A scalar-valued infinitely divisible random field with Pólya autocorrelation

We construct and characterize a stationary scalar-valued random field with domain Rd or Zd, d∈Z+, which is infinitely divisible, can take any (univariate) infinitely divisible distribution with finite variance at any single point of its domain, and has autocorrelation function between any two points in its domain expressed as a product of arbitrary positive and convex functions equal to 1 at the origin. Our method of construction-based on carefully chosen sums of independent and identically distributed random variables-is simple and so lends itself to simulation.