Largest Entries of Sample Correlation Matrices from Equi-correlated Normal Populations

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1 Largest Etries of Saple Correlatio Matrices fro Equi-correlated Noral Populatios Jiaqig Fa ad Tiefeg Jiag Priceto Uiversity ad Uiversity of Miesota Abstract The paper studies the liitig distributio of the largest off-diagoal etry of the saple correlatio atrices of high-diesioal Gaussia populatios with equi-correlatio structure. Assue the etries of the populatio distributio have a coo correlatio coefficiet ρ > 0 ad both the populatio diesio p ad the saple size ted to ifiity with log p = o 3. As 0 < ρ <, we prove that the largest off-diagoal etry of the saple correlatio atrix coverges to a Gaussia distributio, ad the sae is true for the saple covariace atrix as 0 < ρ < /. This differs substatially fro a well-kow result for the idepedet case where ρ = 0, i which the above liitig distributio is a extree-value distributio. We the study the phase trasitio betwee these two liitig distributios ad idetify the regie of ρ where the trasitio occurs. It turs out that the thresholds of such a regie deped o ad coverge to zero. If ρ is less tha the threshold, larger tha the threshold or is equal to the threshold, the correspodig liitig distributio is the extree-value distributio, the Gaussia distributio ad a covolutio of the two distributios, respectively. The proofs rely o a subtle use of the Che-Stei Poisso approxiatio ethod, coditioig, a couplig to create idepedece ad a special property of saple correlatio atrices. The results are the applied to evaluatig the power of a high-diesioal testig proble of idetity correlatio atrix. Keywords: axiu saple correlatio, phase trasitio, ultivariate oral distributio, Gubel distributio, Che-Stei Poisso approxiatio. MSC00: Priary 6H0, 6E0; Secodary 60F05. E-ail address: jqfa@priceto.edu E-ail address: jiag040@u.edu

2 Itroductio The correlatio coefficiet atrix is a iportat statistic i the ultivariate aalysis. It plays pivotal roles i the statistical aalysis of a ultivariate oral data. The axiu likelihood estiator is the saple correlatio atrix. This paper ivestigates the liitig distributio of the largest off-diagoal etry of the saple correlatio atrix i the high-diesioal settig whe the correlatio atrix adits a copoud syetry structure, aely, is of equi-correlatio. Let N p 0, Σ stad for a p-variate oral populatio with the correlatio atrix R = ρ ij p p. Let X,, X be a rado saple fro the populatio N p 0, Σ. We have the data atrix X = X,, X. Write X = x ij p = x, x,, x p, the the Pearso correlatio coefficiet betwee x i ad x j is give by ˆρ ij = x ki x i x kj x j x ki x i x, i, j p,. kj x j where x i = x ki. I particular, ˆρ ii = for all i p. The saple correlatio atrix ˆR is the defied by ˆR = ˆρ ij p p. I cotrast, X X/ is refereed to as the saple covariace atrix. Defie the largest agitude of off-diagoal etries of the saple correlatio atrix by L 0 = ax ˆρ ij.. i<jp Assuig that x ij s are idepedet ad idetically distributed but ot ecessarily Gaussia-distributed, the asyptotic distributio of L 0 have bee extesively studied as both p ad ted to ifiity. The first result o the topic is due to Jiag 9], who uses the Che-Stei Poisso approxiatio ethod to get the liitig distributio of the L 0 as follows. Assue E x 30+ɛ < for soe ɛ > 0. Let p = p ad p, the P L 0 4 log + log log t exp γ e t/ 8π for ay t R, or equivaletly, P L 0 4 log p + log log p t exp γ 0, as 8π e t/..3 Zhou 9] proves that the oet coditio ca be relaxed to that li x x 6 P x x > p x = 0 ad li sup <. Li ad Rosalsky ] cosider the strog liit of L 0 uder soe ore relaxed assuptio. Li et al., 3] have further iproved the assuptio of the result, uder the assuptio that bouded away fro zero or p ifiity. They actually obtai soe ecessary ad sufficiet coditios for which.3 holds. As p/, Liu et al. 5] establish siilar results to.3 uder the assuptio p = O α where α is a costat. Cai ad Jiag 3] cosider the ultra-high

3 diesioal case where p ca be as large as e α for soe 0 < α ad they exted the result to depedet case. Cai ad Jiag 4] derive the liitig distributio of L 0 uder the assuptio that the populatio has a spherical distributio. I fact, a phase log p trasitio pheoeo occur at three differet regies: 0, log p α 0, ad log p. By usig the liitig distributio of L 0, Cai et al. ] work o the asyptotic behavior of the pairwise geodesic distaces aog rado poits that are idepedetly ad uiforly distributed o the uit sphere i the p-diesioal spaces. The sae phase trasitio pheoeo are also uderstood. Without the Gaussia assuptio, Shao ad Zhou 8] obtai siilar results to.3 as log p = o α for soe 0 < α. Assuig the p etries of x are idepedet, ost of the aforeetioed work aily focus o the iproveet of the oet assuptio o x fro the data atrix X = x ij p as well as relaxig the rage of p relative to. The questio of how depedece ipacts o the liitig distributio of the largest correlatios reais largely ukow. I this paper, we will cosider a case that all the etries of x are very depedet. I fact, we assue x N p µ, Σ, ad the correspodig correlatio atrix R has the copoud syetry structure, which is also referred to as the itraclass covariace or equi-correlatio structure i literature, that is, ρ ρ ρ ρ R = ρ ρ It is easy to see that R is positive defiite if ad oly if > ρ > /p. Sice we will be i the sceario that p = p, we will always assue ρ 0 later. Whe ρ > 0, the saple correlatios ˆρ ij, i < j p are highly depedet ad ew techical challeges arise i derivig the liitig distributio of the axiu value of these etries. I additio, we foud soewhat surprisigly that such a liitig distributio is Gaussia. This is i sharp cotrast to the idepedece case ρ = 0 i which the liitig distributio is a Gubel distributio. Where does the phase trasitio occur? I what way the liitig distributio chages over the regie of correlatio ρ? We provide sharp asyptotic results to describe these regies of ρ ad their associated liitig distributios of the axiu correlatio. Related to our study is the axiu spurious correlatio betwee each variable i X ad a idepedet variable Y i which the variables i X are correlated. Fa et al. 8] derived the asyptotic distributio of such a axiu spurious correlatio usig Gaussia approxiatio techiques of Cherozhukov et al. 6]. Uless the correlatio atrix of X is of a specific for, such a liitig distributio ca ot be aalytically derived ad they require a ultiplier bootstrap ethod to estiate the liitig distributio. Their settig relates to our case with the last row of off-diagoal correlatio equal to zero ad oly coputes the axiu saple correlatio i the last row, albeit these saple correlatios are also depedet due to the depedece of X. 3

4 d Soe otatios will be used i the paper. The sybol eas covergece i distributio, ξ = d η iplies that ξ ad η have the sae distributio. Furtherore, b = oa if b /a 0 ad that b = Oa if li sup b /a <. ξ = o p a if ξ /a 0 i probability as. Ad ξ = O p a if li C li sup P ξ /a > C = 0. I additio, we deote C ad C positive costats idepedet of or p, ad their values ay be differet fro lie to lie. The rest of the paper is orgaized as follows. Sectio gives the ai results, discussios ad a applicatio of the result. The proofs are relegated to Sectio 3, where we develop ecessary techical tools for our quests. Mai results ad discussios Let X,, X be a rado saple fro the populatio N p µ, Σ with the populatio correlatio atrix R defied as i.4. The data atrix is give by X = X,, X = x ij p. We will study the followig two statistics i this paper: J = ax i<jp where ˆρ ij is defied as i.. x ki x kj ad L = ax i<jp ˆρ ij,. The first is the axiu of oralized saple covariaces whe µ = 0, whereas the secod oe is the axiu of the saple correlatios. The purpose havig the oralizatio i J is such that J ad L have the sae scale. To ake our aalysis thoroughly, we allow ρ to deped o. We will see fro iii of Theores. ad. later o that J ad L behave differetly as ρ is a costat.. Mai results We first cosider the liitig distributio for the statistics J. Theore. Let ρ 0 for each ad sup ρ <. Assue µ = 0 ad Σ = R, where R is give by.4. Suppose p = p ad log p = o 3 as. Set µ = ρ + The followig holds as. i. If ρ log p 0, the log p log log p 4 ρ log p. 4 log p / J µ d φ where φ has distributio fuctio F x = e Ke x, x R with K = 4 π. 4

