From doubly stochastic representations of K distributions to random walks and back again: an optics tale

A random walk model with a negative binomially fluctuating number of steps is considered in the case where the mean of the number fluctuations, N-bar, is finite. The asymptotic behaviour of the resultant statistics in the large N-bar limit is derived and shown to give the K distribution. The equivalence of this model to the hitherto unrelated doubly stochastic representation of the K distribution is also demonstrated. The convergence to the K distribution of the probability density function generated by a random walk with a finite mean number of steps is examined along with the moments, and the non-Gaussian statistics are shown to be a direct result of discreteness and bunching effects