Small deviations in lognormal Mandelbrot cascades

TL;DR

This paper investigates small deviation probabilities in Mandelbrot cascades and related models, establishing asymptotic relations for the Laplace transform and deriving new estimates for small mass probabilities.

Contribution

It provides the first detailed asymptotic analysis of small deviations in Mandelbrot cascades and extends results to lognormal scale-invariant multiplicative chaos measures.

Findings

Asymptotic relation for the Laplace transform of total mass variable

New estimates for small probability of total mass being close to zero

Extension of results to lognormal $ imes$-scale invariant measures

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 $\log \log 1/P(W \leq x) \sim \gamma \log \log 1/x$ as $x \to 0$ with $\gamma > 1$, then the Laplace transform of $Y$ satisfies $\log \log 1/\E e^{-t Y} \sim \gamma \log \log t$ 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.