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