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On a growth process. (English. Ukrainian original) Zbl 0960.60068
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.
##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G42 Martingales with discrete parameter 60F15 Strong limit theorems