Concentration of the Spectral Measure for Large Random Matrices with Stable Entries

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1 E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol , Paper no. 5, pages Journal URL Concentration of the Spectral Measure for Large Random Matrices with Stable Entries Christian Houdré Hua Xu Abstract We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries, in particular, stable or heavy tails ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value. Key words: Spectral Measure, Random Matrices, Infinitely divisibility, Stable Vector, Concentration. AMS 2 Subject Classification: Primary 6E7, 6F1, 15A42, 15A52. Submitted to EJP on June 12, 27, final version accepted January 18, 28. Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, 3332, houdre@math.gatech.edu Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, 3332, xu@math.gatech.edu 17

2 1 Introduction and Statements of Results: Large random matrices have recently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics e.g., see Mehta [24], Bai and Silverstein [4], Johnstone [18], Anderson, Guionnet and Zeitouni [2]. For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral measure or of the largest eigenvalue is understood. Many of these results hold true for matrices with independent entries satisfying some moment conditions Wigner [35], Tracy and Widom [33], Soshnikov [28], Girko [8], Pastur [25], Bai [3], Götze and Tikhomirov [9]. There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [3] who, using ideas from perturbation theory, studied the distribution of the largest eigenvalue of Wigner matrices with entries having heavy tails. Recall that a real or complex Wigner matrix is a symmetric or Hermitian matrix whose entries M i,i, 1 i, and M i,j, 1 i < j, form two independent families of iid complex valued in the Hermitian case random variables. In particular, see [3], for a properly normalized Wigner matrix with entries belonging to the domain of attraction of an α-stable law, lim P λ max x = exp x α here λ max is the largest eigenvalue of such a normalized matrix. Further, Soshnikov and Fyodorov [32], using the method of determinants, derived results for the largest singular value of K rectangular matrices with independent Cauchy entries, showing that the largest singular value of such a matrix is of order K 2 2 see also the survey article [31], where band and sparse matrices are studied. On another front, Guionnet and Zeitouni [1], gave concentration results for functionals of the empirical spectral measure of, self-adjoint, random matrices whose entries are independent and either satisfy a logarithmic Sobolev inequality or are compactly supported. They obtained, for such matrices, the subgaussian decay of the tails of the empirical spectral measure when it deviates from its mean. They also noted that their technique could be applied to prove results for the largest eigenvalue or for the spectral radius of such matrices. Alon, Krivelevich and Vu [1] further obtained concentration results for any of the eigenvalues of a Wigner matrix with uniformly bounded entries see, Ledoux [2] for more developments and references. Our purpose in the present work is to deal with matrices whose entries form a general infinitely divisible vector, and in particular a stable one without independence assumption. As well known, unless degenerated, an infinitely divisible random variable cannot be bounded. We obtain concentration results for functionals of the corresponding empirical spectral measure, allowing for any type of light or heavy tails. The methodologies developed here apply as well to the largest eigenvalue or to the spectral radius of such random matrices. Following the lead of Guionnet and Zeitouni [1], let us start by setting our notation and framework. Let M C be the set of Hermitian matrices with complex entries, which is throughout equipped with the Hilbert-Schmidt or Frobenius or entrywise Euclidean norm: M = trm M = M i,j 2. Let f be a real valued function on R. The function f can be viewed as mapping M C to M C. Indeed, for M = M i,j 1 i,j M C, so that M = UDU, where D is a 18 i,j=1

