ASYMPTOTIC RESULTS FOR EMPIRICAL MEASURES OF WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

Transcription

1 Elect. Comm. i Probab. (007), ELECTRONIC COMMUNICATIONS i PROBABILITY ASYMPTOTIC RESULTS FOR EMPIRICAL MEASURES OF WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES BERNARD BERCU Istitut de Mathématiques de Bordeaux, Uiversité Bordeaux, 35 cours de la libératio Talece cedex, Frace Berard.Bercu@math.u-bordeaux.fr W lodzimierz BRYC Departmet of Mathematical Scieces, Uiversity of Ciciati, 855 Campus Way, PO Box 005, Ciciati, OH , USA Wlodzimierz.Bryc@UC.edu Submitted Jauary 4, 007, accepted i fial form May AMS 000 Subject classificatio: Primary: 60F5; Secodary: 60F05, 60F0 Keywords: almost sure cetral limit theorem, large deviatios,ormal approximatio, periodogram Abstract We ivestigate the asymptotic behavior of weighted sums of idepedet stadardized radom variables with uiformly bouded third momets. The sequece of weights is give by a family of rectagular matrices with uiformly small etries ad approximately orthogoal rows. We prove that the empirical CDF of the resultig partial sums coverges to the ormal CDF with probability oe. This result implies almost sure covergece of empirical periodograms, almost sure covergece of spectral distributio of circulat ad reverse circulat matrices, ad almost sure covergece of the CDF geerated from idepedet radom variables by idepedet radom orthogoal matrices. I the special case of trigoometric weights, the speed of the almost sure covergece is described by a ormal approximatio as well as a large deviatio priciple. Itroductio Let (X ) be a sequece of idepedet ad idetically distributed radom variables with E[X ] = 0 ad E[X ] = ad deote S = X t. RESEARCH PARTIALLY SUPPORTED BY NSF GRANT #DMS

2 Empirical measures of weighted sums 85 The almost sure cetral limit theorem (ASCLT) states that, for ay x R, lim log t {S t x} = Φ(x) with probability oe, where Φ stads for the stadard ormal distributio [3], [7], [4], []. Oe ca observe that the ASCLT does ot hold for the Césaro averagig. Assume ow that (X ) has all momets fiite ad cosider the weighted sum S,k = X t cos ( πkt ). (.) It was recetly established by Massey, Miller ad Sisheimer [7] that, for ay x R, lim {S,k x} = Φ(x) (.) with probability oe together with the uiform covergece. This result is closely related to the limitig spectral distributio of radom symmetric circulat matrices [6] sice the eigevalues of theses matrices are exactly give by (.). The goal of this paper is to aswer to several atural questios. (a) Is it possible to remove the assumptio of idetical distributio? (b) Ca the momet coditio be relaxed? (c) Ca the trigoometric coefficiets be replaced by otheumbers? (d) Ca the multivariate versio of covergece (.) be established? (e) Is it possible to prove a CLT or a LDP associated with (.)? We shall propose positive aswers to these questios extedig [7, Theorem 5.] i various directios. First of all, we shall show that it is possible to deal with a sequece (X ) of idepedet radom variables defied o a commo probability space (Ω, A, P) without assumig that they share the same distributio. I additio, we shall oly require that the third momets of (X ) are uiformly bouded. Next, we shall allow more geeral weights ad we will prove the multivariate versio of covergece (.). Fially, we shall provide a CLT as well as a LDP. Mai results Let (U () ) be a family of real rectagular matrices where ( ) is a icreasig sequece of itegers which goes to ifiity with. We shall assume that there exist some costats C, δ > 0 which do ot deped o such that (A ) max k, t u() k,t (A ) max k, l u () k,t u() l,t δ k,l C (log( + )) +δ, C (log( + )) +δ.

