Deviation results for Mandelbrot's multiplicative cascades with (stretched) exponential tails
Résumé
Let $W$ be a nonnegative random variable with expectation $1$. For all $r \geqslant 2$, we consider the total mass $Z_r^\infty$ of the associated Mandelbrot multiplicative cascade in the $r$-ary tree. For all $n \geqslant 1$, we also consider the total mass $Z_r^n$ of the measure at height $n$ in the $r$-ary tree. Liu, Rio, Rouault \cite{lrr,liu2000limit,Rouault04} established large deviation results for $(Z_r^n)_{r \geqslant 2}$ for all $n \in \intervallentfo{1}{\infty}$ (resp.\ for $n = \infty$) in the case $W$ has an everywhere finite cumulant generating function $\Lambda_W$ (resp.\ $W$ is bounded). Here, we extend these results to the case that $\Lambda_W$ is only assumed finite on a neighborhood of zero, and even to the case that $W$ has a stretched exponential tail. In addition, we study deviations of all orders. It is noticeable that we obtain recursive definitions of rate functions and that we resort to the moments bound instead of the standard Chernoff bound to establish the upper bounds of deviation in the infinite tree.
Mots clés
- Mandelbrot multiplicative cascades
- large deviation principles
- stretched exponential tails
- moment bound
- Mandelbrot multiplicative cascades, large deviation principles, moments bound, stretched exponential tails AMS MSC 2010: 60F10 60G57 60J80
- Mandelbrot multiplicative cascades, large deviation principles, moments bound, stretched exponential tails AMS MSC 2010: 60F10
- 60G57
- 60J80
Domaines
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