Multiply stochastic representations for K distributions and their Poisson transforms

The K distribution is used in a number of areas of scientific endeavor. In optics, it provides a useful statistical description for fluctuations of the irradiance (and the electric field) of light that has been scattered or transmitted through random media (e.g., the turbulent atmosphere). The Poisson transform of the K distribution describes the photon-counting statistics of light whose irradiance is K distributed. The K-distribution family can be represented in a multiply stochastic (compound) form whereby the mean of a gamma distribution is itself stochastic and is described by a member of the gamma family of distributions. Similarly, the family of Poisson transforms of the K distributions can be represented as a family of negative-binomial transforms of the gamma distributions or as Whittaker distributions. The K distributions have heretofore had their origins in random-walk models; the multiply stochastic representations provide an alternative interpretation of the genesis of these distributions and their Poisson transforms. By multiple compounding, we have developed a new transform pair as a possibly useful addition to the K-distribution family. All these distributions decay slowly and are difficult to calculate accurately by conventional formulas. A recursion relation, together with a generalized method of steepest descent, has been developed to evaluate numerically the photon-counting distributions and their factorial moments with excellent accuracy.