Let θ$_{pj}$ = h$_{j}$/ 2 c$_{p}$ (1 + h$_{j}$ ) and The condition h$_{j}$ ∈ H$_{δ}$ for some δ > 0 implies that 2 θ$_{pj}$ ∈ Θ$_{εη}$ for some ε > 0 , η > 0 and p sufficiently large. Below, we are only interested in non-negative values of h$_{j}$ , and assume that h$_{j}$ ∈ (0 $_{,}$√$_{c}$ − δ ] under the alternative hypothesis. The corresponding θ$_{pj}$ , thus, is positive. With the above setting for θ$_{pj}$, we have v$_{pj}$ = 1 + h$_{j}$ and K MP p (2 θ$_{pj}$ ) = ( c$_{p}$ + h$_{j}$ ) (1 + h$_{j}$ ) /h$_{j}$ = z$_{j}$$_{0}$, say, as in Theorem 7 in OMH. Define Theorem 3 Suppose that the null hypothesis is true ( h = 0 ). Let δ be any fixed number such that 0 < δ < √$_{c}$$_{, and let}$ C [0 $_{,}$√$_{c}$ − δ ] r be the space of real-valued continuous functions on [0 $_{,}$√$_{c}$ − δ ] r equipped with the supremum norm. Then, as p, n$_{p}$ → c ∞ , almost surely, where the o (1) terms are uniform in h ∈ [0 $_{,}$√$_{c}$ − δ ] $^{r}$. Furthermore, ln L$_{p}$ ( h ; λ$_{p}$ ) and ln L$_{p}$ ( h ; µ$_{p}$ ) , viewed as random elements of C [0 $_{,}$√$_{c}$ − δ ] $^{r}$, converge weakly to L $_{λ}$( h ) and L $_{µ}$( h ) with Gaussian finite-dimensional distributions such that E ( L $_{λ}$( h )) = − 1 2 Var ( L $_{λ}$( h )) , E ( L $_{µ}$( h )) = − 1 2 Var ( L $_{µ}$( h )) , and, for any L$_{p}$ ( h ; λ$_{p}$ ) = r ∏ j =1 exp { − 1 2 Δ$_{p}$ ( z$_{j}$$_{0}$ )+ 1 2 j ∑ s =1 ln ( 1 − h$_{j}$h$_{s}$ c$_{p}$ ) } (1+ o (1)) and (10) L$_{p}$ ( h ; µ$_{p}$ ) = L$_{p}$ ( h ; λ$_{p}$ ) exp    1 4 c$_{p}$ ( r ∑ j =1 h$_{j}$ ) 2 − S$_{p}$ − p 2 c$_{p}$ r ∑ j =1 h$_{j}$    (1+ o (1)) , (11) H$_{δ}$ =    $^{[}$$^{−}$$^{1 +}$ δ, 0) ∪ (0 $_{,}$√$_{c}$ − δ ] for c > 1 , [ − √$_{c}$ + δ, 0) ∪ (0 $_{,}$√$_{c}$ − δ ] for c ≤ 1 . (8) 11