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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Abstract

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 $n\to\infty$.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820848

Digital Object Identifier
doi:10.1214/18-AIHP915

Mathematical Reviews number (MathSciNet)
MR3949970

Zentralblatt MATH identifier
07097348

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

Keywords
Mandelbrot cascades Branching random walk Derivative martingale Conditioned random walk

Citation

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. https://projecteuclid.org/euclid.aihp/1557820848


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References

  • [1] E. Aidékon. Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 (3A) (2013) 1362–1426.
  • [2] E. Aidékon and Z. Shi. The Seneta–Heyde scaling for the branching random walk. Ann. Probab. 42 (3) (2014) 959–993.
  • [3] J. Barral, X. Jin, R. Rhodes and V. Vargas. Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys. 323 (2) (2013) 451–485.
  • [4] J. Barral, A. Kupiainen, M. Nikula, E. Saksman and C. Webb. Critical Mandelbrot cascades. Comm. Math. Phys. 325 (2) (2014) 685–711.
  • [5] J. Barral and B. Mandelbrot. Introduction to infinite products of independent random functions (Random multiplicative multifractal measures. I). In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 3–16. Proc. Sympos. Pure Math. 72. Amer. Math. Soc., Providence, RI, 2004.
  • [6] J. Barral and B. Mandelbrot. Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures. II). In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 17–52. Proc. Sympos. Pure Math. 72. Amer. Math. Soc., Providence, RI, 2004.
  • [7] J. Barral, R. Rhodes and V. Vargas. Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 (9–10) (2012) 535–538.
  • [8] N. Berestycki. Diffusion in planar Liouville quantum gravity. Ann. Inst. Henri Poincaré Probab. Stat. 51 (3) (2015) 947–964.
  • [9] N. Berestycki. An elementary approach to Gaussian multiplicative chaos. 2015. Arxiv preprint, Available at arXiv:1506.09113.
  • [10] J. Bertoin and R. Doney. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (4) (1994) 2152–2167.
  • [11] J. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1) (1977) 25–37.
  • [12] J. Biggins and A. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. 36 (2) (2004) 544–581.
  • [13] J. D. Biggins. Random walk conditioned to stay positive. J. Lond. Math. Soc. (2) 67 (01) (2003) 259–272.
  • [14] L. Borland, J. P. Bouchaud, J. F. Muzy and G. Zumbach. The dynamics of financial markets–Mandelbrot’s multifractal cascades, and beyond. Wilmott Magazine, 2005.
  • [15] L. Breiman. Probability. Addison-Wesley Publishing Company, Reading, Mass.–London–Don Mills, Ont, 1968.
  • [16] D. Buraczewski, E. Damek, Y. Guivarc’h and S. Mentemeier. On multidimensional Mandelbrot cascades. J. Difference Equ. Appl. 20 (11) (2014) 1523–1567.
  • [17] D. Buraczewski and K. Kolesko. Linear stochastic equations in the critical case. J. Difference Equ. Appl. 20 (2) (2014) 188–209.
  • [18] F. Caravenna. A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 (4) (2005) 508–530.
  • [19] B. Derrida and H. Spohn. Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 (5–6) (1988) 817–840.
  • [20] R. Durrett and T. Liggett. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (3) (1983) 275–301.
  • [21] W. Feller. An Introduction to Probability Theory and Its Applications, 2. John Wiley & Sons, New York, 2008.
  • [22] C. Garban, R. Rhodes and V. Vargas. Liouville Brownian motion. Ann. Probab. 44 (4) (2016) 3076–3110.
  • [23] B. Hambly, G. Kersting and A. Kyprianou. Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative. Stochastic Process. Appl. 108 (2) (2003) 327–343.
  • [24] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 (2) (1976) 131–145.
  • [25] J. P. Kahane. Sur le chaos multiplicatif. Prepublications mathematiques d’Orsay. Departement de mathematique, 1985.
  • [26] Q. Liu. On generalized multiplicative cascades. In Stochastic Processes and Their Applications 263–286, 86, 2000.
  • [27] T. Madaule. Convergence in law for the branching random walk seen from its tip. J. Theoret. Probab. 30 (2015) 27–63.
  • [28] B. Mandelbrot. Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence 333–351. M. Rosenblatt and C. Van Atta (Eds) Lecture Notes in Physics 12. Springer, Berlin, Heidelberg, 1972.
  • [29] B. Mandelbrot. Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. In Multifractals and 1/f Noise 317–357. Springer, New York, 1999.
  • [30] J. Neveu. Arbres et processus de Galton–Watson. In Annales de l’IHP Probabilités et statistiques 199–207, 22, 1986.
  • [31] R. Rhodes and V. Vargas. Liouville Brownian motion at criticality. Potential Anal. 43 (2) (2015) 149–197.