5 ii. If ρ log p λ 0,, the / J µ ρ d ξ + λ 0 φ, where ξ N0,, λ 0 = 4, φ is as i i ad φ is idepedet of ξ. λ iii. If ρ log p, the / J µ ρ d N0,. The above theore has the followig iplicatio. Corollary. Let ρ 0, be fixed, µ = 0 ad Σ = R, where R is as i.4. Suppose p = p ad log p = o 3 as. The / J µ ρ d N0, as, where µ = ρ + ρ log p. For the largest etry of the saple correlatio atrix ˆR, we have the followig. Theore. Let ρ 0 for each ad sup ρ <. Assue Σ = R, where R is as i.4. Let p = p satisfyig log p = o 3 as. Set µ = ρ + ρ + ρ ρ The followig holds as. i. If ρ log p 0, the 4 log p L µ d φ, log p log log p 4. log p where φ has distributio fuctio F x = e Ke x, x R with K = 4 π. ii. If ρ log p λ 0,, the L µ ρ d ξ + λ 0 φ, where ξ N0,, λ 0 = 4 ad φ is the sae as i i ad φ is idepedet of ξ. λ iii. If ρ log p, the L µ d N0,. ρ ρ 5

6 If ρ is close to zero, presuably the behavior of L is close to a extree-value distributio as i.3; if ρ is relatively large, L is asyptotically the oral distributio as stated i Theore.. Ite ii of the above theore actually gives the phase trasitio betwee the two cases. The followig is a easy cosequece of Theore.. Corollary. Let ρ 0, be fixed ad Σ = R, where R is as i.4. Suppose p = p ad log p = o 3 as. The, L µ /σ d N0, as, where µ = ρ + ρ + ρ ρ log p ad σ = ρ ρ. The above two results are totally differet fro Jiag 9], Zhou 9], Liu et al. 5], Li et al., 3], Cai ad Jiag 3, 4], Cai et al. ], Shao ad Zhou 8]. They all ed up with the Gubel distributio by arguig that ˆρ ij s are roughly idepedet rado variables. I Theores. ad., the appearace of ρ creates a strog depedecy aog the ters x kix kj, i < j p, i the defiitio of J fro.. This is also true for the ters ˆρ ij, i < j p. The occurrece of ρ akes the situatio so delicate that, if ρ is a costat, the liitig distributios of J ad L are o loger the Gubel distributio, they are the oral distributio istead. For J siilarly for L, a key differece betwee the case ρ = 0 ad the case ρ > 0 is explaied as follows. For ρ > 0, each ter of the deoiator i. ca o loger be regarded as roughly as that i the case ρ = 0. I particular, if ρ > 0 is a costat, the depedece really atters, ad the differece ca be see fro Corollary. by coparig the eas ad the variaces.. Discussios The paper ivestigates the liitig distributios of the largest off-diagoal etry of saple covariace/correlatio atrices geerated by a rado saple fro a highdiesioal oral distributio. We assue the oral distributio has the structure of equi-correlatio.4. Uder the assuptio that p ad log p = o /3, the asyptotical distributios of the largest off-diagoal etries of both atrices are established. Their behaviors deped o the value of ρ. The liits are the oral distributio if ρ is reasoably large; the liits are the extree-value distributio if ρ is very sall. We also figure out the regie to differetiate the two scearios. I particular, for ρ i the regie, the liitig distributio is the covolutio of the Gaussia distributio ad the extree-value distributio. Next we ake soe coets.. For the saple correlatio atrix ˆR, we get the liitig distributio of its largest etry for each ρ 0,. The sae result holds for the saple covariace atrix but uder the ore striget restrictio 0 ρ < /, which is required i Lea 3.9. This differece will be easily uderstood by the fact that the saple correlatio atrix ca be regarded as a type of self-oralized statistics. It is kow that self-oralized statistics are ore taed ; see, for exaple, Shao ad Wag 7]. Ad hece the rage of ρ is ore relaxed i the case of the saple correlatio atrix tha that i the 6

7 case of the saple covariace atrix. We do ot kow whether or ot Theore. is still true for the case ρ,. It is a iterestig project for future.. Uder the Gaussia assuptio ad that for the equi-correlatio R i.4, the decopositio structure of 3.77, i.e., X = ρξ,, ξ + ρξ,, ξ p. where ξ, ξ,, ξ p are idepedet stadard Gaussia rado variables, plays a key role i the proofs. Now let us reove the Gaussia assuptio. Istead, we assue the decopositio. cotiues to hold with ξ, ξ,, ξ p relaxed to idepedet rado variables with ea zero, variace oe, ad a sub-gaussia tail. The Theores. ad. ay also hold except ii fro each theore. The coclusio of ii is derived by usig the Gaussia assuptio essetially. 3. The paper deals with the equi-correlatio atrix. If R = r ij has aother ˆr special structure, oe ay like to work o ax i<jp ˆr ij or ax ij i<jp r ij. It sees that, to get good properties for these two quatities, R ca ot be assued to be too arbitrary. 4. Assuig ρ = 0, Jiag 0] obtais the liitig spectral distributio of the saple correlatio atrix ˆR. Whe /p c 0,, the author proves that the epirical spectral distributio of ˆR asyptotically obeys the Marcheko-Pastur law. If 0 < ρ <, by usig the decopositio 3.77, it ca be show easily that the spectral distributio of the saple covariace atrix also takes the Marcheko-Pastur law as its liit. A siilar result is expected for correlatio atrix ˆR for the case ρ > 0 by eployig the approxiatio ethod fro Jiag 0]. 5. Methodology of our proofs. The key eleets i our proofs are a special property for saple correlatio atrices uder Gaussia assuptios, the Che-Stei Poisso approxiatio ethod, coditioig arguets ad a couplig to create idepedece. Let us take L fro Theore. to elaborate this ext through a few steps. a. The special property Lea 3. for saple correlatio atrices allows us to reove x i ad x j fro the expressio ˆρ ij i.. So we get a easier for of the target to work with. b. With soe efforts, we are able to write L = α + β Q + γ R.3 where α, β, γ are costats, Q goes to N0,, R the quatity M fro Propositio 3. is the axiu of sus of idepedet but o-idetically distributed rado variables; see c. We use the Che-Stei Poisso approxiatio ethod to work o R. However, due to the strog depedecy, we are ot able to apply the ethod directly. I particular, the ethods for derivig the liitig distributio of R uder the assuptio ρ = 0 i all earlier literature are o loger valid. We will use a coditioig trick. I fact, coditioig o certai evet, we obtai the asyptotic distributio of R by the Che-Stei ethod. After takig the expectatio of the coditioal probability, we fially derive the liitig distributio of R Propositio 3.. d. We costruct R such that it is idepedet of Q i.3 see Lea 3.7 ad it has the sae asyptotic distributio as that of R. Furtherore, we show that 7

