Finally, combining (44), (45), and (46), we obtain for the following upper and lower bounds: Let τ > 0 be an arbitrarily small number. Equations (37) and (41) imply that there exist ¯ M and ¯ M ′ such that, for any M > ¯ M and M ′ > ¯ M ′ , for all sufficiently large p. Let us choose M > ¯ M and M ′ > ¯ M ′ so that and for all sufficiently large p, a.s.. Then, (48) implies that for all sufficiently large p , a.s.. Since τ can be chosen arbitrarily, we have, from (47) and (49), where o (1) → 0 as p → ∞ uniformly in { 2 θ$_{pj}$ ∈ Ω$_{εη}$ , j ≤ r } , a.s.. □ I$_{p}$ (Θ$_{p}$ , Λ$_{p}$) = e p ∑ r j $_{=1}$[ θ$_{pj}$v$_{pj}$ − 1 2 p ∑ p i $_{=1}$ln(1+2 θ$_{pj}$v$_{pj}$ − 2 θ$_{pj}$λ$_{pi}$ ) ] × ( r ∏ j =1 j ∏ s =1 √ 1 − 4 ( θ$_{pj}$v$_{pj}$ ) ( θ$_{ps}$v$_{ps}$ ) c$_{p}$ + o (1) ) , ∣ J$_{p}$ − r ∏ j =1 j ∏ s =1 √ 1 − 4 ( θ$_{pj}$v$_{pj}$ ) ( θ$_{ps}$v$_{ps}$ ) c$_{p}$ ∣ < τ (49) [ ( 1 − 2 r 2 $_{e}$− M 2 16 ) − 1 − 1 ] sup { 2 θ$_{pj}$ ∈ Ω$_{εη}$ ,j ≤ r } r ∏ j =1 j ∏ s =1 √ 1 − 4 ( θ$_{pj}$v$_{pj}$ ) ( θ$_{ps}$v$_{ps}$ ) c$_{p}$ < τ 4 ( 1 − 2 r 2 $_{e}$− M 2 16 ) − 1 < 2 , ( 1 − 2 r 2 $_{e}$− M 2 16 ) − 1 4 r 2 $_{e}$− ( M ′ ) 2 16 ˜ λ 2 max + β$_{1}$r $^{2}$MM ′ < τ 4 , ∣ J M,M ′ p − r ∏ j =1 j ∏ s =1 √ 1 − 4 ( θ$_{pj}$v$_{pj}$ ) ( θ$_{ps}$v$_{ps}$ ) c$_{p}$ ∣ < τ 4 J M,M ′ p ≤ J$_{p}$ ≤ ( 1 − 2 r 2 $_{e}$− M 2 16 ) − 1 ( J M,M ′ p + 4 r 2 $_{e}$− ( M ′ ) 2 16 ˜ λ 2 max + β$_{1}$r $^{2}$MM ′ ) . (48) J$_{p}$ = I$_{p}$ (Θ$_{p}$ , Λ$_{p}$) $_{e}$− p ∑ r j $_{=1}$[ θ$_{pj}$v$_{pj}$ − 1 2 p ∑ p i $_{=1}$ln(1+2 θ$_{pj}$v$_{pj}$ − 2 θ$_{pj}$λ$_{i}$ ) ] (47) 32