Verifique la batería

Proof of Lemma 2.1. De…ne gn( ) = (n1=2jj jj; jj jj; =jj jj; ; ; ); where by de- …nition =_{jj jj = 1}d =jj1d jj if = 0 and 1d = (1; :::; 1)0 2 Rd : De…ne G1 = fg : gn( n)! g for some f ng 2 ( 0; 0; b) with jjbjj < 1g; G2 =fg : gn( n)! g for some f ng 2 ( 0;1; !0)g; and G = G1[ G2:

First, we show AsySz min_finfh2HCP (h); CP_{1g: Let f n 2} : n 1_{g be a} sequence such that lim inf_n!1CPn( n) = lim infn!1inf 2 CPn( ) (= AsySz): Such a sequence always exists. _{Let fw}n : n 1g be a subsequence of fng such that lim_n!1CPwn( wn) exists and equals AsySz: Such a sequence always exists. Below we

show there exists a subsequence fpng of fwng such that CPpn( pn)! CP (h) for some

h _{2 H or limn!1}CPpn( pn) CP1: In consequence, AsySz = limn!1CPpn( pn)

min_finfh2HCP (h); CP_1g:

Now we show that the claim concerning the subsequence fpng holds. To this end, we show (a) for any sequence f n 2 : n 1g and any subsequence fwng of n; there exists a subsequence fpng of fwng such that gpn( pn) ! g for some g 2 G and

(b) for any subsequence fpng of fng and any sequence f pn 2 : n 1g for which

gpn( pn) ! g for some g 2 G; CPpn( pn) ! CP (h) for some h 2 H if g 2 G1 and

lim inf_n!1CPpn( pn) CP1 if g 2 G2:

To show (a), let _wn;j denote the jth component of _wn and p1;n = wn 8n 1: For j = 1; either (1) lim sup_n!1p1=2_{j;n p}

j;n;j < 1 or (2) lim supn!1p

1=2

j;n pj;n;j = 1: If

(1) holds, then for some subsequence fpj+1;ng of fpj;ng; p 1=2

j+1;n pj+1;n;j ! bj for some

bj 2 R: If (2) holds, then for some subsequence fpj+1;ng of fpj;ng; p 1=2

j+1;n pj+1;n;j ! 1

or _{1: Applying the same argument successively for j = 1; :::; d yields a subsequence} fpng = fpd +1;ng of fwng such that (pn)1=2 pn ! b 2 R

d _{or (p}

n)1=2jj pnjj ! 1:

Because _{is compact, there exists a subsequence fp}n g of fpng such that pn ! 0 2 :

Finally, let fpng be a subsequence of fpn g such that pn=jj pnjj ! !0:By construction,

gpn( pn)! g = (jjbjj; jj 0jj; !0; 0; 0; 0);where b 2 (R [ f 1g)

d _:

It remains to show that the vector g constructed in the previous paragraph is in G: (This is needed because G is de…ned by the limits of full sequences rather than subsequences.) To this end, it su¢ ces to show that there exists a sequence f n 2 : n 1_{g such that g}n( n)! g and pn = pn 8n 1: Such a sequence f k : k 1g can

be constructed as follows: (i) 8k = pn; de…ne k = pn and (ii) 8k 2 (pn; pn+1); de…ne

k = (pn=k)1=2 pn when jjbjj 2 R and k= pn when jjbjj = 1, and (iii) k = pn; k =

pn; and k = pn in both cases. Note that when jjbjj 2 R; k = ( k; k; k; k)2 for

k _{large by Assumption ACP(iv). When jjbjj 2 R; g}n( n)! g because k1=2 k = p 1=2 n pn

8k 2 [pn; pn+1); p 1=2

n pn ! b as n ! 1; and pn=jj pnjj ! !0 as n ! 1 imply that

k1=2 jj kjj ! jjbjj and k=jj kjj ! !0 as k ! 1: When jjbjj = 1; k1=2jj kjj p 1=2 n jj pnjj 8k 2 [pn; pn+1):Thus, p 1=2 n jj _pnjj ! 1 as n ! 1 implies jjk1=2 _kjj ! 1 as k ! 1: In addition, when jjbjj = 1; k=jj kjj = pn=jj pnjj 8k 2 [pn; pn+1) and pn=jj pnjj ! !0

as n ! 1 implies that k=jj kjj ! !0 as k ! 1:

To show (b), note that we have shown that for any subsequence fpng of fng and any sequence f pn 2 : n 1g for which gpn( pn) ! g for some g 2 G; there exists

a sequence f n 2 : n 1g such that gn( n) ! g 2 G and pn = pn 8n 1:

This and Assumptions ACP(i) and ACP(ii) imply (b). This completes the proof of AsySz min_finfh2HCP (h); CP_1g:

Next, we show AsySz min_finfh2HCP (h); CP_{1g: First, we show that H equals} H =_{fh = (b; 0}) : n1=2 _{n ! b 2 R}d ; _{n ! 0 for some f n2} : n 1_gg: (9.1)

We have H H because 0 in H has 0 = 0 since n1=2jj njj ! jjbjj < 1: To show H H ; _{we need to show that for all b 2 R}d and 0 2 with 0 = 0 there exists a sequence f n2 : n 1g such that n1=2 n! b and n! 0:Take n= ( n; 0; 0; 0) with _n = b=n1=2 _{for n 1: Then, n}1=2

n = b for all n; n ! 0; and n 2 for n su¢ ciently large that b=n1=2_{< by Assumption ACP(iv).}

Given that H = H ; for any h 2 H; there exists a sequence f n2 : n 1g such that f ng 2 f 0; 0; b) by the de…nition of H : Then, AsySz = lim infn!1inf 2 CPn( ) lim inf_n!1CPn( n) = CP (h); where the last equality holds by Assumption ACP(i). There also exists a sequence f ng 2 ( 0;1; !0) such that CPn( n) ! CP1 by As- sumption ACP(iii). Thus, AsySz lim inf_n!1CPn( n) = CP1: Hence, AsySz min_finfh2HCP (h); CP_{1g as desired.}