content_image/297044.jpg content_image/297043.jpg content_image/297040.jpg content_image/297041.jpg content_image/297042.jpg
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 

<img src='content_image/297044.jpg'>

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, 

<img src='content_image/297043.jpg'>

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). □ 

## Proof of Theorem 1 

Proposition 2 and Lemma 3 imply that 

<img src='content_image/297040.jpg'>

As is shown in OMH (see their Lemma 11 and (A8)) $^{3}$, for h$_{i}$ ≤ ¯ h , 

<img src='content_image/297041.jpg'>

<img src='content_image/297042.jpg'>

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 

