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MODERATE DEVIATIONS FOR THE EIGENVALUE COUNTING FUNCTION OF WIGNER MATRICES 3 1 0 2 Hanna Do¨ring1, Peter Eichelsbacher2 n a J 1 1 R] Abstract: We establish a moderate deviation principle (MDP) for the number P of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymp- . h totics of the variance of the eigenvalue counting function of GUE matrices due t a to Gustavsson. The extension to certain families of Wigner matrices is based m on the Tao and Vu Four Moment Theorem and applies localization results by [ Erdös, Yau and Yin. Moreover we investigate families of covariance matrices as 2 v well. 1 2 2 AMS 2000 Subject Classification: Primary 60B20; Secondary 60F10, 15A18 0 . 4 0 Key words: Large deviations, moderate deviations, Wigner random matrices, covariance 1 matrices, Gaussian ensembles, Four Moment Theorem 1 : v i X r a 1 Ruhr-Universität Bochum, Fakulta¨t fu¨r Mathematik, NA 3/68, D-44780 Bochum, Germany, [email protected] 2 Ruhr-Universität Bochum, Fakulta¨t fu¨r Mathematik, NA 3/66, D-44780 Bochum, Germany, [email protected] Both authors have been supported by Deutsche Forschungsgemeinschaftvia SFB/TR 12. The first author was supported by the international research training group 1339 of the DFG. 2 HANNADO¨RINGANDPETER EICHELSBACHER 1. Introduction Recently, in [4] the Central Limit Theorem (CLT) for the eigenvalue counting function of Wigner matrices, that is the number of eigenvalues falling in an interval, was established. This universality result relies on fine asymptotics of the variance of the eigenvalue counting function, on the Fourth Moment Theorem due to Tao and Vu as well as on recent localization results due to Erdös, Yau and Yin. See also [3]. Our paper is concerned with the moderate deviation principle (MDP) of the eigenvalue counting function. We will start with the MDP for Wigner matrices where the entries are Gaussian (the so-called Gaussian unitary ensemble (GUE)), first proven by the authors in [11]. Next we establish a MDP for individual eigenvalues in the bulk of the semicircle law (which is an MDP corresponding to the Gaussian behaviour proved in [14]). This MDP will be extended to certain families of Wigner matrices by means of the Four Moment Theorem (see [21], [20]). It seem to be the first application of the Four Moment Theorem to be able to obtain not only universality of convergence in distribution but also to obtain deviations results on a logarithmic scale, universally. Finally a strategy based on the MDP for individual eigenvalues in the bulk will be shown to imply the MDP for the eigenvalue counting function, universally for certain Wigner matrices. In the meantime, we successfully apply the Four Moment Theorem to obtain MDPs also at the edge of the spectrum as well as for the determinant of certain Wigner matrices, see [9], [10]. Consider two independent families of i.i.d. random variables (Z ) (complex-valued) i,j 1 i<j ≤ and (Y ) (real-valued), zero mean, such that EZ2 = 0,E Z 2 = 1 and EY2 = 1. Consider i 1≤i 1,2 | 1,2| 1 the (Hermitian) n n matrix M with entries M (j,i) = M (i,j) = Z /√n for i < j and × n n∗ n i,j M (i,i) = M (i,i) = Y /√n. Such a