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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. 

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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 

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