Small deviations in lognormal Mandelbrot cascades

Abstract

We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if $\log \log 1/P(W\le x)\sim \gamma \log \log 1/x$ as $x\to 0$ with $\gamma >1$, then the Laplace transform of $Y$ satisfies $\log \log 1/\E e^{-t Y} \sim γ\log \log t$ as $t\to \infty$. As an application, this gives new estimates for $\Prob(Y \leq x)$ for small $x>0$. As another application of our methods, we prove a similar result for a variable arising as a total mass of a lognormal $\star$-scale invariant multiplicative chaos measure.

arXiv comment13 pages