Analysis of Financial Time Series, Third Edition

Appendix B: Multivariate Normal Distributions

A k-dimensional random vector follows a multivariate normal distribution with mean and positive-definite covariance matrix if its probability density function (pdf) is

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We use the notation to denote that follows such a distribution. This normal distribution plays an important role in multivariate statistical analysis and it has several nice properties. Here we consider only those properties that are relevant to our study. Interested readers are referred to Johnson and Wichern (1998) for details.

To gain insight into multivariate normal distributions, consider the bivariate case (i.e., k = 2). In this case, we have

Using the correlation coefficient ρ = σ12/(σ1σ2), where is the standard deviation of xi, we have and . The pdf of then becomes ...