## The Annals of Applied Probability

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

Simeon M. Berman

#### Abstract

Let $\mathbf{Z}$ be a normal random vector in $R^k$ and let $\mathbf{1}$ be the element of $R^k$ with equal components 1. Let $X$ be a random variable that is independent of $\mathbf{Z}$ and consider the sum $\mathbf{Z} + X\mathbf{1}$. The latter has a normal distribution in $R^k$ if and only if $X$ has a normal distribution in $R^1$. The first result of this paper is a formula for a uniform bound on the difference between the density function of $\mathbf{Z} + X\mathbf{1}$ 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.

#### Article information

Source
Ann. Appl. Probab., Volume 4, Number 4 (1994), 968-980.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177004899

Digital Object Identifier
doi:10.1214/aoap/1177004899

Mathematical Reviews number (MathSciNet)
MR1304769

Zentralblatt MATH identifier
0844.92021

JSTOR