On the generation of Tikhonov variates

A novel, simple and efficient method for the generation of Tikhonov (a.k.a. von Mises) random variates is proposed. In the proposed method, circular variates of a prescribed Tikhonov distribution p_T(x;alpha,xi) are generated via the transformation of variates selected randomly, on a one-for-one basis, from a bank of K distinct Cauchy and Gaussian generators. The mutually exclusive probabilities of sampling from each of the Cauchy or Gaussian generators, as well as the variance and half-width parameters that specify the latter, are derived directly from the Cauchy, Gaussian and Tikhonov characteristic functions, all of which are either known or given in closed form. The proposed random mixture technique is extremely efficient in that a single pair of uniform random numbers is consumed in the generation of each Tikhonov (or von Mises) sample, regardless of the prescribed concentration and centrality parameters (alpha, xi), all requiring neither the rejection of samples, nor the repetitive evaluation of computationally demanding functions. Additional attractive features of the method are as follows. By construction, the first (dominant) N circular moments of Tikhonov variates generated with the proposed random mixture technique are the ones that best approximate their corresponding theoretical values, with errors measured exactly. The exact distribution of generated Tikhonov variates is determined analytically, and its (Kullback-Leibler) divergence to the exact Tikhonov PDF is shown also analytically to be negligible. Finally, the technique establishes a connection between Tikhonov and Gaussian variates which can be exploited, e.g., in the generation of piecewise-continuous pseudo-random functions with Tikhonov-distributed outcomes.