Distributional properties of portfolio weights

In this paper, we prove several distributional properties for optimal portfolio weights. The weights are estimated by replacing the parameters with the sample counterparts. All results for finite samples are made assuming normally distributed returns. We calculate the exact covariances for the weights obtained by the expected quadratic utility. Additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio. We obtain the conditional density for the Sharpe ratio optimal weights and show that the expectations of the Sharpe ratio optimal weights do not exist. Moreover, we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary Gaussian process.