matrix is called Hermitian Wigner matrix. An important n∗ n i example of Wigner matrices is the case where the entries are Gaussian, giving rise to the so- called Gaussian Unitary Ensembles (GUE). GUE matrices will be denoted by M . In this n′ case, the joint law of the eigenvalues is known, allowing a good description of their limiting behaviour both in the global and local regimes (see [1]). In the Gaussian case, the distribution of the matrix is invariant by the action of the group SU(n). The eigenvalues of the matrix M are independent of the eigenvectors which are Haar distributed. If (Z ) are real- n′ i,j 1 i<j ≤ valued the symmetric Wigner matrix is defined analogously and the case of Gaussian variables with EY2 = 2 is of particular importance, since their law is invariant under the action of the 1 orthogonal group SO(n), known as Gaussian Orthogonal Ensembles (GOE). Denoteby λ ,...,λ the real eigenvalues of the normalised Hermitian (or symmetric) Wigner 1 n matrix W = 1 M . The Wigner theorem states that the empirical measure n √n n n 1 L := δ n n λi Xi=1 MODERATE DEVIATIONS FOR WIGNER MATRICES 3 on the eigenvalues of W converges weakly almost surely as n to the semicircle law n → ∞ 1 d̺ (x) = √4 x21 (x)dx, sc [ 2,2] 2π − − (see [1, Theorem 2.1.21, Theorem 2.2.1]). Consequently, for any interval I R, ⊂ n 1 1 N (W ) := 1 ̺ (I) n I n n {λi∈I} → sc Xi=1 almost surely as n . At the fluctuation level, it is well known that for the GUE, W := → ∞ n′ 1 M satisfies a CLT (see [18]): Let I be an interval in R. If V(N (W )) as n , √n n′ n In n′ → ∞ → ∞ then N (W ) E[N (W )] In n′ − In n′ N(0,1) V(N (W )) → In n′ as n in distribution. p → ∞ In [14] the asymptotic behavior of the expectation and the variance of the counting function N (W ) for intervals I = [y(n), ) with y(n) = G 1(k/n) (where k = k(n) is such that In n′ n ∞ − k/n a (0,1) – strictly in the bulk–, and G denotes the distribution function of the → ∈ semicircle law) was established: logn 1 E[N (W )] = n k(n)+O and V(N (W )) = +o(1) logn. (1.1) In n′ − n In n′ 2π2 (cid:0) (cid:1) (cid:0) (cid:1) The proof applied strong asymptotics for orthogonal polynomials with respect to exponential weights, see [5]. In particular the CLT holds for N (W ) if I = [y, ) with y ( 2,2), and I n′ ∞ ∈ − moreover in this case one obtains N (W ) n̺ (I) I n′ − sc N(0,1) → 1 logn 2π2 q as n (called the CLT with numerics). These conclusions were extended to non-Gaussian → ∞ Wigner matrices in [4]. Certain deviations results and concentration properties for Wigner matrices were considered. Our aim is to establish certain moderate deviation principles. Recall that a sequence of laws (P ) on a Polish space Σ satisfies a large deviation principle (LDP) with good rate function n n 0 ≥ I : Σ R and speed s going to infinity with n if and only if the level sets x : I(x) M , + n → { ≤ } 0 M < , of I are compact and for all closed sets F ≤ ∞ limsups 1logP (F) inf I(x) n −n n ≤ −x F →∞ ∈ whereas for all open sets O liminfs 1logP (O) inf I(x). n −n n ≥ −x O →∞ ∈ We say that a sequence of random variables satisfies the LDP when the sequence of measures induced by these variables satisfies the LDP. Formally a moderate deviation principle