8 the differece betwee L ad L := α + β Q + γ R is egligible. So, basically speakig, L is reduced to a liear cobiatio of two idepedet rado variables such that oe goes to the oral distributio ad aother goes to the extree-value distributio..3 A applicatio to a high-diesioal test Let X,, X be a rado saple fro the populatio N p µ, Σ. We are iterested i testig whether Σ is diagoal. A atural oparaetric test is to use the test statistic L, which is powerful for sparse alteratives. The ull distributio of such a test statistic correspods to the liitig distributio for case ρ = 0 i regie i of Theore.. A questio arises aturally how powerful it is uder the dese alteratives. The specific alterative of iterest is H 0 : ρ = 0 vs H : ρ = ρ where ρ 0, is give. Assue the diesio p ad saple size are all very large such that log p = o /3. By i of Theore., uder H 0, 4 log p L µ 0 d φ, where µ 0 = log log p log p 4 ad φ has distributio fuctio F x = x log p e Ke, x R ad K = 4. For 0 < α <, deote q π α the α-quatile of the distributio F x, that is, q α = log3π log log α. The, a rejectio regio of the asyptotic size-α test is give by { X 0 = L log p + q α log log p/4 } log p. Usig Theore. i agai, whe ρ = o/ log p, the asyptotic power is still α, like a rado guess, as the asyptotic distributio uder such a cotiguous alterative hypothesis is the sae as that uder the ull hypothesis. Now the power starts to eerge whe ρ = λ/ log p for λ 0, i regie ii. I this case, it ca be calculated that µ i this regie is The power fuctio is µ := λ log p + λ log p + O log p 3/ ] µ 0. βρ = P { L µ 0 + q α /4 log p ρ } = P { 4 log p L µ q α 4λ + 6λ + o ρ }. Accordig to Theore.ii, the power teds to for each fixed λ. By usig a siilar arguet, it is easy to show that the power i regio iii has also asyptotic power. 8

9 3 Proofs The proofs of Theores. ad. are quite ivolved. We break the ito sall sectios fro each of which probles with a coo feature are hadled together. See the detail of each sectio give at the ed of the Itroductio. 3. A result o saple correlatio atrices I the followig, we will preset a special property of the saple correlatio atrix ˆR as defied below.. A auxiliary fact has to be derived first. Lea 3. Let X,, X be i.i.d. rado vectors ad X N p 0, Σ, where Σ is a p p o-egative defiite atrix. Set X = X,, X. The, for ay orthogoal atrix O, we have OX d = X. Proof. Let Y,, Y be i.i.d. ad Y N p 0, I p. The X i ad Σ / Y i have the sae distributio for each i. By idepedece, As a cosequece, X = X,, X d = Y,, Y Σ /. 3. OX d = OY,, Y Σ / for ay orthogoal atrix O. Write Y,, Y = y ij p. The y ij s are i.i.d. N0, -distributed rado variables. Hece OY,, Y d = Y,, Y by the orthogoal ivariace of idepedet Gaussia rado variables. It follows that OX d = Y,, Y Σ / d = X by 3.. The followig lea provides a siple expressio for the saple correlatio atrix. Lea 3. Let X,, X be i.i.d. rado vectors ad X N p µ, Σ where µ R p ad Σ is a positive defiite atrix. Let ˆρ ij be as i.. Suppose Y,, Y are i.i.d. ad Y N p 0, Σ. Write Y,, Y = V,, V p p. The ˆρ ij d V i V j p p = V i V j Proof. Sice ˆρ ij is ivariat uder traslatio ad scalig of the vectors X,, X, without loss geerality, we assue µ = 0 ext. Deotes I =,,, R ad A = I II. Trivially, A is a idepotet atrix with tra =, the there exists a orthogoal atrix O such that A = O I 0 O p p.

10 Write x j x j x j x j. = A x j x j. x j x j x j for each j p. Write X = X,, X = x ij p. The x x x x x p x p x x x x x p x p H :=... x x x x x p x p = O I 0 OX 0 0 d = O I 0 X 0 0 by Lea 3.. The I 0 X := X = 0 0 xij p 0 where 0 above is a p-diesioal row vector with all etries equal to zero. Therefore, H d = O X ad hece H H d = X X = xij px ij p. Defie x ij p = V,, V p p. The above iplies H H d = V i V j p p. 3. For a positive defiite atrix M = ij p p, defie hm to be a p p atrix such that its i, j-etry is equal to ij / ii / jj. Let M p p be the set of all p p positive defiite atrices. The, h : M p p M p p is cotiuous ap, ad therefore is Borel-easurable ap. Fro 3. we coclude hh H = d h V i V j p p. The desired coclusio the follows. 3. Soe techical tools We will collect ad prove soe techical tools for the proof of Theores. ad.. The first oe is the Che-Stei Poisso approxiatio ethod, which is a special case of Theore fro Arratia, Goldstei ad Gordo ]. Lea 3.3 Let η α be rado variables o a idex set I ad {B α, α I} be a set of subsets of I, that is, for each α I, B α I. For ay t R, set λ = α I P η α > t, The we have P ax η α t e λ λ b + b + b 3, α I 0

11 where b = P η α > tp η β > t, α I β B α b = P η α > t, η β > t, α I α β B α b 3 = P {ηα > t ση β, β / B α } P η α > t, α I ad ση β, β / B α is the σ-algebra geerated by {η β, β / B α }. I particular, if η α is idepedet of {η β, β / B α } for each α, the b 3 vaishes. The ext lea is o the oderatio deviatio of the partial su of i.i.d. rado variables. It ca be see, for istace, fro Liik 4]. Lea 3.4 Suppose {ζ, ζ, ζ, } is a sequece of i.i.d. rado variables with Eζ = 0 ad Eζ =. Defie S = i= ζ i. If Ee t 0 ζ α < for soe 0 < α ad t 0 > 0, the S li log P x = for ay x, x = o x α α. The followig lea is o the oderatio deviatio of the partial su of the idepedet but ot ecessarily idetically distributed rado variables. It ca be see i Propositio 4.5 fro Che et al. 5]. Lea 3.5 Let η i, i be idepedet rado variables with Eη i = 0 ad Ee h ηi < for soe h > 0 ad i. Assue that i= Eη i =. The P i= η i x Φx = + C + x 3 γe 4x3 γ for all 0 x h ad γ = i= E η i 3 e x η i, where sup C C ad C is a absolute costat. I our fraework, η i above is a quadratic for of two idepedet orals for each i. We first eed to cotrol E η i 3 e x η i. Lea 3.6 Let U ad V be i.i.d. N0, -distributed rado variables. Let a, b, c, d, e, f be real ubers. Set η = au + buv + cv + du + ev + f. The E η 3 e x η C a 3 + b 3 + c 3 + d 3 + e 3 + f 3 e d +e x + f x as 0 < x, where C is costat ot depedig o a, b, c, d, e or f. a + b + c

12 Proof. First, use UV U + V to see I particular, η a + b U + b + c V + du + ev + f. 3.3 E η E a + b 9 U 8 + b + c 9 V 8 + du + ev 9 + f 9] C a + b 9 + b + c 9 + d + e 9/ + f 9] C a + b + c 9 + d + e 9 + f 9] where C is a costat ot depedig o a, b, c, d, e or f. We also use the facts EU 8 + V 8 < ad du + ev d = d + e U. It follows that Fro 3.3, E η 9 /3 C /3 a + b + c + d + e + f 3. E η 3 e x η E η 9 /3 E exp 3x a + b U + 3x b + c V ] /3 E exp3x du + ev ] /3 e x f. First, Ee 3x du+ev = Ee 3x d +e U Ee 3x d +e U + Ee 3x d +e U = e 9x d +e / by usig the idetity Ee tn0, = e t / for all t R. Secod, settig α = 3x a + b ad β = 3x b + c, ad reviewig Ee su = s / for all s <, we have E exp 3x a + b U + 3x b + c V = α β 4 if α 4 ad β 4 by idepedece. Fially, cobiig the above, we see E η 3 e x η C a + b + c + d + e + f 3 e d +e x + f x as 0 < x. The coclusio the coes fro a iequality o covex fuctio fx := x 3 for x 0. a + b + c I our settig, the paraeter γ fro Lea 3.5 eeds a special care. This will be doe below with the help of Lea 3.6. Lea 3.7 Let {ξ k ; k } be i.i.d. N0, -distributed rado variables. Set τ = E ξ 3 +. Assue p = p satisfies that p ad log p = o /3. Let {y > 0; } be real ubers such that y = Olog p. The, P + ξ k 3 e yξ k / τ as is sufficietly large. exp 4 / log

