Precise tail asymptotics of fixed points of the smoothing transform with general weights
| Autor: | Ewa Damek, Jacek Zienkiewicz, Dariusz Buraczewski |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Condensed Matter::Quantum Gases
Statistics and Probability Independent and identically distributed random variables smoothing transform High Energy Physics::Lattice Mathematics - Statistics Theory Statistics Theory (math.ST) Fixed point large deviations Combinatorics FOS: Mathematics regular variation High Energy Physics::Experiment Large deviations theory linear stochastic equation Limit (mathematics) Constant (mathematics) Random variable Smoothing Mathematics |
| Zdroj: | Bernoulli 21, no. 1 (2015), 489-504 |
| Popis: | We consider solutions of the stochastic equation $R=_d\sum_{i=1}^NA_iR_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent both from $A_i$'s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}|A_1|^{\alpha}=1/N$ and $\mathbb{E}|A_1|^{\alpha}\log|A_1|>0$, the limit $\lim_{t\to\infty}t^{\alpha}\mathbb{P}[|R|>t]=K$ exists. In the present paper, we prove positivity of $K$. Comment: Published at http://dx.doi.org/10.3150/13-BEJ576 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm) |
| Databáze: | OpenAIRE |
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