On Dominating an Average Associated with Dependent Gaussian Vectors

Abstract

Mehler's identity is used to obtain a bound for the integral of the absolute difference between the bivariate gaussian density function and its corresponding marginal densities. In a sense, this integral measures the contribution of the dependency between the gaussian random variables to an expectation. It is shown that the integral is dominated by $|\rho|/ (1 - |\rho|)$, where $\rho$ is the correlation coefficient between the random variables. Using the Hotelling canonical decomposition of a variance-covariance matrix, the result is extended to the case of dependent gaussian vectors with the bound now given in terms of the canonical correlations, i.e., the roots of a characteristic equation related to the variance-covariance matrix of the vectors. As an application of the results, a bound is obtained for the variance of the function $F(X_1, \cdots, X_n) = n^{-1} \sum^n_{i=1} f(X_i)$. The $\{X_i\}$ denote a sequence of dependent, non-stationary, gaussian random variables (or vectors), and $f(x)$ is any bounded measurable function. For the stationary case, the rate of convergence of the variance is easily expressed in terms of the summability properties of the correlation coefficients. The paper concludes with some comments on extending the results to the class of $\varphi^2$-bounded bivariate density functions.