content_image/313404.jpg content_image/313405.jpg content_image/313406.jpg
so that 

Combining this with (40), we get (19). □ 

## Proof of Lemma 2 

<img src='content_image/313404.jpg'>

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. □ 

## Proof of Lemma 3. 

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 

<img src='content_image/313405.jpg'>

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 

<img src='content_image/313406.jpg'>

