content_image/312034.jpg content_image/312033.jpg content_image/312032.jpg content_image/312031.jpg
allows us to use the method of steepest descent in the asymptotic analysis of the corresponding integrals in (18) and (19). The next lemma shows that the change of contours in (18) and (19) does not lead to a change in the value of the corresponding integrals. 

Lemma 2 Suppose that the null hypothesis is true, and let ¯ h be an arbitrary num- ber such that 0 < ¯ h < √$_{c}$$_{. Suppose further that}$ h$_{i}$ ≤ ¯ h for all i = 1 , ..., r . Then, as n, p → ∞ so that c$_{p}$ → c ∈ (0 , + ∞ ) , 

<img src='content_image/312034.jpg'>

<img src='content_image/312033.jpg'>

Our next lemma establishes Laplace approximations to the contour integrals in (18) and (19) after the change of the contours. The lemma uses some new notation that we introduce now. When f$_{i}$ ( z ) is analytic at z$_{i}$$_{0}$, let f$_{is}$ with s = 0 , 1 , ... be the coefficients in the power series representation 

<img src='content_image/312032.jpg'>

When f$_{i}$ ( z ) is not analytic at z$_{i}$$_{0}$ , let the coefficients f$_{is}$ be arbitrary numbers for all s ∈ N . 

Lemma 3 Under the conditions of Lemma 2, 

<img src='content_image/312031.jpg'>

