Perturbation approach applied to the asymptotic study of random operators.

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1 Perturbatin apprach applied t the asympttic study f rm peratrs. André MAS, udvic MENNETEAU y Abstract We prve that, fr the main mdes f stchastic cnvergence (law f large numbers, CT, deviatins principles, I) asympttic results fr selfadjint rm peratrs yield equivalent results fr their eigenvalues assciated prjectrs. Statistical applicatins are mentined. 1 Intrductin et H be a separable Hilbert space (with nrm scalar prduct < :; : >). We dente by (H) the separable Banach space f bunded linear peratrs frm H t H endwed with the nrm : x 2 (H) 7! sup x(h) : h1 We cnsider als the space f Hilbert-Schmidt peratrs 9 < S = : s 2 (H) : X = s (e p ) 2 < 1 ; ; p2n where (e p ) p2n is any cmplete rthnrmal system in H: It is well nwn (see [6]) that if we de ne the scalar prduct hs; ti S = X p2n hs (e p ) ; t (e p )i ; S becmes a separable Hilbert space. et C be a self-adjint peratr cnsider a sequence C n f rm self-adjint element (H) de ned n a cmmn prbability space (; A; P) : Since C (resp. C n ) is bunded self-adjint, its eigenvalues ( ) 1 (resp. ( ;n ) 1 ) are unifrmly bunded real numbers. Withut lss f generality, we assume that ( ) 1 ( ;n ) 1 are nn-increasing sequences.fr every 1; we dente by (resp. ;n ) the assciated prjectr f (resp. ;n ). In the fllwing, (C n ) n1 will be cnsidered as a sequence f estimatrs f C ur aim is t study hw several limit therems caracterizing the cnvergence f C n t C can be used t infer infrmatins abut the cnvergence f ( ;n ) n1 t f ( ;n ) n1 t : Many papers deal with this tpic since applicatins are pssible in the area f principal cmpnent r cannical analysis fr rm vectrs r functins. The sequence C n is cnsequently ften the empirical cvariance peratr f a sample fr which several dependence assumptins are needed. Fr instance, in [3] the authrs cnsidered the case when C n is the empirical cvariance peratr f a sample f i.i.d. rm functins (C n then becmes a sum f i.i.d. rm peratrs). In [2], the case f an autregressive prcess (ARH(1)) is investigated by martingale techniques. In [10], nally, a linear Hilbertian prcess. All these authrs btained transfer results in the case f almst sure wea cnvergence. In [11] in [12] these results are generalized t mderate deviatins principles cmpact laws f the iterated lgarithm fr the ARH(1) prcess. The main gal f this article is t shw that these principles cnsisting in transferring asympttic results frm the peratrs t the eigenelements may be generalized. Nte anyway that the methds f the prf d nt rely n an imprved versin f the delta-methd since rst we deal with C n C nt separatedly with C n C secnd we d nt use Taylr expansins in prbability. The nly bacgrund needed are very basic facts in perturbatin thery (see[9] r the rst chapter f [7]). We nally refer t [13]. These authrs, in a quite similar framewr, investigated the nite dimensinal case (matrices instead f peratrs) restricting themselves t the central limit therem. University Tuluse III - CREST-Insee, Paris y University Mntpellier 2 1

