On a growth process. (English. Ukrainian original) Zbl 0960.60068

Khusanboev, Ya. M.; Rakhimov, G. M.

Theory Probab. Math. Stat. 59, 173-176 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 167-170 (1998).

The asymptotic behavior as \(x\to+\infty\) of a solution to the stochastic difference equation \(X_{n+1}=X_{n}+g(X_{n})+\xi_{n+1}\), is investigated for the case of \(g(x)=ax^{p},\) \(p<1/3,\) where \(\xi_{n}\) is a sequence of square integrated martingale-differences with zero mean and \(E(\xi_{k+1}^{2}/\mathcal {F}_{k})=\sigma^{2}(X_{k})\) for some function \(\sigma(x)>0.\) The rate of growth for the process \(X_{n}\) was investigated and the conditions under which \(X_{n}/a_{n}\to 1\) as \(n\to+\infty\) were proposed by G. Keller, G. Kersting and U. Rösler [Ann. Probab. 15, 305-343 (1987; Zbl 0616.60079)], where the deterministic process \(a_{n}\) is defined by the difference equation \(a_{n+1}=a_{n}+g(a_{n}).\) F. C. Klebaner [ibid. 17, No. 1, 178-188 (1989; Zbl 0674.60077)] investigated the case \(g(x)=ax^{\alpha},\) \(\sigma(x)= bx^{1+\alpha},\) where \(\alpha<1,\) and obtained conditions under which \(X_{n}^{1-\alpha}/n\) converges weakly to \(\Gamma\)-distribution.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G42	Martingales with discrete parameter
60F15	Strong limit theorems

Keywords:

process of growth; stochastic difference equation; quadratic integrated martingale-difference

Citations:

Zbl 0616.60079; Zbl 0674.60077