is noth- ing else but the LDP. However, we speak about a moderate deviation principle (MDP) for a 4 HANNADO¨RINGANDPETER EICHELSBACHER sequence of random variables, whenever the scaling of the corresponding random variables is between that of an ordinary Law of Large Numbers (LLN) and that of a CLT. Large deviation results for the empirical measures of Wigner matrices are still only known for the Gaussian ensembles since their proof is based on the explicit joint law of the eigenvalues, see [2] and [1]. A moderate deviation principle for the empirical measure of the GUE or GOE is also known, see [6]. This moderate deviations result does not have yet a fully universal version for Wigner matrices. It has been generalised to Gaussian divisible matrices with a deterministic self-adjoint matrix added with converging empirical measure [6] and to Bernoulli matrices [8]. Our first result is a MDP for the number of eigenvalues of a GUE matrix in an interval. It is a little modification of [11, Theorem 5.2]. In the following, for two sequences of real numbers (a ) and (b ) we denote by a b the convergence lim a /b = 0. n n n n n n n n n ≪ →∞ Theorem 1.1. Let M be a GUE matrix and W := 1 M . Let I be an interval in R. If n′ n′ √n n′ n V(N (W )) for n , then for any sequence (a ) of real numbers such that In n′ → ∞ → ∞ n n 1 a V(N (W )) (1.2) ≪ n ≪ In n′ p the sequence (Z ) with n n N (W ) E[N (W )] Z = In n′ − In n′ n a V(N (W )) n In n′ satisfies a MDP with speed a2 and rate functpion I(x) = x2. n 2 Remark 1.2. Let I = [y, ) with y ( 2,2). An easy consequence of Theorem 1.1 is that ∞ ∈ − the sequence (Zˆ ) with n n N (W ) n̺ (I) Zˆ = I n′ − sc n a 1 logn n 2π2 q satisfies the MDP with the same speed, the same rate function, and in the same regime (1.2) (called the MDP with numerics). Inourpaperweextendtheseconclusionstocertainnon-GaussianHermitianWignermatrices. Tail-condition (T): Say that M satisfies the tail-condition (T) if the real part η and the n imaginary part η of M are independent and have a so-called stretched exponential decay: n there are two constants C and C such that ′ P η tC e t and P η tC e t − − | | ≥ ≤ | | ≥ ≤ (cid:0) (cid:1) (cid:0) (cid:1) for all t C . ′ ≥ We say that two complex random variables η and η match to order k if 1 2 E Re(η )mIm(η )l = E Re(η )mIm(η )l 1 1 2 2 (cid:2) (cid:3) (cid:2) (cid:3) for all m,l 0 such that m+l k. ≥ ≤ MODERATE DEVIATIONS FOR WIGNER MATRICES 5 The following theorem is the main result of our paper: Theorem 1.3. Let M be a Hermitian Wigner matrix whose entries satisfy tail-condition (T) n and match the corresponding entries of GUE up to order 4. Set W := 1 M . Then, for any n √n n y ( 2,2) and I(y) = [y, ), with Y := N (W ), for any sequence (a ) of real numbers n I(y) n n n ∈ − ∞ such that 1 a V(Y ) (1.3) n n ≪ ≪ p the sequence (Z ) with n n Y E[Y ] n n Z = − n a V(Y ) n n p satisfies a MDP with speed a2 and rate function I(x) = x2. Moreover the sequence n 2 Y n̺ (I) ˆ n sc Z = − n a 1 logn n 2π2 q satisfies the MDP with the same speed, the same rate function, and in the same regime (1.3) (called the MDP with numerics). Before we will prove the MDP for the GUE, we describe the organisation of the