A scalar valued random field is called operator-scaling if for some dxd matrix E with positive real parts of the eigenvalues and some H0 we have where denotes equality of all finite-dimensional marginal distributions. We present a moving average and a harmonizable representation of stable...

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays,...

Ultraslow diffusion is a physical model in which a plume of diffusing particles spreads at a logarithmic rate. Governing partial differential equations for ultraslow diffusion involve fractional time derivatives whose order is distributed over the interval from zero to one. This paper develops...

A stochastic process on a finite-dimensional real vector space is operator-self-similar if a linear time change produces a new process whose distributions scale back to those of the original process, where we allow scaling by a family of affine linear operators. We prove a spectral decomposition...

For data belonging to the domain of normal attraction of nonnormal operator stable laws we present a strongly consistent estimate of the s pectral measure. The cases of a known or unknown exponent are considered.

Continuous time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. When the time between particle jumps has a slowly varying probability tail, the resulting plume disperses at a slowly varying rate. The limiting stochastic...

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions,...

If An[mu]kn * [delta](an) = [nu] where [nu] is full and kn+1/kn -- c [greater-or-equal, slanted] 1, we say that [mu] belongs to the generalized domain of semistable attraction (GDOSA) of [nu]. In this paper we describe the structure of GDOSA, and we give concise necessary and sufficient...

Bounds on the norming operators for distributions in the domain of semi-stable attraction of an operator semi-stable distribution are found. These bounds are used to establish the existence and nonexistence of moments of distributions in the domain of semi-stable attraction of an operator...

We consider the asymptotics of certain symmetric k-tensors, the vector analogue of sample moments for i.i.d. random variables. The limiting distribution is operator stable as an element of the vector space of real symmetric k-tensors.