2 2 Main results In the fllwing, we use the assumptin (H:) (resp. ;n ) is a eigenvalue f rder ne f C (resp. C n ). Fr all such that (H:) hlds,we dente by S the bunded linear peratr frm H t H de ned in the basis f eigenvectrs f C by : S = X p6= ( p ) 1 p ; (1) we set ' : s 2 S 7! S s + ss 2 S; (2) p : s 2 S 7! h ; si S 2 R: (3) Nte fr further reference that ' p are linear cntinuus since their nrms are respectively bunded by 2 inf p6= j p j. Here are ur main results. Therem 2.1 If (C n C) n1 fllws the law f large numbers in S; then, fr all such that (H:) hlds, i) ( ;n ) n1 fllws the law f large numbers in S: ii) ( ;n ) n1 fllws the law f large numbers in R: Therem 2.2 If fr sme b n " 1; (b n (C n C)) n1 cnverges in law in S t the limit G C ; then, fr all such that (H:) hlds, i) (b n ( ;n )) n1 cnverge in law in S t the limit ii) (b n ( ;n G : A 2 B (S) 7! G C ' 1 (A) : )) n1 cnverges in law in R t the limit G : B 2 B (R) 7! G C p 1 (B) : Therem 2.3 If fr sme b n " 1; (b n (C n C)) n1 fllws the large deviatin principle in S with speed (v n ) rate functin J C ; then, fr all such that (H:) hlds, i) (b n ( ;n )) n1 fllws the large deviatin principle in S with rate functin J : t 2 S 7! inf fj C (s) : ' (s) = tg : ii) (b n ( ;n )) n1 fllws the large deviatin principle in R with rate functin J : 2 R 7! inf fj C (s) : p (s) = g : Therem 2.4 If fr sme b n " 1; (b n (C n C)) n1 is almst surely cmpact in S with limit set K C then : i) (b n ( ;n )) n1 is almst surely cmpact in S with limit set K = ' (K C ) : ii) (b n ( ;n )) n1 is almst surely cmpact in R with limit set K = p (K C ) : Remar 2.1 The main interest f assumptin (H:) is t mae all the previus results mre easily readible. In fact, when (H:) is nt ful lled we btain fr each mde f cnvergence very similar results. It su ces t replace (resp. ;n ) by m (resp. m ;n ;n ) where m (resp. m ;n ) dentes the rder f multplicity f (resp. ;n ). Nte that m ;n is a rm integer. We refer t the remar in the prf f the ii) f the frthcming Prpsitin

3 3 Prfs In the next lemma we give tw results related t perturbatin thery fr linear peratrs useful fr ur needs. De nitin 3.1 et be a self-adjint element f (H) ; be an islated pint f the spectrum f we call an admissible cntur fr whenever is a cntur arund which cntains n ther eigenvalues f : emma 3.1 i) et be a self-adjint element f (H) be an islated pint f the spectrum f then fr every ; admissible cntur fr ; the mapping = 1 Z (zid H ) 1 dz; (4) where i 2 = 1; is the rthgnal prjectin nt er ( Id H ) : ii) et be a self-adjint element f (H) z is nt an eigenvalue f A, then (zid H ) 1 n 1 sup jz j 1 : eigenvalue f : (5) Prf. i) See e.g. Prpsitin 6.3 f [] ii) See e.g. Therem 5. f []. Fr all such that (H:) hlds, set = inf p6= j p j = min ( +1 ; 1 ) : et be the riented circle with center radius = =2. Nte that is an admissible cntur fr C: Mrever, de ne the event O ;n = fc n C S < =4g : (6) Since (see e.g. [7] p.99), we can prve : sup j p;n p j C n C S (7) p2n emma 3.2 i) Fr all! 2 O ;n ; is an admissible cntur fr ;n (!) C n (!). ii) iii) n (zidh sup z2 n (zidh sup z2 Prf. i) Set! 2 O ;n : By (6) (7) ; C n ) 1 1 O;n 4 1 : () C) : (9) j ;n (!) j =4 < (10) inf j p;n (!) p6= j inf j p j sup j p;n (!) p6= p6= > =4 = 3 =4 > : (11) p j Hence, the result hlds by (10) (11). ii) Set! 2 O ;n : By (5) ; n (zidh sup z2 C n (!)) 1 n sup jz p;n (!)j 1 : p 2 N; z 2 : (12) 3

4 Mrever, fr all z 2 ; jz ;n (!)j jz j j ;n (!)j =4 = =4; inf jz p6= p;n (!)j = inf p6= j( p ) + ( z) + ( p;n (!) p )j inf j p j j zj sup j p;n (!) p6= =4 = =4: p6= p j Therefre, which, cmbined with (12) ; give the result. iii) Nte that, fr all z 2 ; inf inf jz p;n (!)j =4; z2 p2n j zj = =2; inf p6= j p zj inf p6= j p j j zj =2 = =2: Therefre, using (5) ; we get, n (zidh sup z2 C) 1 n sup jz 2 1 : p j 1 : p 2 N; z 2 Nw, we can state the main tls used in the prf f ur therems. Prpsitin 3.1 Fr all such that (H:) hlds : i) There exists a S-valued rm variable R ;n such that, fr every n 1; ;n = ' (C n C) + R ;n ; (13) R ;n S 1 O;n 2 C n C 2 S : (14) ii) There exists a real valued rm variable r ;n such that fr every n 1;, fr sme > 0 > 0; ;n = p (C n C) + r ;n ; (15) jr ;n j 1 O;n C n C 2 S + C n C 3 S : (16) Prf. i)set! 2 O ;n : Since, by the rst part f emma 3.2, is an admissible cntur fr C n (!) ( als fr C), (5) implies that ;n (!) = 1 Z (zid H C n (!)) 1 (zid H C) 1 dz: (17) Fr cnvenience, set Nte that a = zid H C n (!) b = zid H C: a 1 b 1 = a 1 (b a) b 1 = b 1 (b a) b 1 + a 1 b 1 (b a) b 1 = b 1 (b a) b 1 + b 1 (b a) a 1 (b a) b 1 : (1) 4