next sections. In a first step, we will apply Theorem 1.1 to obtain a MDP for eigenvalues in the bulk of the semicircle law. Next we extend this result to certain families of Hermitian Wigner matrices satisfying tail-condition (T) by means of the Four Moment Theorem due to Tao and Vu. This is the content of Section 2. In Section 3, we show the MDP with numerics for the counting function of Wigner matrices. Moreover we apply recent results of Erdös, Yau and Yin [12] on the localization of eigenvalues and of Dallaporta and Vu [4] in order to prove the MDP without numerics. Section 4 is devoted to discuss the case of symmetric real Wigner matrices as well as the symplectic Gaussian ensemble applying interlacing formulas due to Forrester and Rains, [13]. Finally, in Section 5 we present results for covariance matrices. We prove a universal MDP with numerics for the counting eigenvalue function of covariance matrices. In [11] we proved a MDP for certain determinantal point processes (DPP), including GUE. Theorem 1.1 follows immediately from an improvement of Theorem 5.2 in [11], which can be easily observed applying the proof of [1, Theorem 4.2.25]. Let Λ be a locally compact Polish space, equipped with a positive Radon measure µ on its Borel σ-algebra. Let (Λ) denote + M the set of positive σ-finite Radon measures on Λ. A point process is a random, integer-valued χ (Λ), and it is simple if P( x Λ : χ( x ) > 1) = 0. Let χ be simple. A locally + ∈ M ∃ ∈ { } integrable function ̺ : Λk [0, ) is called a joint intensity (correlation), if for any mutually → ∞ disjoint family of subsets D ,...,D of Λ 1 k k E χ(D ) = ̺ (x ,...,x )dµ(x ) dµ(x ), i k 1 k 1 k (cid:0)Yi=1 (cid:1) ZQki=1Di ··· 6 HANNADO¨RINGANDPETER EICHELSBACHER where E denotes the expectation with respect to the law of the point configurations of χ. A simple point process χ is said to be a determinantal point process with kernel K if its joint intensities ̺ exist and are given by k ̺ (x ,...,x ) = det K(x ,x ) . (1.4) k 1 k i j i,j=1,...,k (cid:0) (cid:1) An integral operator : L2(µ) L2(µ) with kernel K given by K → (f)(x) = K(x,y)f(y)dµ(y),f L2(µ), K Z ∈ is admissible with admissible kernel K if is self-adjoint, nonnegative and locally trace-class K (for details see [1, 4.2.12]). A standard result is, that an integral compact operator with K admissible kernel K possesses the decomposition f(x) = n η ϕ (x) ϕ ,f , where the K k=1 k k h k iL2(µ) functions ϕ are orthonormal in L2(µ), n is either finite or iPnfinite, and η > 0 for all k, leading k k to n K(x,y) = η ϕ (x)ϕ (y), (1.5) k k ∗k X k=1 an equality in L2(µ µ). Moreover, an admissible integral operator with kernel K is called × K good with good kernel K if the η in (1.5) satisfy η (0,1]. If the kernel K of a determinantal k k ∈ point process is (locally) admissible, then it must in fact be good, see [1, 4.2.21]. The following example is the main motivation for discussing determinantal point processes in this paper. Let (λn,...,λn) be the eigenvalues of the GUE (Gaussian unitary ensemble) 1 n of dimension n and denote by χ the point process χ (D) = n 1 . Then χ is a n n i=1 {λni∈D} n determinantal point processwithadmissible, goodkernel