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

Khusanbaev, Ya. M.

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 \(\varepsilon_{\infty}\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

branching process; weak convergence; Wiener process; limit theorem

Citations:

Zbl 0576.60078