The perturbation methods and the Taylor expansions are very often used to obtain test statistics approximations in multivariate analysis (Specially in Principal Component and Canonical Analyses). These approximations are then used to obtain formal Edgeworth expransions of the distribution functions of the statistics. BHATTACHARYA and GHOSH 1978 have justified these practices under suitable assumptions. In this paper a non classical perturbation problem is solved in order to obtain almost surely expansions of test statistics