Infinitely Divisible Positive Definite Sequences

Conditionally Positive Definite Kernels and Infinitely Divisible Distributions

We give a precise characterization of two important classes of conditionally positive definite (CPD) kernels in terms of integral transforms of infinitely divisible distributions. In particular, we show that for any stationary CPD kernel A(x, y) = f(x − y), f is the log-characteristic function of a uniquely determined infinitely divisible distribution; further, for any additive CPD kernel A(x, ...

Infinitely Divisible Matrices

Infinitely divisible cascades

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic, to name but two. It was recently shown and explained how and why multifractal analysis could be interestingly and fruitfully placed in the more general framework of infinitely divisible cascades. The aim of this contribution is to design processes, c...

Log-infinitely divisible multifractal processes

We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson “product of cynlindrical pulses” [7]. Their construction involves some “continuous stochastic m...

Scale Invariant Infinitely Divisible Cascades

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic. It was recently shown and explained how and why multifractal analysis could be fruitfully placed in the general framework of infinitely divisible cascades. The aim of this contribution is to design processes, called Infinitely Divisible Cascading (ID...