3 diagonal matrix, with real entries λ 1,..., λ, and U is a unitary matrix, set fλ 1 fm = UfDU fλ 2, fd = fλ Let trm = i=1 M i,i be the trace operator on M C and set also tr M = 1 M i,i. For a random Hermitian matrix with eigenvalues λ 1, λ 2,..., λ, let F x = 1 i=1 1 {λ i x} be the corresponding empirical spectral distribution function. As well known, if M is a Hermitian Wigner matrix with E[M 1,1 ] = E[M 1,2 ] =, E[ M 1,2 2 ] = 1, and E[M 2 1,1 ] <, the spectral measure of M/ converges to the semicircle law: σdx = 4 x 2 1 { x 2} dx/2π [2]. We study below the tail behavior of either the spectral measure or the linear statistic of fm for classes of matrices M. Still following Guionnet and Zeitouni, we focus on a general random matrix X A given as follows: X A = X A i,j 1 i,j, X A = X A, X A i,j = 1 A i,j ω i,j, with ω i,j 1 i,j = ωi,j R + 1ωi,j I 1 i,j, ω i,j = ω j,i, and where ω i,j, 1 i j is a complex valued random variable with law P i,j = Pi,j R + 1Pi,j I, 1 i j, with P I i,i = by the Hermite property. Moreover, the matrix A = A i,j 1 i,j is Hermitian with, in most cases, non-random complex valued entries uniformly bounded, say, by a. Different choices for the entries of A allow to cover various types of ensembles. For instance, if ω i,j, 1 i < j, and ω i,i, 1 i, are iid, 1 random variables, taking A i,i = 2 and A i,j = 1, for 1 i < j gives the GOE Gaussian Orthogonal Ensemble. If ωi,j R, ωi i,j, 1 i < j, and ωr i,i, 1 i, are iid, 1 random variables, taking A i,i = 1 and A i,j = 1/ 2, for 1 i < j gives the GUE Gaussian Unitary Ensemble see [24]. Moreover, if ωi,j R, ωi i,j, 1 i < j, and ωr i,i, 1 i, are two independent families of real valued random variables, taking A i,j = for i j large and A i,j = 1 otherwise, gives band matrices. Proper choices of non-random A i,j also make it possible to cover Wishart matrices, as seen in the later part of this section. In certain instances, A can also be chosen to be random, like in the case of diluted matrices, in which case A i,j, 1 i j, are iid Bernoulli random variables see [1]. On R 2, let P be the joint law of the random vector X = ωi,i R, ωr i,j, ωi i,j, 1 i < j, where it is understood that the indices for ωi,i R are 1 i. Let E be the corresponding expectation. Denote by ˆµ A the empirical spectral measure of the eigenvalues of X A, and further note that tr fx A = 1 trfx A = fxˆµ Adx, 19 i=1 R

4 for any bounded Borel function f. For a Lipschitz function f : R d R, set fx fy f Lip = sup, x y x y where throughout is the Euclidean norm, and where we write f Lipc whenever f Lip c. Each element M of M C has a unique collection of eigenvalues λ = λm = λ 1,, λ listed in non increasing order according to multiplicity in the simplex S = {λ 1 λ : λ i R, 1 i }, where throughout S is equipped with the Euclidian norm λ = i=1 λ2 i. It is a classical result sometimes called Lidskii s theorem [26], that the map M C S which associates to each Hermitian matrix its ordered list of real eigenvalues is 1-Lipschitz [11], [19]. For a matrix X A under consideration with eigenvalues λx A, it is then clear that the map ϕ : ωi,i R, ωr i,j, ωi i,j 1 i<j λx A is Lipschitz, from R 2, to S,, with Lipschitz constant bounded by a 2/. Moreover, for any real valued Lipschitz function F on S with Lipschitz constant F Lip, the map F ϕ is Lipschitz, from R 2, to R, with Lipschitz constant at most a F Lip 2/. Appropriate choices of F [19], [2] ensure that the maximal eigenvalue λ max X A = λ 1 X A and, in fact, any one of the eigenvalues is a Lipschitz function with Lipschitz constants at most a 2/. Similarly, the spectral radius ρx A = max λ i and 1 i tr fx A, where f : R R is a Lipschitz function, are themselves Lipschitz functions with Lipschitz constants at most a 2/ and f Lip /, respectively. These observations and our results are also valid for the real symmetric matrices, with proper modification of the Lipschitz constants. ext, recall that X is a d-dimensional infinitely divisible random vector without Gaussian component, X IDβ,, ν, if its characteristic function is given by, ϕ X t = Ee i t,x { = exp i t, β + R d } e i t,u 1 i t, u 1 u 1 νdu, 1.1 where t, β R d and ν the Lévy measure is a positive measure on BR d, the Borel σ-field of R d, without atom at the origin, and such that R d 1 u 2 νdu < +. The vector X has independent components if and only if its Lévy measure ν is supported on the axes of R d and is thus of the form: νdx 1,...,dx d = d dx 1... dx k 1 ν k dx k dx k+1... dx d, 1.2 k=1 for some one-dimensional Lévy measures ν k. Moreover, the ν k are the same for all k = 1,...,d, if and only if X has identically distributed components. The following proposition gives an estimate on any median or the mean, if it exists of a Lipschitz function of an infinitely divisible vector X. It is used in most of the results presented in this paper. The first part is a consequence of Theorem 1 in [14], while the proof of the second part can be obtained as in [14]. 11

