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Zbl 1014.60042
Barral, Julien; Mandelbrot, Benoît B.
Multifractal products of cylindrical pulses. (English)
[J] Probab. Theory Relat. Fields 124, No.3, 409-430 (2002). ISSN 0178-8051; ISSN 1432-2064/e

Consider a Poisson process $S=\{(s_j,\lambda_j)\}$ on $\Bbb{R}\times(0,1]$ 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 $\Bbb{R}$ 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 $\mu$.
[Ilya S.Molchanov (Glasgow)]

MSC 2000:: *60G18 Self-similar processes
60G44 Martingales with continuous parameter
60G55 Point processes
60G57 Random measures
28A80 Fractals

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

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