∮ K$_{ρ}$$_{(1)}$ ... ∮ K$_{ρ}$$_{(}$$_{r}$$_{)}$ q$_{ρ}$ ( z ) r ∏ j =1 { e − nf$_{ρ}$$_{(}$$_{j}$$_{)}$ ( z$_{j}$ $^{)}$g$_{j}$ ( z$_{j}$ ) } d z$_{r}$... d z$_{1}$ = q$_{ρ}$ ( z$_{0}$ ) r ∏ j =1 e − $_{nf$_{ρ}$$_{(}$$_{j}$$_{)0}$}$g$_{j}$ ( z$_{ρ}$$_{(}$$_{j}$$_{)0}$ ) π 1 / 2 f 1 / 2 ρ ( j $_{)2}$n 1 / 2 + O$_{p}$ (1) n r ∏ j =1 e − nf$_{ρ}$$_{(}$$_{j}$$_{)0}$ h j ρ ( j $_{)}$n 1 / $_{2}$, (24) L ( h ; λ ) = exp { − r ∑ i =1 Δ$_{p}$( z$_{i}$$_{0}$ )+ 1 2 r ∑ i,j =1 ln ( 1 − h$_{i}$h$_{j}$ c$_{p}$ ) } + O$_{p}$ ( 1 n ) and (25) L ( h ; µ ) = exp { − r ∑ i =1 Δ$_{p}$( z$_{i}$$_{0}$ )+ 1 2 r ∑ i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c$_{p}$ ) + h$_{i}$h$_{j}$ c$_{p}$ ) − S − p c$_{p}$ r ∑ j =1 h$_{j}$ } + O$_{p}$ ( 1 n ) , (26) 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 ∈ 17 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