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Article Dans Une Revue Scientia Sinica Mathematica Année : 2019

Weighted moments of solutions of the stochastic linear recursive distributional equation and applications

Xu Li
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Résumé

Let $(a_k,b_k)$ be a sequence of independent and identically distributed $\R^2$- valued random variables. We consider asymptotic properties of the random series $X= \sum_{k=1}^\infty \pi_{k-1} b_k$, where$\pi_0=1$£¬ $\pi_k= \prod_{i=1}^{k} a_i$. When the series converges almost surely, it is the limit in law of the sequence $(X_n)$ defined by the recursive stochastic linear equations $X_n = a_nX_{n-1} +b_n$, starting with $X_0=x \in \R$, and is the unique solution of the distributional equation $X \stackrel{d}{=} aX+b$ (equality in law), where $(a,b)= (a_1,b_1)$ is independent of $X$. We give a criterion for the existence of weighted moments of the form $ \E (|X|^\alpha \ell (|X|)$, where $\alpha >0$, $\ell$ is a function slowly varying at $\infty$. As an application, we obtain a criterion for the existence of weighted moments of solutions of the fixed point equation $Z=\sum_{i=1}^N A_i Z_i$ of the smoothing transform, where $(N,A_1,A_2,\ldots)$ is a given sequence of random variables with $N\in \mathbb{N}\cup\{\infty\}$ and $A_i\in \mathbb{R}_+$, $Z_i$ are random variables independent of each other and independent of $(N,A_1,A_2,\ldots)$, each has the same distribution as $Z$ which is unknown. Applications are also given to limit variables of general branching processes and branching random walks.
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Dates et versions

hal-02487876 , version 1 (21-02-2020)

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Yanqing Wang, Xu Li, Quansheng Liu. Weighted moments of solutions of the stochastic linear recursive distributional equation and applications. Scientia Sinica Mathematica, 2019, 2019 (11), pp.1687-1706. ⟨10.1360/N012019-00045⟩. ⟨hal-02487876⟩
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