Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components

This study derives a new property of the Wishart distribution when the degree-of-freedom and the size of the matrix parameter of the distribution grow simultaneoulsy. Particularly, the asymptotic normality of the product of four independent Wishart matrices is shown under a high dimensional asymptotic regime. As an application of the result, a statistical test procedure for the common principal components hypothesis is proposed. For this problem, the proposed test statistic is asymptotically normal under the null hypothesis. In addition, the proposed test statistic diverges to positive infinity in probability under the alternative hypothesis.