Multifractal products of cylindrical pulses. (English) Zbl 1014.60042

Barral, Julien; Mandelbrot, Benoît B.

Probab. Theory Relat. Fields 124, No. 3, 409-430 (2002).

Consider a Poisson process \(S=\{(s_j,\lambda_j)\}\) on \(\mathbb{R}\times(0,1]\) with intensity \(\Lambda(dt d\lambda)=(\delta\lambda^{-2}/2)dt d\lambda\). The cylindrical pulses associated with \(S\) are a denumerable family of functions \(P_j(t)\), such that each \(P_j(t)=W_j\) for \(t\in[s_j-\lambda_j,s_j+\lambda_j]\) and \(P_j(t)=1\) otherwise, where \(W_j\)’s are i.i.d. with \(W\) and independent of \(S\). The multifractal product of the cylindrical pulses is the measure \(\mu\) that appears as the a.s. vague limit as \(\varepsilon\downarrow 0\) of the family of measures \(\mu_\varepsilon\) on \(\mathbb{R}\) with densities proportional to the product of \(P_j(t)\) for \((s_j,\lambda_j)\in S\) with \(\lambda_j\geq\varepsilon\). The authors formulate conditions for non-degeneracy of \(\mu\), existence of the moments and describe the whole multifractal spectrum of \(\mu\).

Reviewer: Ilya S.Molchanov (Glasgow)

MSC:

60G18	Self-similar stochastic processes
60G44	Martingales with continuous parameter
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57	Random measures
28A80	Fractals

Keywords:

random measures; multifractal analysis; self-similar Poisson point processes; cylindrical pulses; measure-valued martingales