5 Therefre, if we set U ;n = 1 O ;n Z we get, by (17) (1) ; ;n (!) = 1 (zid H C) 1 (C n C) (zid H C n ) 1 (C n C) (zid H C) 1 dz; Z Nw, in [3] p.145, it is shwn that Hence, if we de ne ' : s 2 S 7! 1 (zid H C) 1 (C n (!) C) (zid H C) 1 dz + U ;n (!) : Z (zid H C) 1 s (zid H C) 1 dz : R ;n = U ;n + ( ;n ' (C n C)) 1 O c ;n ; (13) hlds. Mrever, fllwing [3] p.142 (lines 2 3), we btain, using () (9) ; that U ;n S (zidh sup C) 1 (C n C) (zid H C n ) 1 (C n C) (zid H C) 1 S z2 (zidh 2 C n C 2 S sup C) 1 2 (zid H C n ) 1 1 O;n z2 2 C n C 2 S : ii) Observe that = h ; Ci S ;n = h ;n ; C n i S : Indeed the fllwing decmpsitin fr C: is well nwn (Schmidt decmpsitin f selfadjint peratrs) : C = P p p p hence 1 O;n h ; Ci S = X p p h ; p i S = h ; i S = The last line stems frm the fact that is an eigenvalue f rder ne. The same elementary calculatin wuld lead t an equivalent result fr C n : Remar 3.1 If we suppse that (H:) is nt true denting as previusly m (resp. m ;n ) the rder f multplicity f (resp. ;n ). h ; Ci S = h ; i S = m since h ; i S = tr = m (trt dentes the trace f peratr T). Therefre, ;n = h ;n ; C n i S h ; Ci S = h ; C n Ci S + h ;n ; C n i S = h ; C n Ci S + h ;n ; Ci S + h ;n ; C n Ci S = p (C n C) + h' (C n C) ; Ci S + hr ;n ; Ci S + h ;n ; C n Ci S : (19) Furthermre, let (e p ) p1 be an rthnrmal basis f H such that e is an eigenvectr f C assciated with : Then, by (??) ; (2) (1) ; fr all s 2 S; h' (s) ; Ci S = X p hs s (e p ) ; C (e p )i + X p h ss (e p ) ; C (e p )i = hs s (e ) ; e i + X p6= p p h s (e p ) ; e p i = 0: (20) 5

6 Hence, if we cmbine (19) (20) ; we get ;n = p (C n C) + r ;n ; where satis es r ;n = hr ;n ; Ci S + h ;n ; C n Ci S (21) jr ;n j 1 O;n C S R ;n S 1 O;n + C n C S ;n S 1 O;n 2 C S + ' (S) C n C 2 S + 2 C n C 3 S : Prf f Therem 2.1 : i) By (13) the linearity bundedness f ' yields the fllwing : ' (C n C) cnverges t 0 as n tends t in nity. We just need t prve that R ;n decays almst surely.fr xed but : R ;n S = R ;n S 1 O;n + R ;n S 1 O c ;n 2 C n C 2 S + R ;n S 1 O c ;n The rst term tends almst surely t zer. Nw let us tae an! in N (N is the cnvergence set f C n C P (N) = 1). It is clear that fr a su ciently large n C n (!) C S < =5 hence 1 O c ;n (!) = 0 This prves the desired result. ii) Fr the eigenvalues (we refer t (15)) we may prve exactly the same way that p (C n C) decays t zer as well as R ;n. The previus result ensures via (21) that r ;n als tends t zer. Prf f Therem 2.2 : i) The linearity f ' (13) entail b n ( ;n ) = ' [b n (C n C)] + b n R ;n : (22) Therefre, by emma?? stard cnsideratins, we just have t shw that b n R ;n tends t zer in prbability i.e., fr all > 0; lim P b n R ;n S = 0: T this aim, bserve that, fr all > 0; (14) leads t, fr n large enugh; P b n R ;n S P b n C n C 2 S + P O;n c! 1=2 b n P b n C n C S + P b n C n C S 4 b n 2P (b n C n C S ) : (23) Hence, using the cnvergence in law f b n (C n C) we get lim sup P b n R ;n S 2lim sup ii)once mre we just have t prve that fr all > 0; lim supp (b n C n C S )!1 2lim supg C (fs 2 S : s S g)!1 = 0: lim P b n r ;n S = 0: but bserve that b n r ;n is the sum f tw real rm variable hb n R ;n ; Ci S b n h ;n ; C n Ci S : We prved just abve that the rst ne tends t zer in prbability. the secnd ne is bunded by b n C n C S ;n S but b n C n C S is bunded in prbability since it cnverges wealy we prved that ;n S cnverges in prbability t zer which nishes the prf. 6