13 Proof. By assuptio, we assue y N 0 log p for all, where N 0 > 0 is a costat. For ɛ > 0, set Θ ɛ = {ax k ξk ɛ/y }. By the iequality P N0, y πy e y / e y / for all y, there exists a costat such that P Θ c ɛ P ξ > ɛ/y / exp ɛ y as, which is agai bouded by exp ɛ N 0 e /3 log p as ɛ, where ɛ is a iteger depedig o ɛ. It follows that P P + ξ k 3 e yξ k / τ + ξ k 3 e ɛ τ + e /3 as ɛ. Take ɛ = log 4. The 3 e ɛ τ = 3 τ. Cosequetly, P = P P + ξ k 3 e ɛ τ ξ k 3 E ξ k 3 τ ζ k x as is sufficietly large, where ζ k = ξ k 3 E ξ k 3 / Varξ 3 ad x = /4 / log. Set σ = Varξ. 3 Observe that σ /3 ζ k /3 ξ k + E ξ k 3 /3. This iplies E exp 4 σ/3 ζ k /3 < sice ξ k N0,. Take α = i Lea 3.4 to see 3 P as is sufficietly large. I suary, P + ξ k 3 e yξ k / τ This iplies the desired iequality. ζ k x exp 4 / log exp 4 / log + e /3. The followig result provides us with a equivalet expressio o a liit theore. It will be applied to the proofs of Propositios 3. ad 3. later, i which F x is a extree-value distributio. 3

14 Lea 3.8 Let M be a rado variable for each satisfyig li P M 4 log p log log p + x = F x for ay x R, where F x is a cotiuous distributio fuctio o R. The M = log p log log p 4 log p + 4 log p U, where U coverges weakly to a probability easure with distributio fuctio F x. Proof. Easily, + t / = + t + rt where sup t <ɛ rt t for soe ɛ > 0. Fix x 0 R. Let A 0 > 0 be give. For ay x x 0 A 0, x 0 + A 0 ], / log log p 4 log p log log p + x = log p 4 log p + x 4 log p = log log p log p 8 log p + x ] 8 log p + rp, x where log log p sup rp, x sup x x 0 A 0 x x 0 A 0 4 log p x 4 log p log log p 5log p as is large eough. By the give coditio, li P M log log p log p 4 log p + x 4 log p + sp, x = F x 3.4 as, where sp, x := rp, x log p ad as is sufficietly large. Defie The 3.4 iplies that sup sp, x x x 0 A 0 U = 4 log p M log p + log log p 7log p 3/ 3.5 log log p log p li P U x + tp, x = F x 3.7 where tp, x := 4sp, x log p. Easily, fro 3.5, sup tp, x x x 0 A 0 4 log log p log p

15 as is sufficietly large. Therefore, for ay δ > 0, P U x δ P U x + tp, x P U x + δ as is sufficietly large. Fro 3.7, li sup P U x δ F x li if P U x + δ for ay x x 0 A 0, x 0 + A 0 ]. For δ 0, A 0, takig x = x 0 + δ ad x = x 0 δ, respectively, we have li sup P U x 0 F x 0 + δ; li if P U x 0 F x 0 δ. Lettig δ 0, we obtai li P U x 0 = F x 0. Sice x 0 R is arbitrary, the liit together with 3.6 cocludes the proof. 3.3 A propositio o the largest etry of a saple covariace atrix I this sectio, we will use the Che-Stei Poisso approxiatio ethod to get the asyptotical distributio of a statistic M defied i Propositio 3. later. The quatity M will serves as a key buildig block to uderstad the largest etry of a saple covariace atrix. Literally, it will be used i the proof of Theore.. For coveiece, the followig otatio will be used throughout the rest of the paper.. The rado variables {ξ k, ξ k, ξ ki ; k, i =,, } are i.i.d. with N0, -distributio Give ρ 0, for each, set ρ = ρ, ρ a = ρ ad b = ρ + ρ 3. For x R ad iteger p, set s p = 4 log p log log p + x. 3.0 I our theores we assue p, so s p is well-defied as p is large. This clarificatio will ot repeated i the future. 4. Let ξ i s be as i 3.8. we write ξ = ξ,, ξ ad ξ = ξ + + ξ /. 3. Before statig the ai result i this sectio, we will first establish a techical tool, which will play a key role i the proof of the Lea 3. i the sequel. 5

16 Lea 3.9 Review the otatios i Assue ρ 0 for all ad sup ρ <. Defie Z = b ξ kξ k. If p = p ad log p = o /3 as, the there exists a costat δ 0, such that as. E exp + ρ ] + δ Z s p = o p 3 Proof. If ρ = 0 for soe, the Z = 0 ad the expectatio i the lea is idetical to exp 4 log p log log p + x/ + δ, which, by takig δ 0, sall eough, is bouded by p 3.5 as is sufficietly large. Therefore, to prove the lea, w.l.o.g., we assue ρ > 0 for all. First, we show Ee α ξ β = for ay α > 0 ad β R. I fact Ee α ξ β = π α + exp α β α + e α x β x dx. 3. Write α x β x = α + x α β α β α + α +. Now, defie y such that y = α + x α β. α + It follows that π = exp α β α + e α x β x dx π e y / dy α +. Thus, 3. holds. Recall the otatio 3.. By Propositio 7.3 fro Eato 7] or Theore.5.6 fro Murihead 6], we kow ξ ad ξ are idepedet. Also, ξ ξ ξ kξ k N0, by idepedece. Cosequetly, Z d = b ξ ξ

17 I particular, ξ ad ξ are idepedet. Let τ = +δ +ρ. Observe E exp Z s p ] = Ee α ξ β = E ] E e α ξ β τ where E stads for the coditioal expectatio give ξ, 3.4 α = b ξ τ ad β = sp b ξ. By usig 3., we obtai E e α ξ β exp { exp s p τ + b ξ s p + δ + ρ + b ] } 3.5 if ξ < + δ. Observe that + ρ + b = +ρ +ρ +ρ < 4 for all, where +ρ 3 +ρ by assuptio. Take δ 0, such that θ := + δ < 4. +ρ 3 ρ := sup ρ < Hece, give ξ < + δ, E e α ξ β log p5/θ p 4/θ as is sufficietly large. By the large deviatios for i.i.d. rado variables, there exists a costat C > 0 depedig o τ oly such that P ξ + δ < e C for all. Cobiig the above iequality, 3.4 ad 3.5, we arrive at E exp Z s p ] τ = E E e α ξ β ξ ] I < + δ log p5/θ + e C = o p 4/θ p 3 ξ + P + δ where the last equality follows fro the assuptio log p = o /3. Now we state the ai result i the sectio. Review the otatios i Defie for all i < j p. η kij = a ξ ki ξ kj + b ξ k ξ ki + ξ kj ; 3.6 M ij = η kij 3.7 7