3 86 Electroic Commuicatios i Probability For applicatio to periodograms, we will also eed to cosider pairs (U (),V () ) of such matrices ad we shall assume that (A 3 ) max u () k, l k,t v() l,t C (log( + )) +δ. These assumptios are ot really restrictive ad they are fulfilled i may situatios. For example, cosider the sequece of real rectagular matrices (U (),V () ) with give, for all k ad t, by ( ) ( ) u () πkt k,t = cos, v () πkt k,t = si. (.) The (A ) holds trivially while (A ) ad (A 3 ) follow from < ad the followig well kow trigoometric idetities where k < l. ( ) ( ) { πkt πlt 0 if k + l, cos cos = (.) / if k + l =, ( ) ( ) { πkt πlt 0 if k + l, si si = (.3) / if k + l =. I additio, cos ( πkt ( πkt cos ) = ) si ( ) πlt = 0, (.4) ( ) { πkt si / if k, = if k =. (.5) I order to avoid cumbersome otatio, we oly state our first result i the uivariate ad bivariate cases. The d-variate extesio requires itroducig d sequeces of matrices that satisfy assumptios (A ) ad (A ) with each pair from the -th level satisfyig coditio (A 3 ) ad the proof follows essetially the same lies. Theorem (ASCLT). Assume that (X ) is a sequece of idepedet radom variables such that E[X ] = 0, E[X ] = ad sup E[ X 3 ] <. Let (S,k ) be the sequece of weighted sums S,k = u () k,t X t (.6) with k =,,..., where (U () ) is a family of real rectagular matrices satisfyig (A ) ad (A ). The, we have the almost sure uiform covergece lim sup {S,k x} Φ(x) = 0. (.7) x R I additio, let (T,k ) be the sequece of weighted sums T,k = v () k,t X t (.8)

4 Empirical measures of weighted sums 87 with k =,,..., where (V () ) is a family of real rectagular matrices such that (A ) ad (A ) hold with v () k,t i place of u() k,t. Assume that the sequece of pairs (U(),V () ) satisfies (A 3 ). The there is a measurable set Ω of probability oe such that for all x, y R, we have the almost sure covergece o lim {S,k x,t,k y} = Φ(x)Φ(y). (.9) For the uivariate case with trigoometric coefficiets give by (.), we also have the compaio weak limit theorem ad the large deviatio priciple uder the restrictios o the rate of growth of the sequece ( ). The most attractive case = [( )/] which correspods to the spectral measures of radom circulat matrices, is ufortuately ot covered by our result. Some LDP for spectra of other radom matrices ca be foud i [, Chapter 5]. Theorem (CLT). Assume that (X ) is a sequece of idepedet radom variables such that E[X ] = 0, E[X ] = ad satisfyig for some costat τ > 0, sup E[ X 3 exp( X /τ)] τ. (.0) Cosider the sequece of weighted sums (S,k ) give by (.6), where (U () ) correspods to the trigoometric weights give by (.). If ( ) is such that the for all x R, (log ) r 3 0, (.) ( {S,k x} Φ(x) ) D N (0, Φ(x)( Φ(x))). (.) r + I additio, we also have r sup x R {S,k x} Φ(x) where L stads for the Kolmogorov-Smirov distributio. D + L (.3) Remark. The coclusios (.), (.3) also hold if coditios (.0) ad (.) are replaced by the assumptio that there is p > 0 such that sup E[ X +p ] < as soo as r 3 p/(+p) 0. This follows from our proof, usig [0, Sectio 5, Corollary 5] istead of Lemma 3. For example, if p =, it is ecessary to assume that r 9 = o().

5 88 Electroic Commuicatios i Probability Remark. The Kolmogorov-Smirov distributio is the distributio of the supremum of the absolute value of the Browia bridge. The large deviatio priciple was motivated by the LDP from the Brosamler-Schatte almost sure CLT, see [6, Theorem ]. To formulate the result, we eed to itroduce additioal otatio. Let M (R) deote the Polish space of probability measures o the Borel sets of R with the topology of weak covergece. For the sequece of weighted sums (S,k ) give by (.6), cosider the empirical measures µ = δ S,k (.4) The rate fuctio I : M (R) [0, ] i our LDP is the relative etropy with respect to the stadard ormal law. More precisely, if φ(x) deotes the stadard ormal desity ad ν M (R), we have I(ν) = log f(x) φ(x) f(x)dx R if ν is absolutely cotiuous with respect to the Lebesgue measure of R with desity f ad the itegral exists ad I(ν) = + otherwise. It is well kow that the level sets I [0, a] are compact for a <. The coclusio of ouext result is the LDP for the empirical measures µ with speed ad good rate fuctio I. Theorem 3 (LDP). Assume that (X ) shares the same assumptios as i Theorem. Cosider the sequece of weighted sums (S,k ) give by (.6), where (U () ) correspods to the trigoometric weights give by (.). If ( ) is such that r 4 0 ad log 0, the for all closed sets F ad ope sets G i M (R), ad lim sup lim if log P(µ F) if ν F I(ν) log P(µ G) if ν G I(ν). All techical proofs are postpoed to sectio 4. We shall ow provide several applicatios of our results. 3 Applicatios 3. Applicatio to periodograms The empirical periodogram associated with the sequece (X ) is defied, for all λ i the torus T = [ π, π[, by I (λ) = e itλ X t.

