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On the WeakLimitof the LargestEigenvalue
e atrix
Sla m p l eC o v a r i a n cM
a L a r g eD i m e n s i o n a
Jncr W. SnvnRsrEIN*
North Carolina State Uniuersity
Communicatedbv the Editors
Let {r,i}, i, i : I, 2, ..., be i.i.d. random variables and for each n let
, ; i : 1 , 2 , . . . n, ; p : p ( n ) , a n d
M , : ( l l n ) W " W f , , w h e r e W n : ( , u ) , i : 1 , 2 , . . .p
-*
-+
pln y > 0 as n co. The weak behaviorof the largesteigenvalueol M , is studied.
The primary aim of the paper is to show that the largest eigenvalueconverges
in probability to a nonrandom quantity if and only if E(wrt ) : Q and
P ress,
c Inc .
o tssqA cademi
n a P l l a rrl > n ): o (1),the l i mi t bei ng(1 + ,/ y)' E (r1).
1. INrnopucrloN
Let {*,i}, i, j:1,2, ..., be i.i.d. random variablesand for each n let
, i p - p(n),
M , : t i t r | W , W : , w h e r e W , : ( * , ) , i : 1 , 2 , . . . ,p ; i : 1 , 2 , , . . . n
a n d p l n + y > 0 a s n + 6 . T h e m a t r i x M n c a n b e v i e w e da s t h e s a m p l e
covariancematrix of r?samplesof a p-dtmensionalvector containing i.i.d.
components,wherep and n &re large but on the same order of magnitude.
Denote the largesteigenvalueof M, by ,t-"*( M,). In [3] it is proven that
as
if E(*1)<oo and E(r,,):0 then I^*(Mn)--"'(1+,/y)'E(*1)
n -+ @, while in [ 1] it is shown that lim supnl^u*(M) : .c almost surely
if E(wf ,) : .o. Thus the almost sure behavior of l^^*(M) is essentially
understood.The aim of this paper is to establishits weak limiting behavior.
We will prove the following
TueonBu. We haue
(a) l^u*(M n) conuergestn probability to a nonrandom quantity tf
a n d noP(l r, , l 7 n) - o(l), the limit being
and only if E(r,,):0
(1+,fi1'E(r?,).
ReceivedMay 17, 1988;revisedJune 1988.
AMS 1980subjectclassifrcation:Primary 60F05; Secondaty62H99.
Key words and phrases:largest eigenvalueof sample covariancematrix, convergencein
probability, boundedin probability.
* Supportedby the National ScienceFoundation under Grant DMS-8603966,
307
0047-2s9xl89
$3.00
Copyright O 1989by AcademicPress,Inc.
All rights of reproductionin any form reserved'
308
JACK W. SILVERSTEIN
(b) ff
lim SUp,,noP(lrttl 7 n) > 0, then fo,
lim sup, P(A^^*(M") > K) > 0.
(c) If lim SUp,,noP|r,rl ) n) : @, then .fo,
lim sup, P(I^^*(M,)2 K) - 1.
any
K > 0
any
K > 0
( d ) I f E ( w r , ) : 0 a n d l i m s u pn, o P ( l r , r l) n ) < c o , t h e n { A ^ ^ * ( M " ) } f : t
is boundedin probability.
Thus convergencein probability holds only under a slight weakeningof
For case(d) we remark
the condition neededfor almost sure convergence.
here that, due tc the known limiting behavior of the empiricaldistribution
function of the eigenvaluesof M, l2l, we have lim inf, A^u*(M,) 7
( 1 + ",,5f E@?) almost surely, implying any limiting distribution of a
weakiy tonu.tging subsequenceof {A^^*(M, ) } f: , must have mass in
t ( 1 + ,/ yY E(*1), .o ). Other than that, no additional information is
known, for example,whether L^u*(M,) convergesin distributionor how
of {2,,,u*(M)}f:t
the distributionof any weakly convergentsubsequence
dependson lr r .
The proof of the theorem will be given in the next section.It relies on
part of the proof of the main theorem in [3], as well as the fact that
A^u*(M,) is boundedbelow by each diagonalelementof M,.
2. Pnoor oF rHE TunonrPt
A s s u m eE ( r , , ) : 0 a n d n o P ( l r , , l 2 n ) - o ( 1 ) . W i t h o u tl o s sg f g e n e r a l i t y
w e m a y a s s u m eE ( * 1 ) : 1 . I t t h e n f o l l o w st h a t n ' P ( l w , , l 2 e r f n ) - o ( 1 ) f o r
everye >0. From this it is straightforwardto constructa sequence{d,}f:t
ofpositivenumberssuchthat6,-+0,6,,Iogn--}@,andn,P(|w.'|>
6,r/n) -0 asn-+cc.
Let fi,,:fii@) -*,i^I(l,r,4t
< 6^,fft^(1,being the indicator function on the
set A), and define U,,-- (lln\ W,W:, where fI/, is p x n with (i, i)th
elementilu. Then
P ( I ^ ^ * ( M , ) * 1 ^ u * ( l ; [ )) ( n p P ( l r , t l > 6 " t / n ) - 0
as n -+@. (2.1)
Letfi ,,: fru(n): fru- E(fr,),and define frfn- Qlfi fi/,trV;,wttereY, r,
p xn wiitr (i, i)th element,1u.Let llBll denotethe spectralnorm of any
matrix.B(that is, llBll: S"!!-1AAr)),and let 1- denotethe m-dimensional
vectorconsistingof 1's.Then
1
<+ lE(]t,,)l
,,)1.
lllol'nll:'[plE(la
l1'J?-6[,)-r'J:*(fu,,)l
Jn
WEAK LIMIT OF LARGEST EIGENVALUE
309
S i n c eE ( u , , , :)0 , E ( w l , ) : 1 , w e h a v e
< P t''(l* rrl2 5,r,G).
lE(r0r,
)l : lE(wrrI r,rrt>-6,uG))l
Therefore,
lA'J:-(tr
)| * o
) - A'J\-Qrt"
as n+ @.
