Multifractal products of cylindrical pulses. (English) Zbl 1014.60042

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 \(\mu\).


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