content_image/297815.jpg content_image/297816.jpg content_image/297817.jpg
<img src='content_image/297815.jpg'>

where O$_{p}$ (1) is uniform in h$_{1}$, ..., h$_{r}$ ∈ [ 0 , h ] , and z$_{0}$ = ( z$_{ρ}$$_{(1)0}$, ..., z$_{ρ}$$_{(}$$_{r}$$_{)0}$ ) . The branch of the square root in formulae (23) and (24) is chosen so that ( − 1) 1 / 2 = − i . 

Using Lemma 3, we establish the following theorem. 

Theorem 1 Suppose that the null hypothesis is true ( h = 0 ). Let ¯ h be any fixed number such that 0 < ¯ h < √$_{c}$ and let C [ 0 , h ] r be the space of real-valued contin- uous functions on [ 0 , h ] r equipped with the supremum norm. Then, as n, p → ∞ so that p/n = c$_{p}$ → c ∈ (0 , + ∞ ) , we have 

<img src='content_image/297816.jpg'>

<img src='content_image/297817.jpg'>

where the O$_{p}$ ( n − $^{1}$) terms are uniform in h ∈ ( 0 , ¯ h ] $^{r}$. Furthermore, ln L ( h ; λ ) and ln L ( h ; µ ) , viewed as random elements of C [ 0 , h $^{r}$, converge weakly to L $_{λ}$( h ) and L $_{µ}$( h ) with Gaussian finite-dimensional distributions such that, for any h, ˜ h ∈ 

