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,...

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...

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...

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling...

Self-similar processes are useful models for natural systems that exhibit scaling. Operator scaling allows a different scale factor in each coordinate. This paper develops practical methods for modeling and simulation. A simulation method is developed for operator scaling Lévy processes, based...

Let X= X(t),t[set membership, variant]R+ be an operator stable Lévy process in Rd with exponent B, where B is an invertible linear operator on Rd. We determine the Hausdorff dimension and the packing dimension of the range X([0,1]) in terms of the real parts of the eigenvalues of B.

Periodic ARMA, or PARMA, time series are used to model periodically stationary time series. In this paper we develop the innovations algorithm for periodically stationary processes. We then show how the algorithm can be used to obtain parameter estimates for the PARMA model. These estimates are...