In this paper the asymptotic expansion of a percentage point of Hotelling's generalized $T^2_0$ distribution is derived in terms of the corresponding percentage point of a $\chi^2$ distribution. Our result generalizes Hotelling's and Frankel's asymptotic expansion for the generalized Student $T$ , . The technique used in this paper for obtaining the asymptotic expansion of $T^2_0$ is an extension of the previous methods of Welch  and of James , , who used them to solve the distribution problem of various statistics in connection with the Behrens-Fisher problem. An asymptotic formula for the cumulative distribution function (c.d.f.) of $T^2_0$ is also given together with an upper bound for the error committed when all but the first few terms are omitted in the series. This formula is a sort of multivariate analogue of Hartley's formula of "Studentization" .
"Asymptotic Formulae for the Distribution of Hotelling's Generalized $T^2_0$ Statistic." Ann. Math. Statist. 27 (4) 1091 - 1105, December, 1956. https://doi.org/10.1214/aoms/1177728075