K(n)(x,yP) = kn=−01Ψk(x)Ψk(y),where the functions Ψ are the oscillator wave-functions, that is Ψ (x) := e−xP2/4Hk(x), where H (x) := k k √√2πk! k ( 1)kex2/2 dk e x2/2 is the k-th Hermite polynomial; see [1, Def. 3.2.1, Ex. 4.2.15]. − dxk − We will apply the following representation due to [15, Theorem 7]: Suppose χ is a determinantal process with good kernel K of the form (1.5), with η < . Let (I )n be k k ∞ k k=1 independent Bernoulli variables with P(I = 1) = η . Set K (x,yP) = n I ϕ (x)ϕ (y), and k k I k=1 k k ∗k let χ denote the determinantal point process with random kernel KP. Then χ and χ have I I I the same distribution, interpreted as stating that the mixture of determinental processes χ I has the same distribution as χ. In the following let K be a good kernel and for D Λ we ⊂ write K (x,y) = 1 (x)K(x,y)1 (y). Let D be such that K is trace-class, with eigenvalues D D D D η , k 1. Then χ(D) has the same distribution as ξ where ξ are independent Bernoulli k ≥ k k k random variables with P(ξ = 1) = η and P(ξ = 0P) = 1 η . k k k k − Theorem 1.4. Consider a sequence (χ ) of determinantal point processes on Λ with good n n kernels K . Let D be a sequence of measurable subsets of Λ such that (K ) is trace-class. n n n Dn MODERATE DEVIATIONS FOR WIGNER MATRICES 7 Assume that (a ) is a sequence of real numbers such that n n n 1 a ηn(1 ηn) 1/2, ≪ n ≪ k − k (cid:0)Xk=1 (cid:1) where ηn are the eigenvalues of K . Then (Z ) with k n n n 1 χ (D ) E(χ (D )) n n n n Z := − n a V(χ (D )) n n n p satisfies a moderate deviation principle with speed a2 and rate function I(x) = x2. n 2 Remark 1.5. In [16], functional moderate deviations for triangular arrays of certain independent, not identically distributed random variables are considered. Our result, Theorem 1.4, seem to follow from Proposition 1.9 in [16]. Anyhow we prefer to present a direct proof. Proof of Theorem 1.4. We adapt the proof of [1, Theorem 4.2.25]. We write K for the kernel n (K ) and let S := V(χ (D )). χ (D ) has the same distribution as the sum of inde- n Dn n n n n n pendent Bernoulli variapbles ξn whose parameters ηn are the eigenvalues of K . We obtain k k n S2 = ηn(1 ηn) and since K is trace-class we can write, for any θ real n k k − k n P θa2(ξn ηn) logE eθa2nZn = logE exp n k − k (cid:20) a S (cid:21) (cid:2) (cid:3) Xk (cid:0) n n (cid:1) θa2 ηn = n k k + log 1+ηn ea2nθ/(anSn) 1 . − aPS (cid:18) k − (cid:19) n n Xk (cid:0) (cid:1) Forany realθ and nlargeenoughsuch that ηn(eθan/Sn 1) [0,1] we apply Taylor for log(1+x) k − ∈ and obtain 1 θ2a2 ηn(1 ηn) a3 ηn(1 ηn) logE eθa2nZn = n k k − k +o n k k − k . a2 P2a2S2 (cid:18) P a2S3 (cid:19) n (cid:2) (cid:3) n n n n The last term is o an . Applying the Theorem of Ga¨rtner-Ellis, [7, Theorem 2.3.6], the result Sn follows. (cid:0) (cid:1) (cid:3) Proof of Theorem 1.1. Now the first statement of Theorem 1.1 follows since V(χ (D )) , n n → ∞ see [1, Cor. 4.2.27]. In particular for any I = [y, ) with y ( 2,2) ∞ ∈ − N (W ) E[N (W )] Z˜ := I n′ − I n′ n a V(N (W )) n I n′ p satisfies the MDP. The MDP with numerics (see Remark 1.2) follows, since the sequences (Z˜ ) n n and(Zˆ ) areexponentially equivalent inthe sense of definition [7, Definition 