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**Stochastic difference equations and generalized gamma distributions.**
*(English)*
Zbl 0674.60077

Author’s summary: “We study the asymptotic growth rates of discrete-time stochastic processes \((X_ n)\), where the first two conditional moments of the process depend only on the present state. Such processes satisfy a stochastic difference equation \(X_{n+1}=X_ n+g(X_ n)+R_{n+1}\), where g is a positive function and \((R_ n)\) is a martingale difference sequence. It is known that a large class of such processes diverges with positive probability, and when properly normalized converges almost surely or converges in distribution to a normal or a log normal distribution.

Here we find a class of processes that when properly normalized converges in distribution to a generalized gamma distribution. Applications of this result to state dependent random walks and population size-dependent branching processes yield new results and reprove some of the known results.”

The model is where g(x) is of order \(x^{\alpha}\) and the variance of \(R_{n+1}\) given \(X_ n=x\) is of order \(x^{1+\alpha}\) for some \(\alpha <1\). The proof uses the method of moments, for which higher order moment assumptions are required. The paper may be regarded as one of a sequence by a number of authors, notably G. Keller, G. Kersting and U. Rösler [ibid. 15, 305-343 (1987; Zbl 0616.60079)].

Here we find a class of processes that when properly normalized converges in distribution to a generalized gamma distribution. Applications of this result to state dependent random walks and population size-dependent branching processes yield new results and reprove some of the known results.”

The model is where g(x) is of order \(x^{\alpha}\) and the variance of \(R_{n+1}\) given \(X_ n=x\) is of order \(x^{1+\alpha}\) for some \(\alpha <1\). The proof uses the method of moments, for which higher order moment assumptions are required. The paper may be regarded as one of a sequence by a number of authors, notably G. Keller, G. Kersting and U. Rösler [ibid. 15, 305-343 (1987; Zbl 0616.60079)].

Reviewer: D.R.Grey

### MSC:

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

60G50 | Sums of independent random variables; random walks |