## The Annals of Applied Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60E99: None of the above, but in this section

Secondary: 60G15: Gaussian processes 62E99: None of the above, but in this section 92A07

**Keywords**

Gaussian process HIV latency time nonnormal translation normal random vector posterior density

#### Citation

Berman, Simeon M. Perturbation of Normal Random Vectors by Nonnormal Translations, and an Application to HIV Latency Time Distributions. Ann. Appl. Probab. 4 (1994), no. 4, 968--980. doi:10.1214/aoap/1177004899. https://projecteuclid.org/euclid.aoap/1177004899