5 Proposition 1.1. Let X = ω R i,i, ωr i,j, ωi i,j 1 i<j IDβ,, ν in R 2. Let V 2 x = u x u 2 νdu, νx = u >x νdu, and for any γ >, let p γ = inf { x > : < V 2 x/x 2 γ }. Let f Lip1, then for any γ such that νp γ 1/4, i any median mfx of fx satisfies γ mfx f G 1 γ := p γ + 3kγ 1/4 + E γ, ii the mean E [fx] of fx, if it exists, satisfies γ E [fx] f G 2 γ := p γ + kγ 1/4 + E γ, where k γ x, x >, is the solution, in y, of the equation y y + γln 1 + y = lnx, γ and where E γ = 2 k=1 1/2 2 e k, β e k, y νdy + e k, y νdy, 1.3 p γ< y 1 1< y p γ with e 1, e 2,...,e 2 being the canonical basis of R 2. Our first result deals with the spectral measure of a Hermitian matrix whose entries on and above the diagonal form an infinitely divisible random vector with finite exponential moments. Below, for any b >, c >, let Lip b c = { f : R R : f Lip c, f b }, while for a fixed compact set K R, with diameter K = sup x y, let x,y K where suppf is the support of f. Lip K c := {f : R R : f Lip c, suppf K}, Theorem 1.2. Let X = ωi,i R, ωr i,j, ωi i,j 1 i<j be a random vector with joint law P IDβ,, ν such that E [e t X ] < +, for some t >. Let T = sup{t : E [ e t X ] < + } and let h 1 be the inverse of hs = u e s u 1 νdu, < s < T. R 2 i For any compact set K R, P sup tr fx A E [tr fx A ] f Lip K 1 8 K for all > such that 2 < 8 K ht /. exp { } 2 8 K h 1 sds,

6 ii P sup tr fx A E [tr fx A ] f Lip b 1 { C, b exp } 2 C,b h 1 sds, 1.5 for all > such that 2 C, bht /, where C, b = C G 2 γ + ht + b, with G 2 γ as in Proposition 1.1, C a universal constant, and with t the solution, in t, of tht t hsds ln12b/ =. Remark 1.3. i The order of C, b in part ii can be made more specific. Indeed, it will be clear from the proof of this theorem see 2.39, that for any fixed t, < t T, ln 12b C, b C t + t hsds t + G 2 γ. ii As seen from the proof see 2.38, in the statement of the above theorem, G 2 γ can be replaced by E [ X ]. ow E [ X ] is of order, since min j=1,2,..., 2E[ X j ] E [ X ] max E [ ] X 2 j=1,2,..., 2 j, 1.6 where the X j, j = 1, 2,..., 2 are the components of X. Actually, an estimate more precise than 1.6 is given by a result of Marcus and Rosiński [22] which asserts that if E[X] =, then 1 4 x E [ X ] 17 8 x, where x is the solution of the equation: V 2 x x 2 + Ux x with V 2 x as defined before and Ux = u x u νdu, x >. = 1, 1.7 iii As usual, one can easily pass from the mean E [tr f] to any median mtr f in either 1.4 or 1.5. Indeed, for any 2b, if sup tr f mtr f, f Lip b 1 there exist a function f Lip b 1 and a median mtr f of tr f, such that either tr f mtr f or tr f mtr f. Without loss of generality assuming the former, otherwise dealing with the latter with f, consider the function gy = mindy, A, /2, y R 2, where A = {tr f mtr f}. Clearly 112