4.2.10], andhence n n the result follows from [7, Theorem 4.2.13]: Let N (W ) n̺ (I) Z := I n′ − sc . n′′ a V(N (W )) n I n′ p 8 HANNADO¨RINGANDPETER EICHELSBACHER Since |Zñ − Zn′′| = (cid:12)E[aNnI√(WVn′(N)]−I(nW̺sn′c)()I)(cid:12) → 0 as n → ∞, (Zñ)n and (Zn′′)n are exponentially equivalent. Moreove(cid:12)r by Taylor we(cid:12)obtain |Zn′′ − Zˆn| = oa(n1)N√I(WV(n′N)−I(nW̺sn′c)()I) and the MDP for (Z ) implies limsup 1 logP Z Zˆ > ε = for any ε > 0. (cid:3) n′′ n n→∞ a2n | n′′ − n| −∞ (cid:0) (cid:1) 2. Moderate deviations for eigenvalues in the bulk Under certain conditions on i it was proved in [14] that the i-th eigenvalue λ of the GUE i W satisfies a CLT. Consider t(x) [ 2,2] defined for x [0,1] by n′ ∈ − ∈ t(x) 1 t(x) x = d̺ (t) = √4 x2dx. sc Z 2π Z − 2 2 − − Then for i = i(n) such that i/n a (0,1) as n (i.e. λ is eigenvalue in the bulk), i → ∈ → ∞ λ (W ) satisfies a CLT: i n′ 4 t(i/n)2λ (W ) t(i/n) − i n′ − N(0,1) (2.1) r 2 √logn → n for n . Remark that t(i/n) is sometimes called the classical or expected location of the i-th → ∞ eigenvalue. The standard deviation is √logn 1 . Note that from the semicircular law, π√2 n̺sc(t(i/n)) the factor 1 is the mean eigenvalue spacing. n̺sc(t(i/n)) The proof in [14] is achieved by the tight relation between eigenvalues and the counting function expressed by the elementary equivalence, for I(y) = [y, ), y R, ∞ ∈ N (W ) n i if and only if λ y. (2.2) I(y) n i ≤ − ≤ Hence the theorem due to Costin and Lebowitz as well as Soshnikov, see [18], can be applied. Moreover the proof in [14] relies on fine asymptotics for the Airy function and the Hermite polynomials due to [5]. Our first result in this Section is a corresponding MDP for λ in the bulk: i Theorem 2.1. Consider the GUE matrix W = 1 M . Consider i = i(n) such that i/n n′ √n n′ → a (0,1) as n . If λ denotes the eigenvalue number i in the GUE matrix W it holds that ∈ → ∞ i n′ for any sequence (a ) of real numbers such that 1 a √logn the sequence (X ) with n n ≪ n ≪ n′ n 4 t(i/n)2λ (W ) t(i/n) X = − i n′ − n′ r 2 a √logn n n satisfies a MDP with speed a2 and rate function I(x) = x2. n 2 Interesting enough, thisresult canbeextended tolargefamiliesofHermitianWigner matrices satisfying tail-condition (T) by means of the Four Moment Theorem of Tao and Vu: MODERATE DEVIATIONS FOR WIGNER MATRICES 9 Theorem 2.2. Consider a Hermitian Wigner matrix W = 1 M whose entries satisfy tail- n √n n condition (T) and match the corresponding entries of GUE up to order 4. Consider i = i(n) such that i/n a (0,1) as n . If λ denotes the eigenvalue number i of W it holds that i n → ∈ → ∞ for any sequence (a ) of real numbers such that 1 a √logn, the sequence (X ) with n n n n n ≪ ≪ 4 t(i/n)2λ (W ) t(i/n) i n X = − − n r 2 a √logn n n satisfies a MDP with speed a2 and rate function I(x) = x2. n 2 Remark 2.3. In [14] a CLT at the edge of the spectrum was considered also. The proof applies the result of Costin and Lebowitz as well as fine asymptotics presented in [5]. Consider i → ∞ such that i/n 0 as n and consider λ , eigenvalue number n i in the GUE or n i → → ∞ − − Hermitian Wigner case. A CLT for the rescaled λ is stated in [14, Theorem 1.2]. We would n i − be able to formulate and