18 Propositio 3. Let ρ 0 for each ad sup ρ <. Let s p be as i 3.0. Set M = ax i<jp M ij. If p = p ad log p = o /3, the for ay x R, where K = 4 π. li P M s p = e Ke x/ Proof. I the ext we will assue p is large eough such that s p > 0. Set I = {i, j; i < j p}. For α = i, j I, defie X α = M ij ad B α = {k, l I; either k {i, j} or l {i, j}, but k, l α}. Let P ad E stad for the coditioal probability ad the coditioal expectatio give {ξ k ; k }, respectively. The crucial poit is that, give {ξ k ; k }, rado variable X α is idepedet of {X β ; β / B α }. Sice {X α, α I} are idetically distributed uder P, by Lea 3.3, we have P ax X α s p e λ p w + w, 3.8 α I where λ p = pp P η k > s p ad ad w = P X α > s p P X β > s p α I β B α pp p P η k > s p w = P X α > s p, X β > s p α I β B α pp p P η k > s p, η k3 > s p. Note that P ax α I X α s p = EP ax α I X α s p. Fro 3.8, P ax X α s p Ee λ p E P ax X α s p e λ p α I α I Ew + Ew. 8

19 Now, Ee λ p = E exp Ew p 3 E P Ew p 3 P pp P η k > s p ]; η k > s p ]; η k > s p, η k3 > s p. The followig three leas say that Ee λ p exp 4 e x/, Ew π 0 ad Ew 0. The proof is the copleted. Lea 3.0 Let the assuptios i Propositio 3. hold. Review that P stads for the coditioal probability give {ξ k ; k }. The E exp as for all x R. pp P ] η k > s p exp 4 π e x/ Lea 3. Let the assuptios i Propositio 3. hold. Review that P stads for the coditioal probability give {ξ k ; k }. The as. E P ] η k > s p = o p 3 Lea 3. Let the assuptios i Propositio 3. hold. The as. P η k > s p, Now we start to prove the three results oe by oe. η k3 > s p = o p 3 Proof of Lea 3.0. Write η k = a ξ k ξ k + b ξ k ξ k + ξ k ]. 3.9 Give {ξ k ; k }, it is the su of idepedet rado variables with ea E a ξ k ξ k + b ξ k ξ k + ξ k ] = 0 ad variace Var a ξ k ξ k + b ξ k ξ k + ξ k ] = a + b ξ k. Thus, Var η k = a + b 9 ξk. 3.0

20 Defie { F = ax ξ k ad 6 k 7 ξk 5 }. 4 Set τ = E ξ 3 +. For v > 0, set { } G v = + ξ k 3 e vξ k log p/ τ. The paraeter v will be chose later. By the fact P N0, x π x e x / for all x > 0, the large deviatios for i.i.d. rado variables ad Lea 3.7, we have P F G v c P ξ + P ξk 6 7, 5 ] c 4 + P + ξ k 3 e vξ k log p/ > τ 3 exp 4 / log 3. as v, where v is a costat depedig o v. Defie σ 0 = a +b ξ k. The, o F, = a + b σ 0 a b 8 7, 3. where the last iequality follows fro the idetity a + b =. Next we will use Lea 3.5 to get a precise estiate o P η k > s p. To do so, Lea 3.6 will be applied to cotrol γ defied i Lea 3.5. Reviewig 3.9, we take a = a σ0, b = bξ k σ0. Set η k = aξ k ξ k + bξ k + ξ k. The, it follows fro 3.0 that E η k = 0 ad Var η k = 3.3 for each k. Furtherore, by 3. we have a ad b ξ k 3.4 o F. The, o F, use the Hölder iequality, the facts that ξ ξ ξ + ξ ad ξ + ξ N0,, ad idepedece to see h Ee h ηk E exp ξ ξ + h ξ + ξ h ] / ] E exp ξ + ξ / E exp4 hn0, h = E exp N0, e 6h < 3.5 0

21 for all h, k, satisfyig 0 < h h := 8 ad k. Now, o F, by Lea 3.6 ad 3.4 we have E ηk 3 e x η k C + ξ k 3 4bx e 3/ C + ξ k 3 e 6xξ k / 3.6 3/ for all x 0,. Observe that 0, a 4 0, o F a by 3.4. Thus, 3.6 particularly holds for all x 0,. Now take x 4 0 = sp σ 0. The x 0 s p < o F by the assuptio log p = o /3. We the have γ : = E ηk 3 e x 0 η k C 3/ C 3/ + ξ k 3 e 6x 0 ξ k / + ξ k 3 e 56ξ k log p/ o F. Thus, γ Cτ o F G 56 := H. The iequality i 3. iplies P H c 3 exp 4 / log 3.8 as is sufficietly large. Fro 3.3, 3.5 ad Lea 3.5, we coclude P η k > s p = P η k > x 0 = Φx 0 ] + O + x 3 0γe 4x3 0 γ] 3.9 o H sice x 0 < h = 8 by 3.7. Fially, x 3 0 γ = Os 3 p / 0 o H by the assuptio log p = o /3. Reviewig 3., we have sp sp σ 0 s p o H. Hece, fro the forula P N0, x = π x e x / + o as x we obtai that, o H, P η k > s p = = Φ s p ] log 3/ p ] + O σ 0 σ 0 e s p /σ 0 + o 3.30 π sp

22 as, where the last ter o does ot deped o ξ k s. To prove the lea, it is eough to show p P η k > s p 4 π e x/ i probability as. Sice P H, to fiish the proof, it suffices to check p P η k > s p I H 4 π e x/ i probability as. Now σ 0 i probability as ad s p log p, coparig this with 3.30, it suffices to show p 4 π log p e s p /σ 0 I H 4 π e x/ 3.3 i probability. By the cetral liit theore for i.i.d. rado variables, σ0 = + O p. Hece σ0 = + O p. It follows that s p σ 0 = log p log log p + x + O p ] = log p log log p + x + o p by the coditio log p/ /3 0. This iplies 3.3. Proof of Lea 3.. Review the proof of Lea 3.0. Let H be defied as above 3.8. By 3.9, there exists a costat ot depedig o ξ k s such that P η k > s p I H Φ s p ] I H σ 0 as sice x 0 = sp σ 0. Recall the iequality Φx π x e x / for all x > 0. The, fro 3. we have P ] η k > s p IH C σ 0 e s s p /σ 0 IH p C log p e 7s p /8 as p, where is a costat ot depedig o ξ k s. Therefore, cobiig this with 3.8, we see ] E P η k > s p E P ] η k > s p IH + P H c p exp 4 / log

23 as is sufficietly large sice log p = o /3. This proves the lea. Proof of Lea 3.. Let P 3 ad E 3 stad for the coditioal probability ad the coditioal expectatio give {ξ k, ξ k ; k }, respectively. By idepedece, P η k > s p, = E P 3 η k3 > s p η k > s p ]. 3.3 Recall the otatios i 3.6 ad 3.7. Write η k = b ξ kξ k + a ξ k + b ξ k ξ k. The, give {ξ k, ξ k ; k }, we have fro idepedece that η k Nµ 0, σ where µ 0 = b ξ k ξ k ad σ0 = a ξ k + b ξ k. Trivially, b = ρ ρ +ρ sup +ρ := κ < ad 3 a + b = +ρ, ] for all ρ 0,. Defie A = { µ 0 < 3s p /} ad B δ = { δ < σ 0 a + b } < + δ for δ 0,. Observe a ξ + b ξ d = a + b ξ sice ξ ad ξ are i.i.d. N0, - σ 0 distributed rado variables. Thus, = a +b ξ k. The, by the large deviatios for the su of i.i.d. rado variables, we obtai P Bδ c = P d ξk δ, + δ] c e C δ 3.34 for all δ 0, where C δ > 0 for each δ 0,. Siilarly, {ξ k ξ k ; k } are i.i.d. with ea zero ad variace oe. Notice ξ ξ ξ + ξ. Therefore E exp ξ ξ <. Fro Lea 3.4 ad the fact s p log p = o /6 we see that, for ay ɛ 0,, P A c 3sp P ξ k ξ k κ 3sp = P ξ k ξ k κ exp ɛ 3s p 4κ 3