6 Empirical measures of weighted sums 89 The empirical distributio of the periodogram is the radom CDF give, for all x 0, by F (x) = {I(πk/) x} where =. Theorem stregthes the coclusio of [3, Propositio 4.] to almost sure covergece at the expese of the assumptio that third momets are fiite. Corollary 4. Assume that (X ) is a sequece of idepedet radom variables such that E[X ] = 0, E[X] = ad sup E[ X 3 ] <. The, we have the almost sure uiform covergece lim F (x) ( exp( x)) = 0. sup x 0 Proof. Followig [3, (.)], we ca rewrite F as the CDF of the empirical measure µ = δ (S,k +T,k ) where (S,k, T,k ) are defied by (.6) ad (.8) ad (U (),V () ) are the trigoometric weights give by (.). Cosequetly, Corollary 4 immediately follows from Theorem. As a matter of fact, let h : E F be a cotiuous mappig of Polish spaces. If a sequece of discrete measure (ν ) coverges weakly to some probability measure ν o the Borel sigma-field of E, the (ν h ) coverge weakly to the probability measure ν h, see e.g. [4, Theorem 9.]. We apply it to ν = δ (S,k,T,k ) ad to the cotiuous mappig h : R R give by h(x, y) = (x + y ). O the oe had, we clearly have µ = ν h. O the other had, as ν is the product of two idepedet stadard ormal distributios, the limitig distributio µ = ν h is simply the stadard expoetial distributio. Hece, for all x 0, F(x) = exp( x). Fially, as this limit is a cotiuous CDF, it is well kow, see [4, Exercise 4.8], that the covergece is uiform. 3. Applicatio to symmetric circulat ad reverse circulat matrices Corollary 5. The weak covergece i [5, Theorem 5] ad i [6] holds with probability oe. Proof. Oe ca fid i [5, Theorem 5] ad [6] the aalysis of the limitig spectral distributio of the symmetric radom matrices with typical eigevalues of the form ± (S,k + T,k )/ where (U (),V () ) are the trigoometric weights give by (.) ad = see [6, Lemma ]. After omittig at most two eigevalues which do ot modify the covergece of the spectral measure, Theorem implies that the covergece holds with probability oe by the same argumets as i the proof of Corollary 4.

