L ( h ; λ ) = n r $^{2}$/ 2 r ∏ t $_{=1}$[ ( p − t )! ( c$_{p}$ − h 2 t 2 π ) 1 / 2 ] e $^{rp}$p − rp × exp { − r ∑ i =1 Δ$_{p}$ ( z$_{i}$$_{0}$ ) } r ∏ i>j ( c$_{p}$ − h$_{i}$h$_{j}$ ) ( 1 + O$_{p}$ ( n − $^{1}$)) ( p − t )! = e − $^{p}$pp − t +1 ( 2 π p ) 1 / 2 ( 1 + O ( p − $^{1}$)) L ( h ; λ ) = exp { − r ∑ i =1 Δ$_{p}$ ( z$_{i}$$_{0}$ ) } r ∏ t =1 ( 1 − h 2 t c$_{p}$ ) 1 / 2 r ∏ i>j ( 1 − h$_{i}$h$_{j}$ c$_{p}$ ) ( 1 + O$_{p}$ ( n − $^{1}$)) , L ( h ; µ ) = ( − 1) r ( r − 1) / $^{2}$n − pr + r ( r +1) / 2 r ∏ i>j ( h$_{i}$ − h$_{j}$ ) − 1 × (51) r ∏ t =1 [ h r − p t (1 + h$_{t}$ ) p − n − $^{1}$( p − t )! ] ( nS ) pr − r ( r +1) / 2 Γ ( p ( n − r ) + r ( r + 1) / 2) Γ ( np ) × ∑ ρ sgn ρ (2 π i) $_{r}$q$_{ρ}$ ( z$_{0}$ ) r ∏ j =1 e − $_{nf$_{ρ}$$_{(}$$_{j}$$_{)0}$}$g$_{j}$ ( z$_{ρ}$$_{(}$$_{j}$$_{)0}$ ) π 1 / 2 f 1 / 2 ρ ( j $_{)2}$n 1 / 2 ( 1 + O$_{p}$ ( n − $^{1}$)) q$_{ρ}$ ( z$_{0}$ ) = ( 1 − r ∑ i =1 h$_{i}$ 1 + h$_{i}$ z$_{i}$$_{0}$ S ) − p ( n − r ) − r ( r +1) / 2 exp { − r ∑ i =1 nh$_{i}$z$_{i}$$_{0}$ 1 + h$_{i}$ } = ( 1 − r ∑ i =1 h$_{i}$ + c$_{p}$ S ) − p ( n − r ) − r ( r +1) / 2 exp { − pr − r ∑ i =1 nh$_{i}$ } . (52) Using the definition of q$_{ρ}$ ( z$_{0}$ ) and of z$_{i}$$_{0}$, we get which implies (25). Turning to the proof of (26), Proposition 2 and Lemma 3 imply that we get Applying Stirling’s formula 31