Perturbation of normal random vectors by nonnormal translations, and an application to HIV latency time distributions (Q1345577)

Let \({\mathbf Z}\) be a normal random vector in \(R^k\) and let \textbf{1} be a vector in \(R^k\) with equal components 1. Let \(X\) and \(Y\) be two random variables such that \(X\) and \({\mathbf Z}\) are independent and \(Y\) and \({\mathbf Z}\) are independent. The author studies the bound on the difference between the density functions of \({\mathbf Z} + X \mathbf{1}\) and \({\mathbf Z} + Y\mathbf{1}\). The absolute difference between the two densities is uniformly bounded below by: \[ \bigl[ (2 \pi)^k \text{det} \Sigma \bigr]^{- 1/2} \int^\infty_{- \infty} |E \exp^{i \lambda xX} - E \exp^{i \lambda xY} |\varphi (x) dx, \] where \(\Sigma\) is a nonsingular covariance matrix, \(\varphi (x)\) is the standard normal density and \(\lambda = \sqrt {\mathbf{1}' \Sigma^{-1} \text\textbf{1}}\). This result is applied to a problem related to HIV latency time distributions based on CD4 levels.