7 90 Electroic Commuicatios i Probability Assume ow that (A ) is a family of symmetric radom circulat matrix formed from the sequece of idepedet radom variables (X ) by takig as the first row [A ],t = X t with t =,,..., [( + )/] ad [A ],t = [A ], t for the other idices. The ext corollary stregthes [6, Remark ] to almost sure covergece ad removes the assumptio of itegrability ad idetical distributio i [7, Theorem.5]. To justify the later claim, we ote that a palidromic matrix aalyzed i [7] differs from A by the last row ad colum oly. Thus their raks differ by at most oe ad asymptotically palidromic matrices ad radom circulat matrices have the same spectrum, see [, Lemma.]. Corollary 6. Assume that (X ) is a sequece of idepedet radom variables with commo mea m = E[X ] = 0, commo variace σ = V ar(x ) > 0 ad uiformly bouded third momets sup E[ X 3 ] <. The, the spectral distributio of the radom matrix σ A coverges weakly with probability oe to the stadard ormal distributio. Proof. Subtractig the rak matrix does ot chage the asymptotic of the spectral distributio. Cosequetly, without loss of geerality, we may assume that m = 0. Rescalig the radom variables by σ > 0 we ca also assume that σ =. With the exceptio of at most two eigevalues, the remaiig eigevalues of A / are of multiplicity two ad are give by (.6) with the trigoometric weights give by (.), see [6, Remark ]. Fially, the weak covergece with probability oe of the spectral distributio of A / to N(0, ) follows from Theorem. 3.3 Applicatio to radom orthogoal matrices A well kow result of Poicaré says that if U () is a radom orthogoal matrix uiformly distributed o O() ad x R is a sequece of vectors of orm the the first k coordiates of U () x are asymptotically ormal ad idepedet, see e.g. [4, Exercise 9.9]. Corollary 7. Assume that (X ) is a sequece of idepedet radom variables such that E[X ] = 0, E[X ] = ad sup E[ X 3 ] <. Cosider the sequece of weighted sums (S,k ) give by (.6) where (U () ) is a family of radom orthogoal matrices uiformly distributed o O() ad idepedet of (X ). The, we have the almost sure uiform covergece lim sup {S,k x} Φ(x) = 0. (3.) x R Remark 3. This result has a direct elemetary proof, which we leared from Jack Silverstei. His proof shows that the result holds also for i.i.d. radom variables with fiite secod momets. Here we derive it as a corollary to Theorem. Proof. Orthogoal matrices satisfy (A ) with =. By [, Theorem ], (A ) holds with probability. Therefore, redefiig U () ad (X t ) o the product probability space Ω U Ω X, by [, Theorem ], there is a subset U of probability such that for each ω U, by Theorem, oe ca fid a measurable subset Ω X,ω Ω X of probability oe such that (3.) holds. By Fubii s Theorem, the set of all pairs (ω, ω ) for which (3.) holds has probability oe.

8 Empirical measures of weighted sums 9 4 Proofs 4. Proof of Theorem I order to use the well kow method of the characteristic fuctio, let s, t R ad cosider the radom variable Φ (s, t) = exp(iss,k + itt,k ). Lemma. For all s, t R, oe ca fid some costat C(s, t) > 0 which does ot deped o such that fo large eough [ E Φ (s, t) Φ(s, t) ] C(s, t). (4.) (log( + )) +δ Proof. For all s, t R, deote by L (s, t) the left had side of (4.). We have the decompositio where L (s, t) = r l= l (s, t, k, l) l (s, t, k, l) = E[(exp(isS,k + itt,k ) Φ(s, t))(exp( iss,l itt,l ) Φ(s, t))]. I additio, if we clearly have ϕ (t) = E[exp(itX )], l (s, t, k, l) = a (s, t, k, l) b (s, t, k, l) (4.) where a (s, t, k, l) = E[exp(isS,k + itt,k iss,l itt,l )] Φ (s, t) ( ) = ϕ j s(u () k,j u() l,j ) + t(v() k,j v() l,j ) Φ (s, t), j= b (s, t, k, l) = Φ(s, t)(e[exp(iss,k + itt,k )] + E[exp( iss,l itt,l )] Φ(s, t)) = Φ(s, t) ϕ j (su () k,j + tv() k,j ) + ϕ j ( su () l,j tv() l,j ) Φ(s, t). j= We shall ow proceed to boud l (s, t, k, l) for all k, l. First of all, we clearly have l (s, t, k, k). Moreover, we will show how to boud a (s, t, k, l) iasmuch as the boud for b (s, t, k, l) ca be hadled similarly. We obviously have j= max k l a (s, t, k, l) A (s, t) + B (s, t), (4.3)

