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 \[ \lim _{x \to 0} \frac{\log \log 1/\mathbb {P} (W \leq x)} {\log \log 1/x} = \gamma > 1, \] then the Laplace transform of $Y$ satisfies \[ \lim _{t \to \infty } \frac{\log \log 1/\mathbb {E}e^{-t Y}} {\log \log t} = \gamma . \] This implies the same estimate for $\mathbb{P} (Y \leq x)$ for small $x > 0$. As an application of the method, we prove a similar result for a variable arising as a total mass of a $\star $-scale invariant Gaussian multiplicative chaos measure.