Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Local fluctuations of critical Mandelbrot cascades

Dariusz Buraczewski, Piotr Dyszewski, and Konrad Kolesko

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We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_{n}(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu(B_{n}(x))$ as $n\to\infty$.


Nous étudions les cascades de Mandelbrot généralisées à la température (critique) de freezing. Il est connu qu’après une mise à l’échelle appropriée, une telle suite de cascades multiplicatives converge faiblement vers une certaine mesure aléatoire continue. La question est alors de savoir à quel point la mesure limite $\mu$ fluctue. Pour tout point $x$ donné, et en notant $B_{n}(x)$ la boule de rayon $2^{-n}$ centrée en $x$, nous présentons des bornes supérieures et inférieures optimales pour $\mu(B_{n}(x))$ lorsque $n\to\infty$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1179-1202.

Received: 25 November 2016
Revised: 15 May 2018
Accepted: 21 May 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G57: Random measures

Mandelbrot cascades Branching random walk Derivative martingale Conditioned random walk


Buraczewski, Dariusz; Dyszewski, Piotr; Kolesko, Konrad. Local fluctuations of critical Mandelbrot cascades. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1179--1202. doi:10.1214/18-AIHP915.

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