9 9 Electroic Commuicatios i Probability where A (s, t) = max k =l j= j= ϕ j (s(u () k,j u() l,j ) + t(v() k,j v() l,j )) exp( (s(u() k,j u() l,j ) + t(v() k,j v() l,j )) ) ad B (s, t) = max k l j= exp( (s(u() k,j u() l,j ) + t(v() k,j v() l,j )) ) Φ (s, t). It follows from (A ) that fo large eough 0 maxs(u () j,k,l k,j u() l,j ) + t(v() k,j v() l,j ). (4.4) Moreover, by use of the well kow iequality (7.3) of [4], we deduce from (A ) ad (4.4) that A (s, t) α max s(u () k,j k l u() l,j ) + t(v() k,j v() l,j ) 3 I order to boud B (s, t), set d (k, l, s, t) = j= a(s, t) ( (log( + )) +δ max k j= (u () k,j ) + max l (v () l,j )) j= A(s, t). (4.5) (log( + )) +δ (s(u () j= k,j u() l,j ) + t(v() k,j v() l,j )) (s + t ). Assumptios (A ) ad (A 3 ) imply that d (k, l, s, t) s ((u () k,j ) + (u () l,j ) ) j= + t ((v () k,j ) + (v () l,j ) ) j= + s u () k,j u() l,j j= + t v () k,j v() l,j j= + st u () k,j v() k,j u() l,j v() k,j + u() l,j v() l,j u() k,j v() l,j D(s, t) (log( + )) +δ. j= By the elemetary fact that for all a, b > 0, exp( a) exp( b) a b, we obtai that B (s, t) B(s, t). (4.6) (log( + )) +δ

10 Empirical measures of weighted sums 93 Cosequetly, we deduce from (4.3), (4.5) ad (4.6) that max a C(s, t) (s, t, k, l) k l 3(log( + )) +δ. Oe ca verify that the same iequality holds for b (s, t, k, l). Fially, we obtai from (4.) that L (s, t) C(s, t) + 3(log( + )) +δ, which completes the proof of Lemma. To prove almost sure covergece, we will use the followig lemma. Lemma ([5, Theorem ]). Assume that (Y,k ) is a sequece of uiformly bouded C-valued ad possibly depedet radom variables. Let ( ) be a icreasig sequece of itegers which goes to ifiity. If Z = Y,k ad for some costat C > 0, E[ Z ] the (Z ) coverges to zero almost surely. C, (4.7) (log( + )) +δ Proof of Theorem. I order to prove the secod part of Theorem, we apply Lemma to uiformly bouded radom variables Y,k = exp(iss,k + itt,k ) Φ(s, t) with Φ(s, t) = exp( (s + t )/). It follows from Lemma that the coditio (4.7) of Lemma is satisfied. Therefore, (Z ) coverges to zero almost surely. Sice Z = Φ (s, t) Φ(s, t), this shows that Φ (s, t) Φ(s, t) almost surely as, ad we immediately deduce (.9) from [8, Theorem.6]. The proof of the first part of Theorem is similar, ad essetially cosists of takig t = 0 i the above calculatios. Oce we establish the weak covergece o a set of probability, due to cotiuity of Φ(x), the covergece is uiform i x for every ω, see [4, Exercise 4.8]. 4. Proof of Theorems ad 3 The proofs rely o strog approximatio of the partial sum processes idexed by the Lipschitz fuctios f k (x) = cos(πkx), compare [0, Theorems.,.]. We derive suitable approximatio directly from the followig result. Lemma 3 (Sakhaeko [9, Theorem ]). Cosider a sequece (X ) which satisfies the assumptios of Theorem. The, it exists some costat c > 0 such that for every oe ca realize X, X,..., X o a probability space o which there are i.i.d. N(0, ) radom variables X, X,..., X such that the partial sums S t = t X i ad St = i= t X i i=

11 94 Electroic Commuicatios i Probability satisfy where τ is give by (.0). [ ( )] c E exp τ max S t S t + t τ, (4.8) We use this keystoe Lemma 3 as follows. For every, we redefie X, X,..., X oto a ew probability space (Ω, A, P ) o which we have the i.i.d. stadard ormal radom variables X, X,..., X which satisfy (4.8). The, we defie the sequece of weighted sums (S,k ) by (.6) ad we also deote S,k = ( ) πkt X t cos (4.9) with k =,,...,. The assumptios ad the coclusios of Theorems ad 3 are ot affected by such a chage. Oe ca observe from the trigoometric idetities (. to.5) that for every fixed, the radom variables S,, S,,..., S,r are i.i.d with stadard N(0, ) distributio. Therefore, for all x, y R, if x = x + y/, we have the CLT r ( ) { S,k e x Φ(x } ) D N (0, Φ(x)( Φ(x))). (4.0) + This is just the ormal approximatio for the biomial radom variable B(, p ) where the probability of success p = Φ(x ) coverges to p = Φ(x). I additio, we also have from the Kolmogorov-Smirov Theorem r sup x R { S,k e x} Φ(x) D L, (4.) + where L stads for the Kolmogorov-Smirov distributio. Furthermore, cosider the correspodig empirical measure µ = δ es,k. (4.) By Saov s Theorem, see e.g. [8], we have for all closed sets F ad for all ope sets G i M (R), lim sup log P( µ F) if r I(ν) ν F ad lim if log P( µ G) if ν G I(ν). Our goal is to deduce Theorems ad 3 from these results. Proof of Theorem. For all x R, deote Z (x) = {S,k x} ad F (x) = r {S,k x}.

