The asymptotic non-null distributions of four statistics, used for testing the hypothesis Ho: $\Sigma_1=\Sigma_2$ , are developed. These expansions are obtained by the partial differential equations method of Muirheard (Ann. Math. Statist. 41 1002-1010). The procedure permits a simple extension of the results of Pillai and Nagarsenker (J. Multivariate Anal. 2 96-114).
"On the Asymptotic Distributions of Some Statistics Used for Testing $\Sigma_1=\Sigma_2$." Ann. Statist. 3 (4) 916 - 925, July, 1975. https://doi.org/10.1214/aos/1176343192