A note on scale mixtures of skew normal distribution

Moments of scale mixtures of skew normal distribution and their quadratic forms are derived using the simple stochastic relationship between skew normal distribution and scale mixtures of skew normal distribution. An application to time series is discussed. It is shown that the mean, covariance, and correlation structure of the sample autocovariance function for a particular class of time series depend on a measure of multivariate kurtosis, but not on a shape parameter. Unlike the multivariate normal distribution and the elliptical distributions with Muirhead kurtosis parameter 0, the covariance and the correlation of the sample autocovariance function for a particular class of time series with underlying scale mixtures of normal distribution are not always negative. We show this using an example of the generalized tt distribution. As a by-product, we derived Muirhead kurtosis for some scale mixtures of normal distribution.