Some Asymptotic Results for Strongly Critical Branching Processes with Immigration in Varying Environment

Abstract

Strongly critical branching processes with immigration (BPI) are investigated in varying environment (VE) when the time-varying offspring mean converges to 1 fast enough. Under natural assumptions a diffusion approximation is derived when either the offspring or the immigration variances are strictly positive. In the case of asymptotically vanishing offspring variances a fluctuation limit theorem is proved.

Appendix

In the proofs we need the following result about convergence of random step processes towards a diffusion process, see Ispány and Pap [6, Corollary 2.2].

Theorem 5.3

Let $\beta: \mathbb{R}_{+} \times \mathbb{R} \rightarrow \mathbb{R}$ and $\gamma: \mathbb{R}_{+} \times \mathbb{R} \rightarrow \mathbb{R}$ be continuous functions. Assume that uniqueness in the sense of probability law holds for the SDE

$$\displaystyle{ \mathrm{d}\,\boldsymbol{\mathcal{U}}_{t} =\beta (t,\boldsymbol{\mathcal{U}}_{t})\,\mathrm{d}t +\gamma (t,\boldsymbol{\mathcal{U}}_{t})\,\mathrm{d}\mathcal{W}_{t}\;,\qquad t \in \mathbb{R}_{+}\;, }$$

(5.39)

with initial value $\mathcal{U}_{0} = u_{0}$ for all $u_{0} \in \mathbb{R}$, where $(\mathcal{W}_{t})_{t\in \mathbb{R}_{+}}$ is a standard Wiener process. Let $(\mathcal{U}_{t})_{t\in \mathbb{R}_{+}}$ be a solution of (5.39) with initial value $\mathcal{U}_{0} = 0$.

For each $n \in \mathbb{N}$ , let $(U_{k}^{n})_{k\in \mathbb{N}}$ be a sequence of random variables adapted to a filtration $(\mathcal{F}_{k}^{n})_{k\in \mathbb{Z}_{+}}$ . Let $\mathcal{U}_{t}^{n}:=\sum _{ k=1}^{\lfloor nt\rfloor }U_{k}^{n}$ , $t \in \mathbb{R}_{+},n \in \mathbb{N}$ . Suppose $\mathrm{E}\big(\vert U_{k}^{n}\vert ^{2}\big) <\infty$ for all $n,k \in \mathbb{N}$ . Suppose that, for each T > 0,

$$\displaystyle\begin{array}{rcl} & & \sup \limits _{t\in [0,T]}\left \vert \sum _{k=1}^{\lfloor nt\rfloor }\mathrm{E}{\bigl (U_{ k}^{n}\,\vert \,\mathcal{F}_{ k-1}^{n}\bigr )} -\int _{ 0}^{t}\beta (s,\boldsymbol{\mathcal{U}}_{ s}^{n})\mathrm{d}s\right \vert \stackrel{\mathrm{P}}{\longrightarrow }0\;, \\ & & \sup \limits _{t\in [0,T]}\left \vert \sum \limits _{k=1}^{\lfloor nt\rfloor }\mathsf{Var}{\bigl (U_{ k}^{n}\,\vert \,\mathcal{F}_{ k-1}^{n}\bigr )} -\int _{ 0}^{t}\left (\gamma (s,\boldsymbol{\mathcal{U}}_{ s}^{n})\right )^{2}\mathrm{d}s\right \vert \stackrel{\mathrm{P}}{\longrightarrow }0\;, \\ & & \sum \limits _{k=1}^{\lfloor nT\rfloor }\mathrm{E}\big(\vert U_{ k}^{n}\vert ^{2}\mathbb{1}_{\{ \vert U_{k}^{n}\vert>\theta \}}\,\big\vert \,\mathcal{F}_{k-1}^{n}\big)\stackrel{\mathrm{P}}{\longrightarrow }0\quad \text{for all}\;\theta> 0\;.{}\end{array}$$

(5.40)

Then   $\boldsymbol{\mathcal{U}}^{n}\stackrel{\mathcal{L}}{\longrightarrow }\boldsymbol{\mathcal{U}}$  as  n →∞.