Some limit theorems for controlled branching processes.(Russian, English)Zbl 1224.60219

Teor. Jmovirn. Mat. Stat. 81, 46-52 (2009); translation in Theory Probab. Math. Stat. 81, 51-58 (2010).
Let $$\{\xi_{k,j}(l),\;k, j, l\in\mathbb N\}$$ be a family of nonnegative integer-valued independent random variables, and let the random variables $$\xi_{k,j}(l),\;k,j\in\mathbb N$$, be identically distributed. Define the stochastic process $$X_k,\;k\geq0$$, by the following recurrence relation $$X_0 = 1$$, $$X_k =\sum_{j=1}^{X_{k-1}}\xi_{k,j}(X_{k-1}),\;k\geq1$$. This process is called a branching process dependent on the size of the population or a controlled branching process. The author proposes some sufficient conditions for the convergence as $$n\to\infty$$ of the processes $$\{X_{[nt]},\;t\geq 0\}$$, $$n\geq 1$$, to a deterministic process. The following theorem is proved.
Theorem 1. Let $$m(x) = E\xi_{1,1}(x)=1+\alpha/x$$ for $$x > 0$$ and some $$\alpha>0$$ and let $$x\sigma^2(x)\leq Cx^{\beta}$$ for some $$0\leq\beta<1$$, where $$\sigma^2(x) =\operatorname{Var}\xi_{1,1}(x)$$. Then $\sup_{0\leq t\leq T}\left| \frac{X_{[nt]}}{n}-\alpha t\right| \overset {P}\longrightarrow 0$ as $$n\to\infty$$ on the event $$\varepsilon_{\infty}=\{X_n\to\infty\}$$ (for definitions and more properties, see [P. Küster, Ann. Probab. 13, 1157–1178 (1985; Zbl 0576.60078)]).
Under some additional conditions the author proves that $\frac{X_{[nt]}-n\alpha t}{\sqrt{n}}\overset {D}\longrightarrow W(t),\;t\in[0,T],$ as $$n\to\infty$$, where $$W(t)$$ is a standard Wiener process, in the space $$D[0,T]$$ on the event $$\varepsilon_{\infty}$$.

MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F17 Functional limit theorems; invariance principles 60J65 Brownian motion

Zbl 0576.60078
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