c$_{p}$ ln ( z$_{i}$$_{0}$ ) − h$_{i}$ − c$_{p}$ > f$_{i}$$_{0}$. Indeed, using the definition of z$_{i}$$_{0}$ and the fact, established in OMH’s Lemma 11, that f$_{i}$$_{0}$ = − c$_{p}$ − (1 − c$_{p}$ ) ln (1 + h$_{i}$ ) + c$_{p}$ ln c$_{p}$ $_{h$_{i}$}$, we have The right hand side of this equality equals 0 at h$_{i}$ = 0 and has a non-negative derivative with respect to h$_{i}$ for all 0 ≤ h$_{i}$ ≤ √ c$_{p}$. Therefore, and thus, Ψ$_{3}$ = e − $^{nf$_{i}$$_{0}$}$h − j i n − 3 / $^{2}$O$_{p}$ (1), uniformly in h$_{i}$ ∈ ( 0 , ¯ h ] . Combining this with (43) and (47), we obtain (23). □ Proposition 2 and Lemma 3 imply that As is shown in OMH (see their Lemma 11 and (A8)) $^{3}$, for h$_{i}$ ≤ ¯ h , Moreover, by OMH’s Lemma A2, exp {− Δ$_{p}$ ( z$_{i}$$_{0}$ ) } = O$_{p}$ (1) uniformly in h ∈ ( 0 , ¯ h ] $^{r}$. Using these facts and the definition of k$_{1}$ given in Proposition 2, we get after some algebra $^{3}$Note that the expressions given in OMH are half times the expressions given below because the equivalent of f$_{i}$ in the real-valued data case considered by OMH is f$_{i}$/ 2. Proof of Theorem 1 L ( h ; λ ) = k$_{1}$ exp {− n ∑ r i $_{=1}$f$_{i}$$_{0}$ } (2i) $^{r}$( πn ) r/ 2 det ( z j − 1 i 0 exp {− Δ$_{p}$ ( z$_{i}$$_{0}$ ) } f 1 / 2 i 2 + O$_{p}$ (1) h j $_{i}$n )$_{1}$$_{≤}$$_{i,j}$$_{≤}$$_{r}$ f$_{i}$$_{0}$ = − c$_{p}$ − (1 − c$_{p}$ ) ln (1 + h$_{i}$ ) + c$_{p}$ ln c$_{p}$ h$_{i}$ , and (49) f$_{i}$$_{2}$ = − h 2 i 2 (1 + h$_{i}$ ) $^{2}$( c$_{p}$ − h 2 $_{i}$) . (50) ∣ ∮ K$_{i}$$_{2}$∪ ¯ K$_{i}$$_{2}$ e − nf$_{i}$ ( z $^{)}$g ( z ) d z ∣ < 2 j (3 z$_{i}$$_{0}$ ) j − 1 ( n h$_{i}$ 1 + h$_{i}$ ) − 1 3 − $^{p}$e − $^{nf$_{i}$$_{0}$}$, c$_{p}$ ln ( z$_{i}$$_{0}$ ) − h$_{i}$ − c$_{p}$ − f$_{i}$$_{0}$ = ln (1 + h$_{i}$ ) + c$_{p}$ ln ( c$_{p}$ + h$_{i}$ ) − h$_{i}$ − c$_{p}$ ln c$_{p}$. 30