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Multifractal products of cylindrical pulses. (English) Zbl 1014.60042
Consider a Poisson process S={(s j ,λ j )} on ×(0,1] with intensity Λ(dtdλ)=(δλ -2 /2)dtdλ. 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[s j -λ j ,s j +λ 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 μ that appears as the a.s. vague limit as ε0 of the family of measures μ ε on with densities proportional to the product of P j (t) for (s j ,λ j )S with λ j ε. The authors formulate conditions for non-degeneracy of μ, existence of the moments and describe the whole multifractal spectrum of μ.
MSC:
60G18Self-similar processes
60G44Martingales with continuous parameter
60G55Point processes
60G57Random measures
28A80Fractals