so that Combining this with (40), we get (19). □ Proof of Lemma 2 Proof of Lemma 3. 28 where g ( z ) = exp { − 1 $_{2}$Δ$_{p}$ ( z ) } and O$_{p}$ (1) is uniform in h$_{i}$ ∈ ( 0 , ¯ h ] . A careful inspection of OMH’s proof of their Lemma 5 reveals that a version of (42) remains valid for general functions g ( z ) that are analytic in the open ball B ( z$_{i}$$_{0}$, r$_{i}$ ) with center at z$_{i}$$_{0}$ and radius r$_{i}$ = min { z$_{i}$$_{0}$ − max { ¯ b$_{p}$, λ$_{1}$ } , 1+ h$_{i}$ h$_{i}$ S − z$_{i}$$_{0}$ } with probabil- ity approaching 1 as n, p → ∞ . Precisely, for such general g ( z ) we have To save space, we will only establish (23), relegating a conceptually similar but more technical proof of (24) to the Supplementary Appendix. Lemma 5 in OMH implies that The lemma can be proven using arguments very similar to those in the proof of Lemmas 4 and 6 in Onatski, Moreira and Hallin (2012) (OMH in what follows), and we omit the proof to save space. □ ∫ ∞ 0 y p ( n − r )+ r ( r +1) / 2 − $^{1}$e − ny det R ( y )d y = Γ ( p ( n − r ) + r ( r + 1) / 2) n p ( n − r )+ r ( r +1) / 2 × ∑ ρ sgn ρ (2 π i) r ∮ K ... ∮ K ( 1 − r ∑ j =1 h$_{ρ}$$_{(}$$_{j}$$_{)}$ 1 + h$_{ρ}$$_{(}$$_{j}$$_{)}$ z$_{j}$ S ) − p ( n − r ) − r ( r +1) / 2 × r ∏ j $_{=1}${ z j − 1 j p ∏ s =1 ( z$_{j}$ − λ$_{s}$ ) − 1 } d z$_{r}$... d z$_{1}$ . ∮ K$_{i}$ e − nf$_{i}$ ( z $^{)}$g ( z ) d z = e − nf$_{i}$$_{0}$ [ g ( z$_{i}$$_{0}$ ) π 1 / 2 f 1 / 2 i 2 n 1 / 2 + O$_{p}$ (1) h$_{i}$n 3 / 2 ] , (42) ∮ K$_{i}$ e − nf$_{i}$ ( z $^{)}$g ( z ) d z = e − $_{nf$_{i}$$_{0}$}$g ( z$_{i}$$_{0}$ ) π 1 / 2 f 1 / 2 i 2 n 1 / 2 + Ψ$_{1}$ + Ψ$_{2}$ + Ψ$_{3}$ (43)