Skellam Random Fields and Their Fractional Variants

Abstract

We study some Skellam-type spatial point processes. As a particular case, we consider a Skellam random field (SRF) on the positive quadrant of the plane, which is a two parameter L\'evy process with rectangular increments. A weak convergence result is obtained for the SRF. The Riemann-Liouville integral of the SRF over finite rectangles is analyzed. We derive a scaled compound Poisson field characterization for the Riemann integral of the SRF. Also, an explicit expression of its characteristic function is obtained. Later, we consider three fractional variants of the two parameter SRF. Their point probabilities, associated governing equations, and various other distributional properties are analyzed.