12 Empirical measures of weighted sums 95 Let Z (x) ad F (x) be the correspodig sums associated with S,k give by (4.9) from Lemma 3. Fix ε > 0, ad let ε = ε π/. If { A = max S,k S },k > ε k ad R = A, we clearly have Z (x ε ) R Z (x) Z (x + ε ) + R. Hece, it follows from the trivial boud Φ(x ± ε ) Φ(x) ε / π that { Z (x) Φ(x) Z (x ε ) Φ(x ε ) ε R, Z (x) Φ(x) Z (x + ε ) Φ(x + ε ) + ε + R. (4.3) I additio, we also have { r (F (x) Φ(x)) ( F (x ε ) Φ(x ε )) ε R, which gives r (F (x) Φ(x)) ( F (x + ε ) Φ(x + ε )) + ε + R, r sup F (x) Φ(x) sup F (x) Φ(x) ε + R (4.4) x x We ow claim that (R ) goes to zero i probability. As a matter of fact, we have max S,k S,k = k max ( ) (X k t X πkt t )cos max ( (S t k S t ) cos ( πkt ) cos Sice the cosie is a Lipschitz fuctio, we obtai that max S,k S,k k max k ( πk(t + ) S t S t πk + )) + S S. S S ( + π) max t S t S t. (4.5) From Lemma 3 together with Markov iequality, we obtai for large eough so that / c/τ, P(A ) P max S t S ε ) π t t r ( + π ) ( cε ) π exp τ ( + π ) ( [ ( E exp c/τ max )] S t S t t ( exp log( + τ ) cε ) π τ 0. ( + π )

13 96 Electroic Commuicatios i Probability Cosequetly R 0 i probability. Fially, as ε > 0 is arbitrary, we deduce (.) from (4.0) ad (4.3) while (.3) follows from (4.) ad (4.4), which achieves the proof of Theorem. Our proof of Theorem 3 is based o the followig approximatio lemma. Lemma 4 ([, Theorem 4.9]). Suppose that the sequece of radom variables (S,k ) ad ( S,k ) are such that, for every θ > 0, lim sup [ log E exp(θ ] S,k S,k ). (4.6) If the sequece of empirical measures ( µ ) give by (4.) satisfies a LDP i M (R) with speed ad good rate fuctio I, the the sequece of empirical measures (µ ) give by (.4) satisfies a LDP i M (R) with the same speed ad the same rate fuctio I. Proof of Theorem 3. The large deviatio priciple for the sequece ( µ ) follows from Saov s Theorem. I order to complete the proof, we oly eed to verify assumptio (4.6) of Lemma 4. Iequality (4.5) implies that fo large eough, there is some costat C > 0 such that, for every θ > 0, [ E exp(θ S,k S ],k ) Sice r 4 = o(), we ifer from (4.8) that fo large eough, [ log E exp(θ [ E exp(θ max S,k S ],k ) k [ E exp(cr / max S t S ] t ). t S,k S ] log( + /τ),k ), which completes the proof of Theorem 3 as log() = o( ). Ackowledgemet The secod-amed author (WB) thaks S. Miller for the early versio of [7], ad M. Peligrad for a helpful discussio. The authors thak J. Silverstei for showig us the elemetary direct proof of Corollary 7. Refereces [] Bai, Z. D. Methodologies i spectral aalysis of large-dimesioal radom matrices, a review. Statist. Siica 9, 3 (999), With commets by G. J. Rodgers ad Jack W. Silverstei; ad a rejoider by the author. MR7663 [] Baxter, J. R., ad Jai, N. C. A approximatio coditio for large deviatios ad some applicatios. I Covergece i ergodic theory ad probability (Columbus, OH, 993), vol. 5 of Ohio State Uiv. Math. Res. Ist. Publ. de Gruyter, Berli, 996, pp MR4597 [3] Berkes, I., ad Csáki, E. A uiversal result i almost sure cetral limit theory. Stochastic Process. Appl. 94 (00), MR835848