7 Prf f Therem 2.3 : i) By (22) ; emma?? Therem f Demb Zeituni [4], we just have t shw that, fr all > 0; lim supv n lg P b n R ;n S = 1: But, (23) the large deviatin principle f b n (C n C) give lim supv n lg P b n R ;n S lim sup lim supv n lg P (b n C n C S )!1 lim sup!1 inf fj C (s) : s S g = 1: ii) Fllwing the same argument as abve we have t prve that It su ces t prve that < : lim supv n lg P (b n jr ;n j ) = 1: lim supv n lg P b hr;n n ; Ci S = 1: lim supv n lg P b n h ;n ; C n Ci S = 1: The rst limit is a crllary f the i). The secnd term may be bunded by < lim supv n lg P b n C n C S p b n : max : lim supv n lg P b n ;n S p b n : The limit f the term n the tp is 1 since b n (C n C) satis es the large deviatins principles with speed v n : The in nite limit f the secnd term is btained by cnsidering, with the same arguments, the i) prved immediately abve. Prf f Therem 2.4 : i) By (13) ; emma?? we just need t prve that b n R ;n cnverges almst surely t zer as n tends t in nity. Nw. b n R ;n S 2 b n C n C 2 S + b n R ;n S 1 O c ;n we may inve the methd f prf f Therem 2.1. The rst term n the right tends t zer since b n C n C S is almst surely bunded. S des the secnd term since 1 O c ;n is almst surely null fr large n: ii) Adapting what was dne abve it is bvius that hb n R ;n ; Ci S as well as b n h ;n ; C n Ci S bth tend t zer almst surely. References [1] P. Billingsley, Cnvergence f Prbability measures, Wiley, 196. [2] D. Bsq, inear prcesses in functin spaces, ecture Ntes, Springer-Verlag, [3] J. Dauxis, A. Pusse Y. Rmain, Asympttic thery fr the principal cmpnent analysis f a vectr rm functin : sme applicatins t statistical inference. J. Multivar. Anal , 192. [4] A. Demb O. Zeituni, arge Deviatins Techniques Applicatins, Jnes Bartlett, Bstn ndn,

8 [5] J. Diestel J.J. Uhl, Vectr measures, Mathematical Survey f the A.M.S, 15, [6] N. Dunfrd J.T. Schwartz, inear Operatrs Vl II, Wiley Classics ibrary, 19. [7] I. Ghberg, S. Gldberg M.A. Kaashe, Classes f inear Operatrs Vl I. Operatr Thery : Advances Applicatins Vl 49. Birhaüser Verlag, [] P.D. Hislp, I.M. Sigal, Intrductin t Spectral Thery, Applied Mathematical Sciences, 113, Springer, [9] T. Kat, Perturbatin thery fr linear peratrs, Grundlehren der mathematischen Wissenschaften. 132, Springer-Verlag, [10] A. Mas, Wea cnvergence fr cvariance peratrs f a linear Hilbertian prcess, t appear in Stchastic Prcesses their Applicatins, [11] A. Mas,. Menneteau, arge mderate deviatins fr in nite dimensinal autregressive prcesses, submitted. [12]. Menneteau, Sme laws f the iterated lgarithm in hilbertian autregressive mdels, submitted. [13] F.H. Ruymgaart S. Yang, Sme applicatins f Watsn s perturbatin apprach t rm matrices. J. Multivar. Anal. 60 (1997), 4-60.