24 is large eough. Sice κ <, we choose ɛ = 3 κ. The ɛ = + κ iplies that P A c exp ɛ κ 3s p = o 4 p 3 4κ >. This as. It is easy to see that s p µ 0 o A. By the iequality P N0, y πy e y / / e y for all y, we have fro 3.33 that, o A B δ, P 3 η k s p = P 3 Nµ 0, σ 0 s p = P 3 N0, s p µ 0 σ 0 exp s p µ 0 σ 0. Note that σ 0 < + δa + b = +δ +ρ o B δ. Therefore, o A B δ, P 3 η k > s p Review 3.3. We the have exp + ρ + δ s p µ 0 P η k > s p, E P 3 η k3 > s p IA ] η k > s p c Bδ c P A c + P Bδ c + E exp o + e C δ + E exp p 3 + E exp + ρ + δ s p µ 0 + ρ + δ s p µ 0 + ρ + δ µ 0 s p. By Lea 3.9, choosig δ > 0 sall eough, we kow the last expectatio is idetical to o. The desired coclusio follows fro the assuptio log p = o /3. p A propositio o the largest etry of a saple correlatio atrix Siilar to Sectio 3.3, we ow study a statistic M, which is essetially a key quatity to uderstad the largest etry of a saple correlatio atrix. The ai result is Propositio 3., which will be used i the proof of Theore.. 4

25 Review the otatios fro 3.8 ad 3.9. Throughout this sectio, we assue σ = ρ + ρ a. Set a = a σ ad b = ρ b σ ; 3.35 γ k = ρ a ξ k + b ξ k ξ k 3.36 for k :=. Set V = γ + + γ /. Siilar to Lea 3.9, the followig techical result studies the behavior of the oet geeratig fuctio of a rado variable. It will be used i the proof of Lea 3.6. Lea 3.3 Let ρ 0, be costats. Suppose p = p ad log p = o /3 as. Let s p be as i 3.0. The, there exists δ 0, such that { E I K exp δ ]} V ω s p = o 3.37 p 3 as, where K := {0 < V < 7 ω s p } ad ω := Varγ. Proof. First, if ρ = 0 for soe, the γ k = 0 for all k. Hece V = 0 ad the expectatio i 3.37 is zero by the defiitio of K. So it is eough to prove the coclusio by assuig ρ > 0 for all. The proof is divided ito a few of steps. Step. Reductio of K to a saller set. Fro the defiitios of a ad b i 3.35, it is easy to check that Trivially, we have ω = ρ a + b. Therefore, + ρ a + b = ω = a ρ +ρ = ρ + ρ ρ +ρ = < + ρ ρ 4 because + x x < for all x 0,. I particular, ω 5 ω > 9 5 ω > This iplies that { E I0 < V 5 ω s p exp δ ]} V ω s p exp δ ω 5 ω ] s p ] exp 3. δ log p = o p

26 as if δ > 0 is sall eough. Therefore, it is eough to prove 3.37 with K beig replaced by K = { ω 5 s p < V < 7 ω s p }. Step. The tail probability of V. By the forula ω = ρ a + b agai, ρ a = ω + b ρ a ad b = + ω ρ a b. Recall γ k i Set γ k = γ k/ ω for k. The above iplies that γ k ξ k + + ξ k ξ k ξ k + ξ k +. I additio, γ k s are i.i.d. with ea zero ad satisfy Varγ k =. Also, Eet γ k < if 0 < t < 4. Observe γ := i= E γ k 3 e x γ k E ξ + ξ + 3 e x ξ +ξ+] C for all 0 x 4, where C = E ξ + ξ + 3 e 4 ξ +ξ +] <. By Lea 3.5, P V x = P γ k x Φ x ] 3.40 ω ω i= provided x ω 3 0 ad 0 x ω 4. I particular, by the assuptio log p = o /3 ad the fact P N0, t π t e t / for all t > 0 agai, we have P V x e x /ω 3.4 for all 5 ω s p < x < 7 ω s p. Step 3. The estiate of the expectatio fro Let A, B ad α > 0 be costats. Assue A, B ] 0, s p ]. Notice d e α x sp = α dx s p x e α x s p, we have v e α v s p = e α A s p + α s p x e α x s p dx A e α A s p + α s p e α x s p IA x v dx 0 6

27 for ay s p > v > A. Replacig v with V, the ultiplyig both sides of the above by IA < V < B, we get e α V s p IA < V < B e α A s p + α s p e α x s p IA x V < B dx 0 B e α A s p + α s p IV x dx. A e α x s p Set A = 5 ω s p ad B = 7 ω s p. By takig expectatios o both sides of the above, we obtai fro 3.4 that E e α V s p IA < V < B ] e α A s p + α s p B A e α x s p P V x dx B e α A s p + α s p exp α x s p x Now we evaluate the itegral. Write α x s p x ω Now, defie y such that It follows that A = α + x ω y = α s p B A = α s p exp α + x ω α s p α + ω α s p α + ω exp α x s p x 8πα s p exp α + ω α s p α ω + ω B α s p α ω + dx ω α s p α ω +. dx e y / dy A α + ω π e y / dy where A ad B are the correspodig values of y i 3.43 as x = A ad B, respectively. This cobiig with 3.4 iplies E e α V s p IA < V < B ] e α A s p + s 4πα s p p exp ω + α, 7

28 where the iequality α α + ω / α / is used. Take α = δ. The ω δ < α < 4 by Note 3 ω + α = ω + ω δ + ω δ + ω δ, where ω := sup ω. Fro 3.39, we kow A 4 s 0 p. This cocludes { E I K exp δ ]} V ω s p exp δ ] 0 s p s p + 6πlog p exp δ + ω s p. The first ter o the right had side is op 3 if δ > 3, which is true if 0 < δ < ; the secod ter is 7 op 3 as log as δ > 3, which is equivalet to that +ω 4 0 < δ < ω. The desired coclusio the follows fro the fact ω. 4 Let us cotiue to use the otatios before Lea 3.3. Set = ad η kij = a ξ ki ξ kj ρ ] ξ ki + ξkj + b ξ k ξ ki + ξ kj ; M ij = η kij for k =,,. The ai result of this sectio is give below. Propositio 3. Set M = ax i<jp M ij. Let ρ 0, for each. Let s p be as i 3.0. If p ad log p = o /3, the for ay x R, where K = 4 π. li P M s p = e Ke x/ Proof. The strategy of the proof is siilar to that of Propositio 3.. However, the techical details are ore ivolved. Let I, s p ad B α be as i the proof of Propositio 3.. For α = i, j I, defie X α = M ij. Let P ad E stad for the coditioal probability ad the coditioal expectatio give {ξ k ; k }, respectively. Agai, the key observatio is that, give {ξ k ; k }, rado variable X α is idepedet of {X β ; β / B α }. Sice {X α, α I} are idetically distributed uder P, by Lea 3.3, we have P ax X α s p e λ p w + w, 3.44 α I where λ p = pp P η k > s p 8

29 ad ad w = P X α > s p P X β > s p α I β B α pp p P η k > s p w = P X α > s p, X β > s p α I β B α pp p P η k > s p, η k3 > s p. Note that P ax α I X α s p = EP ax α I X α s p. Fro 3.44, P ax X α s p Ee λ p E P ax X α s p e λ p α I α I Obviously, Ee λ p = E exp Ew + Ew. pp P η k > s p ]; Ew p 3 E P η k > s p ]; Ew p 3 P η k > s p, η k3 > s p. The followig three leas say that Ee λ p exp 4 e x/, Ew π 0 ad Ew 0. The proof is the copleted. Lea 3.4 Let the assuptios i Propositio 3. hold. Review = ad P stads for the coditioal probability give {ξ k ; k }. The as for all x R. E exp pp P exp 4 π e x/ ] η k > s p 3.45 Lea 3.5 Let the assuptios i Propositio 3. hold. Review = ad P stads for the coditioal probability give {ξ k ; k }. The as. E P ] η k > s p = o p 3 9