14 Empirical measures of weighted sums 97 [4] Billigsley, P. Probability ad measure, third ed. Wiley Series i Probability ad Mathematical Statistics. Joh Wiley & Sos Ic., New York, 995. A Wiley-Itersciece Publicatio. MR34786 [5] Bose, A., Chatterjee, S., ad Gagopadhyay, S. Limitig spectral distributios of large dimesioal radom matrices. J. Idia Statist. Assoc. 4, (003), 59. MR0995 [6] Bose, A., ad Mitra, J. Limitig spectral distributio of a special circulat. Statist. Probab. Lett. 60, (00), 0. MR [7] Brosamler, G. A almost sure everywhere cetral limit theorem. Math. Proc. Cambridge Philos. Soc. 04 (988), MR09576 [8] Dembo, A., ad Zeitoui, O. Large deviatios techiques ad applicatios, secod ed., vol. 38 of Applicatios of Mathematics (New York). Spriger-Verlag, New York, 998. MR69036 [9] Fay, G., ad Soulier, P. The periodogram of a i.i.d. sequece. Stochastic Process. Appl. 9, (00), MR8759 [0] Grama, I., ad Nussbaum, M. A fuctioal Hugaria costructio for sums of idepedet radom variables. A. Ist. H. Poicaré Probab. Statist. 38, 6 (00), E l hoeur de J. Bretagolle, D. Dacuha-Castelle, I. Ibragimov. MR [] Hiai, F., ad Petz, D. The semicircle law, free radom variables ad etropy, vol. 77 of Mathematical Surveys ad Moographs. America Mathematical Society, Providece, RI, 000. MR [] Jiag, T. Maxima of etries of Haar distributed matrices. Probab. Theory Related Fields 3, (005), 44. MR05046 [3] Kokoszka, P., ad Mikosch, T. The periodogram at the Fourier frequecies. Stochastic Process. Appl. 86, (000), MR7496 [4] Lacey, M. T., ad Phillip, W. A ote o the almost sure cetral limit theorem. Statist. Probab. Lett. 9 (990), MR04584 [5] Lyos, R. Strog laws of large umbers for weakly correlated radom variables. Michiga Math. J. 35, 3 (988), MR [6] March, P., ad Seppäläie, T. Large deviatios from the almost everywhere cetral limit theorem. J. Theoret. Probab. 0, 4 (997), MR48655 [7] Massey, A., Miller, S. J., ad Sisheimer, J. Distributio of Eigevalues of Real Symmetric Palidromic Toeplitz Matrices ad Circulat Matrices. Jour. Theoret. Probab. (005). (to appear), arxiv:math.pr/0546. [8] Patrizia Berti, L. P., ad Rigo, P. Almost sure weak covergece of radom probability measures. Stochastics: A Iteratioal Joural of Probability ad Stochastic Processes 78 (006), MR36634

15 98 Electroic Commuicatios i Probability [9] Sakhaeko, A. I. Rate of covergece i the ivariace priciple for variables with expoetial momets that are ot idetically distributed. I Limit theorems for sums of radom variables, vol. 3 of Trudy Ist. Mat. (Eglish traslatio i: Limit theorems for sums of radom variables, eds. Balakrisha ad A.A.Borovkov. Optimizatio Software, New York, 985.). Nauka Sibirsk. Otdel., Novosibirsk, 984, pp MR [0] Sakhaeko, A. I. O the accuracy of ormal approximatio i the ivariace priciple [traslatio of Trudy Ist. Mat. (Novosibirsk) 3 (989), Asimptot. Aaliz Raspred. Sluch. Protsess., 40 66; MR 9d:6008]. Siberia Adv. Math., 4 (99), Siberia Advaces i Mathematics. MR38005 [] Schatte, P. O strog versios of the almost sure cetral limit theorem. Math. Nachr. 37 (988), MR