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2020 Small deviations in lognormal Mandelbrot cascades
Miika Nikula
Electron. Commun. Probab. 25: 1-12 (2020). DOI: 10.1214/17-ECP85

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.

Citation

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Miika Nikula. "Small deviations in lognormal Mandelbrot cascades." Electron. Commun. Probab. 25 1 - 12, 2020. https://doi.org/10.1214/17-ECP85

Information

Received: 22 April 2016; Accepted: 7 September 2017; Published: 2020
First available in Project Euclid: 28 January 2020

zbMATH: 1439.60049
MathSciNet: MR4066300
Digital Object Identifier: 10.1214/17-ECP85

Subjects:
Primary: 60G57

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