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Linear stochastic equations in the critical case. (English) Zbl 1292.60069

The authors study the linear stochastic equation (1) \(X=\sum_{i=1}^N A_iX_i+B\), where \(X_i\) are independent copies of \(X\), which are also independent of a given sequence of non-negative random variables \((N,B,A_1,A_2,\dots)\). Here the sign “=” is regarded as equality in distribution. Let the function \(m\) be defined by \(m(t)=\operatorname{E}(\sum_{i=1}^N A_i^t)\). The main result of the paper says that if there exists \(0<\alpha <1\) such that \(m(\alpha )=1\), \(m'(\alpha )=0\) and some additional assumptions are satisfied, then the equation (1) has the minimal solution \(R\), \(\operatorname{P}[R>t]\leq Ct^{-\alpha }\) and \(\lim_{t\to\infty} t^\alpha \operatorname{P}[R>t]=C_+\). Recently the equation (1) was studied by G. Alsmeyer and M. Meiners [J. Difference Equ. Appl. 18, No. 8, 1287–1304 (2012; Zbl 1275.60019)].

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations

Citations:

Zbl 1275.60019
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References:

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