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## 5.2 Proof of Theorem 3 

Setting θ$_{pj}$ = 1 2 c$_{p}$ h$_{j}$ 1+ $_{h$_{j}$}$, we have v$_{pj}$ = 1 + h$_{j}$ , θ$_{pj}$v$_{pj}$ = h$_{j}$ 2 $_{c$_{p}$}$, and 

Further, by Lemma 11 and formula (3.3) of OMH, ∫ ln ( z$_{j}$$_{0}$ − λ ) d F MP p ( λ ) = h$_{j}$ c$_{p}$ − 1 c$_{p}$ ln (1 + h$_{j}$ ) + ln (1+ h$_{j}$ ) c$_{p}$ h$_{j}$ for sufficiently large p, a.s.. With these auxiliary results, formula (10) is a straightforward consequence of (3) and Proposition 2. 

Turning to the proof of (11), consider the integrals 

O ( p ) In what follows, we omit the subscript p in n$_{p}$ to simplify notation. Note that I (0 , ∞ ) is the integral appearing in expression (4) for L$_{p}$ ( h ; µ$_{p}$ ) . Let us now prove that, for some constant α > 0 , 

where o (1) is uniform in h ∈ [0 $_{,}$√$_{c}$ − δ ] $^{r}$. 

Since, by Corollary 1, S$_{p}$/p → 1 a.s., the set H$_{δ}$ is bounded from below, and λ$_{p}$$_{1}$ → (1 + √$_{c}$$_{)}$ 2 a.s., there exists a constant A$_{1}$ > 0 that depends only on δ and r, such that inf [ 0 $_{,}$√$_{c}$ − δ ] r px tr ( D$_{p}$Q ′ Λ$_{p}$ Q ) /S$_{p}$ ≥ − A$_{1}$x/ 2 for all x ≥ 0 and all sufficiently large p, a.s.. Therefore, for all h ∈ [0 $_{,}$√$_{c}$ − δ ] $^{r}$, np 

and, using Stirling’s approximation, we get 

<img src='content_image/307148.jpg'>

<img src='content_image/307147.jpg'>

<img src='content_image/307145.jpg'>

<img src='content_image/307146.jpg'>

<img src='content_image/307144.jpg'>

Next, there exists a constant A$_{2}$ > 0 such that, for all x ≥ 0 and all sufficiently large p, $^{sup$_{h}$}$∈ [ 0 $_{,}$√$_{c}$ − δ ] r px tr ( D$_{p}$Q ′ Λ$_{p}$ Q ) /S$_{p}$ ≤ A$_{2}$x/ 2, a.s.. Therefore, a.s., for all 