30 Lea 3.6 Let the assuptios i Propositio 3. hold. Review =. The as. P η k > s p, Now we start to prove the three results oe by oe. η k3 > s p = o p 3 Proof of Lea 3.4. We will get a sharp estiate o P η k > s p first by usig Lea 3.5. To carry o this, we have to check the required coditios. Step : the behaviors of η k. Write η k = { a ξ k ξ k ρ ] } ξ k + ξk + b ξ k ξ k + ξ k. Give {ξ k ; k }, it is the su of idepedet rado variables. It is easy to check that E η k = 0, Var η kij = a + ρ + b ξ k. So the coditioal variace Var η k = + ρ a + b ξk Set { F = ax ξ k ad 6 k 7 ξk 5 }. 4 Recall the otatio τ = E ξ 3 + defied earlier. For v > 0, defie { G v = } + ξ k 3 e vξ k log p/ τ The paraeter v will be chose later. The iequality 3. says that P F G v c 3 exp 4 / log 3.48 as v, where v is a costat depedig o v oly. Defie Note The, o F, σ = + ρ a + b ξk = + ρ a + b ] σ + ρ a + b

31 Next we will use Lea 3.5 to get a precise estiate o P η k > s p. To do so, set a ρ a = c = ; b = a ; σ σ d = e = b ξ k σ ; f = a ρ σ ad η k = a ξ k + b ξ k ξ k + c ξ k + d ξ k + e ξ k + f. The, it follows fro 3.46 that E η k = 0 for each k ad Var η k =. 3.5 Furtherore, fro 3.49, σ ax{a, b } o F. The ax{ a, b, c, f } ad d = e ξ k 3.5 o F. Hece, o F, η k ξk + ξ k + ξ k + ξ k +. By the fact ξ k + ξ k N0, ad idepedece, Ee h η k h e h/ E exp ξ k + ξk + h ξ + ξ e /6 4h ] / ] E exp ξ k + ξk / E exp hn0, 4h = E exp N0, e h < 3.53 for all h, k, satisfyig 0 < h h := 6 ad k. Now, o F, by Lea 3.6, 3.50 ad 3.5 we have E η k 3 e x η k C + ξ k 3 e 4xξ k / e x/ 3/ C 3/ + ξ k 3 e 4x ξ k / 3.54 if 0 < x, where C here ad later i the proof is a costat ot a + b + c depedig o ξ k s ad ay be differet fro lie to lie. Observe that 0, 36 0, o F a + b + c by 3.5. Thus, 3.54 particularly holds for all x 0,. 36 Now we take x = sp σ. The, by 3.50, x s p <

32 o F by the assuptio log p = o /3. We the have γ : = E η k 3 e x η k C 3/ C 3/ + ξ k 3 e 4x ξ k / + ξ k 3 e 64ξ k log p/ o F as is sufficietly large. Thus, γ Cτ o F G 64 := H by The iequality i 3.48 iplies P H c = o 3.56 p 3 as is sufficietly large sice log p = o /3 by assuptio. Step : a sharp estiate o P η k > s p by Lea 3.5. By 3.55, we see that x < 36 < 6 = h. Fro 3.5, 3.53 ad Lea 3.5, we coclude P η k > s p = P η k > s p σ = Φx ] + O + x 3 γe 4x3 γ] o H. Just otice O is bouded by a absolute costat. Fially, by 3.55, x 3 γ = Os 3 p / 0 o H. Reviewig 3.50, we have s p / x s p o H. Hece, fro the forula P N0, x = π x e x / + o as x we obtai that, o H, P η k > s p = Φ s p ] + o σ = σ π sp e s p/σ + o 3.57 as, where o does ot deped o ξ k s. Step 3: proof of 3.45 by By the bouded covergece theore, to prove the lea, it is eough to show that p P η k > s p 4 π e x/ i probability as. Sice P H, to coplete the proof, it is eough to prove p P η k > s p I H 4 π e x/ 3

33 i probability as. Recallig 3.38 ad 3.49, it is trivial to see σ i probability as. Also, s p log p, coparig this with 3.57, it is eough to prove p 4 π log p e s p /σ I H 4 π e x/ 3.58 i probability. By the cetral liit theore for i.i.d. rado variables, we kow σ = + O p fro 3.38 ad Hece σ = + O p. This leads to that s p σ = log p log log p + x + O p ] = log p log log p + x + o p by the coditio log p/ /3 0. We the get Proof of Lea 3.5. Review the proof of Lea 3.4. Let H be defied as above By 3.57, there exists a costat ot depedig o ξ k s such that as. The P P η k > s p I H σ π sp e s p /σ I H ] η k > s p IH C σ e s s p /σ IH p C log p e 7s p /8 o H as by 3.50, where is a costat ot depedig o ξ k s. Therefore, cobiig this with 3.56, we see ] E P η k > s p ] E P η k > s p IH + P H c p o p 3 as is sufficietly large. This proves the lea. Proof of Lea 3.6. Let P 3 ad E 3 stad for the coditioal probability ad the coditioal expectatio give {ξ k, ξ k ; k }, respectively. By idepedece, P η k > s p, η k3 > s p = E P 3 η k > s p ]

34 Write η k = α U k + β k U k + γ k where U k = ξ k, α = ρ a, β k = a ξ k + b ξ k, ad γ k is defied i Now, P 3 η k > s p ] = P 3 α U k + β k U k > sp γ k We will fiish the proof with a couple of steps. Step : the size of γ k. Ucoditioally, {γ k ; k } are i.i.d. with ea zero ad variace ω = ρ a + b etioed below By 3.39, ω <. 4 Fro 3.40 ad the fact P N0, t π t e t / for all t > 0, P γ k θω s p e θsp / p θ log p θ as is sufficietly large for all θ > 0. Review the short arguet as i gettig 3.40, the above iequality also holds if γ k is replaced by γ k. It follows that P γ k θω s p as is sufficietly large for all θ > 0. Set { 7 } K = γ k < ω s p. p θ log p θ The as. Let P K c = o p 3 s p = s p γ k

35 Set W k = α Uk + β ku k for k. Now we cosider ] P 3 α U k + β k U k > s p = P 3 W k > s p. Step : the behaviors of W k s o typical sets. Observe E 3 W k = 0; 3.6 Var 3 W k = α + βk = ρ a + a ξ k + b ξ k. It follows that σ3 : = Var 3 W k = ρ a + a ξ k + b ξ k d = ρ a + a + b ξk Set { F δ = ax ξ k ad δ k } ξk + δ for ad δ 0,. By the fact P N0, t π t e t / for all t > 0 agai ad 3.34, for ay δ > 0, there is a costat C δ > 0 such that P F δ c e C δ 3.64 as is sufficietly large. Review =. Uder F δ, it is easy to see fro 3.38 that σ 3 σ03 δ 3.65 where σ 03 := ρ + a + b. Evidetly, σ by Now, review the otatio τ = E ξ 3 + defied earlier. For v > 0, defie { G v = } + β k 3 e vβ k log p/ τ. 35

36 The paraeter v will be chose later. Now β,, β = d a + b ξ,, ξ. Fro 3.38 we kow a + b. The, by Lea 3.7, for all v > 0, there exists v > 0 such that P G v c P + ξ k 3 e vξ k log p/ > τ exp 4 / log for all v. Defie H δ, v := F δ G v K. Joi the above with 3.6 ad 3.64 to see P H δ, v c = o p 3 as for all δ 0, ad v > 0. By Hölder s iequality, 3.67 E 3 e h W k / e h α / E 3 exp h / α Uk + h / β k U k e α h E 3 exp ] h / α U / E3 exp h / β k U ] / < 3.68 as log as 0 < h. Fro 3.38 we see α 8 α. Therefore, 3.68 holds for all 0 < h h := 8. Furtherore, by takig a = α, d = β k, f = α ad b = c = e = 0, we have fro Lea 3.6 that Wk 3 E 3 3 ex W k / C β α 3 + β 3/ k 3 exp k x e α x/ Ce β + β k 3 exp k 3/ x 3.69 for all 0 < x sice α. Now take x3 = s p σ 3. The assertios 3.65 ad 3.66 iply that 4 σ 3 o F δ for all δ 0, ]. The x 4 3 s p o H δ, v for all δ 0, ] ad all v > 0. Moreover, due to the fact 0 ω 4 < we see that 0 < s p s p + γ k 7 s p + ω s p s p 3.70 o K. This says that 0 < x 3 s p 4s p 4 as v,δ v for all δ 0, 4 ] 36

