Multifractal products of cylindrical pulses
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 \(\mu\).
scientific article; zbMATH DE number 1883925