prove a MDP for eigenvalue λ , but it is not the main focus of this n i − paper. We omit this. Proof of Theorem 2.1. Theproofis oriented totheproofof [14, Theorem 1.1]andwill applythe precise asymptotic behaviour of the expectation and of the variance of the counting function N (W ),see(1.1),whichisareformulationof[14,Lemma2.1-2.3]. LetP denotetheprobability I n′ n of the GUE determinantal point processes, and set √logn √2 I := t(i/n)+ξa , . n n (cid:20) n 4 t(i/n)2 ∞(cid:19) − p Now we apply relation (2.2) and obtain λ (W ) t(i/n) √logn √2 P i n′ − ξ = P λ (W ) ξa +t(i/n) n(cid:18)a √logn √2 ≤ (cid:19) n(cid:18) i n′ ≤ n n 4 t(i/n)2 (cid:19) n n √4 t(i/n)2 − − p N (W ) E[N (W )] n i E[N (W )] = P N (W ) n i = P In n′ − In n′ − − In n′ . n In n′ ≤ − n(cid:18) a (V(N (W )))1/2 ≤ a (V(N (W )))1/2(cid:19) (cid:0) (cid:1) n In n′ n In n′ Since i/n a (0,1), by definition of t(x) we obtain t(i/n) ( 2,2). Moreover since → ∈ ∈ − a √logn we have ξa √logn √2 +t(i/n) ( 2,2) for n large. Therefore with (1.1) n ≪ n n √4 t(i/n)2 ∈ − − we have for n sufficiently large that logn 1 E[N (W )] = n̺ (I )+O and V(N (W )) = +o(1) logn. In n′ sc n n In n′ 2π2 (cid:0) (cid:1) (cid:0) (cid:1) With b(n) := a √logn √2 and f (t(i/n)) := t(i/n)+ξb(n) we get from symmetry n n √4 t(i/n)2 n − 1 fn(t(i/n)) fn(t(i/n)) ∞ ̺ (I ) = ̺ (x)dx = ̺ (x)dx = 1 ̺ (x)dx. sc n sc sc sc Z 2 −Z −Z fn(t(i/n)) 0 2 − 10 HANNADO¨RINGANDPETER EICHELSBACHER Now fn(t(i/n)) t(i/n) fn(t(i/n)) i fn(t(i/n)) ̺ (x)dx = ̺ (x)dx+ ̺ (x)dx = + ̺ (x)dx sc sc sc sc Z Z Z n Z 2 2 t(i/n) t(i/n) − − and fn(t(i/n)) 1 ̺ (x)dx = ξb(n) 4 t(i/n)2 +O b(n)2). sc Z 2π − t(i/n) p (cid:0) Summarizing we obtain 1 a2 logn n̺ (I ) = n i ξa logn +O n sc n n − − √2π n p (cid:0) (cid:1) and therefore n i E[N (W )] − − In n′ = ξ +ε(n), a (V(N (W )))1/2 n In n′ where ε(n) 0 as n . By Theorem 1.1 we obtain for every ξ < 0 → → ∞ 1 λ (W ) t(i/n) ξ2 lim logP i n′ − ξ = . (2.3) n n a2 (cid:18)a √logn √2 ≤ (cid:19) − 2 →∞ n n n √4 t(i/n)2 − With λ (W ) t(i/n) P i n′ − ξ = P N (W ) n i+1 n(cid:18)a √logn √2 ≥ (cid:19) n In n′ ≥ − n n √4 t(i/n)2 (cid:0) (cid:1) − the same calculations lead, for every ξ > 0, to 1 λ (W ) t(i/n) ξ2 lim logP i n′ − ξ = . (2.4) n n a2 (cid:18)a √logn √2 ≥ (cid:19) − 2 →∞ n n n √4 t(i/n)2 − Hence the conclusion follows: see for example [7, Proof of Theorem 2.2.3]. To be more precise we apply the preceding results (2.3) and (2.4) with Theorem 4.1.11 in [7], which allows us to derive a MDP from the limiting behaviour of probabilities for a basis of topology. For the latter, we choose all open intervals (a,b), where at least one of the endpoints is finite and where none of the endpoints is zero. Denote the family of such intervals by . From (2.3) and (2.4), U it follows for each U = (a,b) , ∈ U b2/2 : a < b < 0 1 := lim logP X U =  0 : a < 0 < b LU n→∞ a2n (cid:0) n′ ∈ (cid:1)  a2/2 : 0 < a < b  By [7, Theorem 4.1.11], (X ) satisfies a weak MDPwith speed a2 and rate function n′ n n t2 t sup = . U 7→ L 2 U ;t U ∈U ∈ With (2.4), it follows that (X ) is exponentially tight, hence by Lemma 1.2.18 in [7], (X ) n′ n n′ n satisfies the MDP with the same speed and the same good rate function. This completes the (cid:3) proof.