(2.2)
From (2.1) and (2.2) we seethat
i'P',
A^u*(Mn)- 1*u*(frt,)
0
as n + @.
At this point the same argumentsin Section4 of [3] can be applied to
l^u*(fu,), sincethe assumptionsneededon fii,,are identical.Therefore,we
have
i^u*(Mn)
u''',
(1 + .,/yY
as n+@,
and the sufficiencypart of (a) is proven.
Then the matrix
S u p p o s en o P ( l r r r l 2 n ) - o ( 1 ) b u t E ( w r ) : a * 0 .
=
the
conditions
of (a).
W,-lolT)'
satisfies
Mn
Ol")(W,-le1;)(
Moreover. we have
1
1X1^(u,,llJ rn "t,t:ll< A';\-w').
However,ll(11.,f,)alrl Ill : ,frtot. Therefore,tr^^*(M
n)lpo'--rip1 as
n -, @. This verifiesthe necessity
of E(w,r):0 in (a).
Assumenow lim sup, noP|w,rl 7n) >0. For any K>0 we have
P(l^^*(M,)>
K)>r (-u-:
,t,*'u,
*)
- 1- ('- " (;
,t,"i,*))',
(2.3)
wherewt,,il2,...are i.i.d.havingthe samedistributionas 'urr.We needthen
to investigate the limiting behavior of nP((Ur) |,i: , *] 7 K). Since
E(*1) : 1 we have nP(wlr>- nK) - o(1) (we remark here the needfor the
secondmoment of w' to be frnite. If E(wi) were infinite, then from the
strong law of large numbers applied to (l ln) >; : , ,?, we would have
tr^^^(M) * "''' oo as n --,@ ). We have
310
JACK W. SILVERSTEIN
'(;,\,*j,4
"*(V(*j>r*r)
nK)-ryp'(*'rr2
2 np(w112
nK)rXrWlr2nK) (2.4)
for n sufficientlylarge.
and if
It is a simple matter to show lim sup,','\(nlrTnK)>0,
cc.
Therefore,
nK):
n2P1wlr2
lim sup, noP(lw,r| 2 n) cc, then lim sup,
from (2.3) and (2.4) we get lim suP, P(A*^*(M")> K) > 0 for any positive
K w h i c h v e r i f i e s( b ) a n d t h e n e c e s s i t yo f n o P ( l r ' l 7 n ) - o ( 1 ) i n ( a ) .
l n)
, ( A ^ ^ * ( M , ) > K ) : 1 i f l i m s u P ,n o P ( l w , , 2
M o r e o v e r ,w e g e t l i m S U p P
- oc, proving (c).
Finally, assumelim sup, noP|rr,l 7n) < oo. Define ff,,,, Wn, Mn, fr,,,
with 6n:1 for eachn. We immediatelyget (2.2),
fr,, [,In as above__but
sincenP(|w r]2 ,/ ,) - o(1). We have
lt'J:^(M
")- A'J:.(u,)l
- fv,ll- l'J:.(!rrr"-Ii/,)(w,-fu,f)
*+ ttw,
Jn
\ 1/2
,v;')
< f1 ft(w.- fr,,)(w,\/?
Therefore,for any K > 0 we get from Chebyshev'sinequality
P(|lil1.(M^)- A' Jl-W,)l
> K)
I 61,>,t)
E ( I f: , 2 ; : , * ? i rI * 1 , > , ) pB(w2rr
<_
K2
(2.s)
Using integrationby parts, it is straightforwardto show for any random
variableX and m> |
n m - t E ( l xI ul * r r , r () +
m
Therefore,with m:2
-
s u p x * P ( l x l >x ) .
r xe[rr,cc)
we concludefrom (2.2) and (2.5) that
}f:,
) - A',11.6t)
{1I1"^W
is boundedin probability.
(2.6)
WEAK LIMIT OF LARGEST EIGENVALUE
311
Now, the main part of the proof in t3] of showingtr^u*(fu)--ras'
(1 + 6f
for an appropriately
chosensequence
{d,}[, involvesproving
f o ra n yz > ( L + r f r ) ' ,
"l)") < .o
in : r n ((^^"-lfri
z
\\
) /
(2.7)
for appropriatelychosen{k"}f: r.Of course,(2.7) implies
P(l^^*(t[)>
z infinitelyoften):0.
However,if 5n:1 and kn: llognl (where [x] is the integerpart of x),
then the same argumentsin t3] can be used to show (2.7) for all z sufficiently large. Therefore, for some z ) 0, lim sup, A^^*(M,) ( z almost
surely.This togetherwith (2.6) being bounded in probability give us (d).
AcTTowLEDGMENT
The authortlranksthe referee
for somehelpfulsuggestions.
RsrenBNcns
[1 ] Be t, 2 .D ., S trv rR s rE tNJ., W ., l N o Y rN , Y . Q. (1988).A note on the l argestei gen value
of a large dimensionalsamplecovariancematrix. J, MultiuariateAnal.26 166-16g.
[2] Ytx, Y. a. (1986). Limiting spectraldistribution for a class of random matrices.
J. MultiuariateAnal.20 50-68.
[3 ] YtN , Y. Q., B a .t,Z .D.,.tN o K R tsH N l rns,P . R . (1983).Onl i mi t of the l argestei gen value
of the large dimensionalsample covariancematrix. probab. Theory RelatedFields 7g
5 0 9 -5 2 1 .
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