By Corollary 1, x S$_{p}$ − 1 = x p − S$_{p}$ p + o ( p − $^{1}$) a.s.. Using this fact, after some algebra, we get and It follows that where the last equality in (55) follows from (3) and Proposition 2. The last equality in (55), (4) and the fact that imply that which establishes (11). The rest of the statements of Theorem 1 follow from (10), (11), and Lemmas 12 and A2 of OMH. □ 37 L$_{p}$ ( h ; µ$_{p}$ ) = (1 + o (1)) L$_{p}$ ( h ; λ$_{p}$ ) e ∑ r j =1 − h j 2 c $_{p}$S$_{p}$ ( 1 − r ∑ j =1 h$_{j}$ nc$_{p}$ ) − np 2 = (1 + o (1)) L$_{p}$ ( h ; λ$_{p}$ ) $_{e}$− S p − p 2 c p ∑ r j $_{=1}$h$_{j}$ + 1 4 c $^{$_{p}$(}$∑ r j $_{=1}$h$_{j}$ ) 2 , ∫ p + α √ p p − α √ p x np 2 − 1 $_{e}$− n $^{$_{2}$x}$e ∑ r j $_{=1}$θ$_{pj}$ v$_{pj}$ ( x − S$_{p}$ $^{)}$d x = e ∑ r j =1 − h j 2 c $_{p}$S$_{p}$ ( n 2 − r ∑ j =1 h$_{j}$ 2 c$_{p}$ ) − np 2 Γ ( np 2 ) (1+ o (1)) I (0 , ∞ ) = ∫ p + α √ p p − α √ p x np 2 − 1 $_{e}$− n $^{$_{2}$x}$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$_{pi}$ ) ] (55) × e ∑ r j $_{=1}$θ$_{pj}$ v$_{pj}$ ( x − S$_{p}$ ) ( r ∏ j =1 j ∏ s =1 √ 1 − 4 ( θ$_{pj}$v$_{pj}$ ) ( θ$_{ps}$v$_{ps}$ ) c$_{p}$ + o (1) ) d x = (1 + o (1)) r ∏ j =1 (1 + h$_{j}$ )$^{n}$ p 2 L$_{p}$ ( h ; λ$_{p}$ ) ∫ p + α √ p p − α √ p x np 2 − 1 $_{e}$− n $^{$_{2}$x}$e ∑ r j $_{=1}$θ$_{pj}$ v$_{pj}$ ( x − S$_{p}$ $^{)}$d x, ˜ $^{θ$_{pj}$}$˜ v$_{pj}$ = θ$_{pj}$v$_{pj}$ + θ$_{pj}$v 2 pj ( x p − S$_{p}$ p ) + O ( ( x p − 1 ) 2 ) , 2 ln ( 2 ˜ θ$_{pj}$ ) =ln (2 θ$_{pj}$ )+ x p − S$_{p}$ p + O x p − 1 2 , p ∑ i =1 ln ( K MP p ( 2 ˜ θ$_{pj}$ ) − λ$_{pi}$ ) = p ∑ i =1 ln ( K MP p (2 θ$_{pj}$ ) − λ$_{pi}$ ) − p ( 1 − 4 c$_{p}$θ 2 $_{pj}$v 2 pj ) ( x p − S$_{p}$ p ) + O x p − 1 ) 2 ) .