Perturbation of Normal Random Vectors by Nonnormal Translations, and an Application to HIV Latency Time Distributions

Let Z be a normal random vector in Rk and let 1 be the element of Rk with equal components 1. Let X be a random variable that is independent of Z and consider the sum Z+X1. The latter has a normal distribution in Rk if and only if X has a normal distribution in R1. The first result of this paper is a formula for a uniform bound on the difference between the density function of Z+X1 and the density function in the case where X has a suitable normal distribution. This is applied to a problem in the theory of stationary Gaussian processes which arose from the author's work on a stochastic model for the CD4 marker in the progression of HIV.