Some uses of Student ’ s t characteristic function and the sum of t and Gaussian random variables . Technical Report 05 : 01

This article derives the probability density function (pdf) of the sum of a normal random variable and a (sphered) Student’s-t distribution on three degrees of freedom. Advice is given on deriving the convolution density for higher degrees of freedom. Apart from its intrinsic interest applications of this result include Bayesian wavelet shrinkage, Bayesian posterior density derivations, calculations in the theoretical analysis of projection indices and computation of certain moments. An R package computing the convolution density and the complex-valued erf function is described. A related result using the characteristic function of Student’s t distribution is used to compute asymptotically optimal thresholds for Haar wavelet shrinkage of double-exponential noise.