigration in a random environment. Preprint
Wang, Y., Liu, Q.: Limit theorems for a supercritical branching process with immigration in a random environment. Sci. China. Math. 60(12), 2481–2502 (2017)
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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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
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
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,
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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