37 ad all v > 0, where v,δ > 0 is a costat depedig o δ ad v. This ad 3.69 yield C 3/ Wk 3 E 3 3 ex 3 W k / + β k 3 exp 8s p β k τc 3.7 o H δ, 8 as δ 8,δ for all δ 0, ], where 4 δ depeds o δ. The last step follows fro the defiitio of G v ad the fact s p 4 log p as is sufficietly large. Step 3: a boud o P 3 η k > s p. Review 3.60 ad the defiitio of W k, we see sice x 3 = s p σ 3. Set W k = P 3 η k > s p W k σ3 = P 3 W k > x 3 σ 3 for k. The, 3.6 ad 3.63 iply EW k = 0 ad Var 3 W k = 3.7 for each k. Sice 4 σ 3 o F δ for all δ 0, ], we see fro that E 3 e h W k E 3 e h W k / < for all 0 < h h :=. Moreover, by 3.7, 6 γ : = = 8 E W k 3 e x 3 W k Wk 3 E 3 exp W k ] x 3 3 σ3 6τC σ 3 3 Wk 3 E 3 3 ex 3 W k / o H δ, 8 for all δ ad δ 0, 4 ]. Trivially, 0 < x 3 4 < h. The iequality fro 3.70 says that x 3 3γ = Os 3 p/ 0 o H δ, 8 by the coditio log p = o /3. After verifyig all coditios required i Lea 3.5, we coclude P 3 W k > x 3 Φx 3 ] 37

38 o H δ, 8 for all δ ad δ 0, 4 ]. The defiitio of s p ad 3.7 yield that P 3 η sp ] k > s p Φ γ k σ 3 σ 3 o H δ, 8 for all δ ad δ 0, 4 ]. O K, 7ω s p γ k σ3 σ sice 0 ω < by By the fact 4 σ 3 o H δ, 8 with δ 0, ]. 4 s Therefore, p σ 3 σ 3 γ k o H δ, 8 as. Sice P N0, x e x / for x, we obtai that, give δ 0, ], 4 P 3 η k > s p 4 exp sp σ 3 s p σ 3 ] γ k σ 3 o H δ, 8 as is sufficietly large. By 3.65 ad 3.67, give δ 0, 4 ], ] E P 3 η k > s p { 4 E I Hδ,8 exp { 4 E I K exp s σ3 p ]} σ03 + δ s p V + o as is sufficietly large, where V = γ k. Now { E IV 0 exp exp sice σ 03 by Deote The, for give δ 0, 5 ], s ] p σ03 + = o 5 { K = 0 < γ k < ] E P 3 η k > s p { 4 E I K exp ]} γ k + o p 3 p 3 ]} σ03 + s p V 5 p 3 7 ω s p }. ]} σ03 + δ s p V + o p 3 as is sufficietly large. By 3.38 ad 3.39, σ03 + ω =. The desired coclusio the follows fro Lea 3.3 ad

39 3.5 Proofs of Theores. ad. Review the otatios Let J ad L be as i.. Defie W = J for all. To ake a suary, we have W = ax x ki x kj ad L = ax ˆρ ij i<jp i<jp The statistics W ad L will be reduced to a su of two rado variables, each of which has a liitig distributio. The lea below, which is a couplig result, eables us to prove that the two rado variables are actually asyptotically idepedet. Lea 3.7 Assue p = p as. Set C ij = / ξ kξ ki + ξ kj for all i < j p. For ay real ubers {λ ; } ad ay set of rado variables {H ij ; i < j p}, we have { } ax Hi,j + λ C ij = ax i<jp i<jp { } H i,j + λ ξ C λ log p ij + O p as. The above also holds if C ij is replaced by C ij with =. Proof. Recall ξ = ξ + + ξ /. The, ξ ξ = ξ + ξ ξ ξk = O p / 3.74 as sice i probability ad ξ ξ k coverges to N0, weakly. For ay real ubers {λ ; } ad ay set of rado variables {H ij ; i < j p}, by a triagle iequality, ax Note that i<jp λ ξ { Hi,j + λ C ij } ax i<jp { } H i,j + λ ξ C ij ax C ij i<jp ax C ij ax i<jp ip. ξ k ξ ki Observe Eξ ξ = 0, Varξ ξ = ad E exp ξ ξ <. By Lea 3.4 ad assuptio log p = o /3 we have P ax C ij A log p i<jp p P ξ k ξ k A log p p e A log p/

40 as log as A > 3. So ax i<jp C ij = O p log p. This joiig with 3.74 ad 3.75 iplies the desired result. Reviewig the arguets above, we see the assertio is still true if is replaced by. Proof of Theore.. By assuptio, µ = 0. Let {ξ k, ξ ki, k, i =,, }, ρ, ξ be as i Write x ki = ρ ξ k + ρ ξ ki, k, i p It is easy to check the rows of the atrix x ij p are i.i.d. rado vectors, x i N0, for each i p ad Covx i, x j = ρ for i < j p. That is, each row follows N p 0, R. As a result, X ad x ij p have the sae distributio. So we assue X = x ij p i the ext. Deote A = ξ k, B ij = ξ ki ξ kj, C ij = ξ k ξ ki + ξ kj for all i j p. The it follows fro the expressio 3.77 that x ki x kj = ρ A + ρ B ij + ρ ρ C ij First, by the cetral liit theore, we are able to write A = + U, where U := ξ k d N0,. Defie M ij := ρ B ij + ρ ρ C ij = ρ a ξ ki ξ kj + b ξ k ξ ki + ξ kj ], where a = ρ +ρ ad b = ρ +ρ. Deote M = ax i<jp M ij. Fro these otatios we have x ki x kj = ρ + ρ U + ρ M ij, 3.79 ad hece ax i<jp x ki x kj = ρ + ρ U + ρ M Review the otatio ξ = ξ,, ξ. Defie M = ax i<jp { = ax H i,j + b i<jp a ξ ki ξ kj + b 40 ξ C ij ] ξ ξ kξ ki + ξ kj }

41 where H i,j = / a ξ ki ξ kj. By Lea 3.7 ad the fact 0 b, M = M + O p log p. 3.8 This ad 3.80 iply that ax i<jp x ki x kj = ρ + ρ U + ρ M + O p log p. 3.8 Sice ξ ad ξ are idepedet see the discussio above 3.3], U ξ = ξ ad M ξ, which is a fuctio of ad ξ ξ ki s, are also idepedet. This is a crucial observatio i the followig arguet. Now, it follows fro Lea 3.8 ad Propositio 3. that M = log p log log p 4 log p + 4 log p U, where U d η with distributio fuctio F η x = e 4 x π e 3.8, for all x R. Fro M = log p log log p 4 log p + 4 log p U + O p log p. The Ũ := 4 log p M log p + log log p 4 d η log p Sice U ad M are idepedet, U ad Ũ are idepedet. Reviewig the defiitio of W as i Solve M fro the first idetity i 3.83 ad the plug it ito 3.8 to see W µ = ρ ρ U + log p 4 log p Ũ + O p = ρ ρ U + 4 log p Ũ + o p log p by the assuptio log p = o /3, where µ = ρ + log p log log p 4 ρ log p. We ow derive the three coclusios by the above relatio. 4