Multifractality of Iterated Pulse Processes With Pulse Amplitudes Generated by a Random Cascade

A previous paper1 has introduced a class of random functions in Rn called iterated random pulse (IRP) processes. IRP processes are sums of pulses whose locations form an iterated clustered point process and whose deterministic shapes, sizes and amplitudes satisfy affine scaling relations. In the simplest case, at each finer scale the pulse support is isotropically contracted by a factor r>1 and the amplitude is multiplied by r-γ, where γ>-n is a given constant and n is the space dimension. We consider extensions in which the pulse amplitudes are random variables with a multiplicative cascade structure. This means that, if a parent pulse at a certain level has amplitude A, its offspring pulses at the next level have amplitudes Ai=Ar-γ ηi, where the ηi are positive iid variables with mean value 1. Interest is in the generalized field X(h)= lim j→∞Xj(h) where h(t) is a test function, Xj(h)=∫ h(t)Xj(t)dt, and Xj(t) is the pulse field at resolution level j. We show that X(h) is multifractal at small scales. We also derive its scaling properties and the fractal dimension of the support. In the case of non=isotropic contraction of the pulse support from level j to level j+1, X(h) has a more general form of multifractality known as generalized scale invariance (gsi). This seems to be the first known construction of gsi random fields.