Convergence of non-homogeneous, asymptotically critical (in the scheme of series) branching processes to processes of the diffusion type. (Russian) Zbl 0628.60088

Yurachkovskij, A. P.

Teor. Veroyatn. Mat. Stat., Kiev 36, 127-135 (1987).

Deeply and widely studying non-homogeneous, asymptotically critical branching processes is given by this paper. With the help of martingale techniques and a theorem of R. Sh. Liptser and A. N. Shiryaev [Mat. Sb., Nov. Ser. 121(163), No.2(6), 176-200 (1983; Zbl 0521.60034)], the author shows the Lindenberg condition: \[ n^{-2}\sum^{n- 2}_{k=0}\sum^{[nx]}_{i=1}E \xi_{nk}^{(i)^ 2} I_{\{| \xi_{nk}^{(i)}| >\epsilon n\}}\to 0\quad for\quad \forall x>0,\quad \epsilon >0, \] for weak convergence \(\eta_ n(.)\to^{w}\eta (.)\), where \(\eta_ n(t)=\eta_{n,[nt]}\), \(t\in [0,1)\), \(\eta_ n(1)=\eta_ n(1_{-0})\), \(\eta_{nk}=\xi_{nk}/n\) and \(\eta\) (.) is the unique weak solution of the equation \[ d\eta (t)=\eta (t)a(t,\eta (t))dt+\sqrt{\eta (t)}b(t,\eta (t))dw(t),\quad \eta (0)=\eta_ 0 \] with some non-random functions b(t,x) (\(\geq 0)\) and a(t,x) and Wiener process W(.).

Reviewer: Ge Yubo

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65	Brownian motion

Keywords:

critical branching processes; martingale techniques; unique weak solution

Citations:

Zbl 0521.60034