7 g Lip b 1, E [tr g] /4, and therefore tr g E [tr g] /4, which indicates that sup tr g E [tr g] g Lip b 1 4. Hence, P sup f Lip b 1 tr f mtr f P sup g Lip b 1 tr g E [tr g] iv When the entries of X are independent, and under a finite exponential moment assumption, the dependency in of the function h above and below can sometimes be improved. We refer the reader to [15] where some of these generic problems are discussed and tackled. ext, recall see [7], [19] that the Wasserstein distance between any two probability measures µ 1 and µ 2 on R is defined by. d W µ 1, µ 2 = sup fdµ 1 fdµ R f Lip b 1 Hence, Theorem 1.2 actually gives a concentration result, with respect to the Wasserstein distance, for the empirical spectral measure ˆµ A, when it deviates from its mean E [ˆµ A ]. As in [1], we can also obtain a concentration result for the distance between any particular probability measure and the empirical spectral measure. Proposition 1.4. Let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be a random vector with joint law P IDβ,, ν such that E [ e t X ] < +, for some t >. Let T = sup{t > : E [ e t X ] < + } and let h 1 be the inverse of hs = R 2 u es u 1νdu, < s < T. Then, for any probability measure µ, R P d W ˆµ A,µ E [d W ˆµ A,µ] { } exp h 1 sds, 1.1 for all < < ht /. Of particular importance is the case of an infinitely divisible vector having boundedly supported Lévy measure. We then have: Corollary 1.5. Let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be a random vector with joint law P IDβ,, ν such that ν has bounded support. Let R = inf{r > : νx : x > r = }, let V 2 = V 2 R = R 2 u 2 νdu, and for x > let lx = 1 + xln1 + x x. 113

8 i For any >, P sup tr fx A E [tr fx A ] f Lip b 1 { where C, b C, b = C exp G 2 γ + V 2 V 2 R 2 l R R 2 C, bv 2 }, 1.11 e t R 1 + b, with G 2 γ as in Proposition 1.1, C a universal constant, and t the solution, in t, of V 2 R 2 tre tr e tr + 1 = ln 12b. ii For any probability measure µ on R, and any >, P d W ˆµ A,µ E [d W ˆµ A,µ] { exp + V 2 R R R 2 ln } 1 + R2 V Remark 1.6. i As in Theorem 1.2, the dependency of C, b in and b can be made more precise. A key step in the proof of 1.11 is to choose τ such that E [tr 1 { XA τ}] /12b, and then C, b is determined by τ. Minimizing, in t, the right hand side of 2.38, leads to the following estimate { } E [tr 1 { XA τ}] exp V 2 R R 2 l τ G 2 γ V 2, where lx = 1 + xln1 + x x. For x 1, 2lx xlnx. Hence one can choose τ to be the solution, in x, of the equation x R xr 12b ln = 2 ln V 2. It then follows that C, b can be taken to be C G 2 γ + τ + b. Without the finite exponential moment assumption, an interesting class of random matrices with infinitely divisible entries are the ones with stable entries, which we now analyze. Recall that X in R d is α-stable, < α < 2, if its Lévy measure ν is given, for any Borel set B BR d, by + νb = σdξ 1 B rξ dr S d 1 r1+α,

9 where σ, the spherical component of the Lévy measure, is a finite positive measure on S d 1, the unit sphere of R d. Since the expected value of the spectral measure of a matrix with α-stable entries might not exist, we study the deviation from a median. Here is a sample result. Theorem 1.7. Let < α < 2, and let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be an α-stable random vector in R 2, with Lévy measure ν given by i Let f Lip1, and let mtr fx A be any median of tr fx A. Then, P tr fx A mtr fx A Cα ασs2 1 α α, 1.14 whenever > [ 1/α, 2σS Cα] 2 1 and where Cα = 4 α 2 α + eα/α2 α. ii Let λ max X A be the largest eigenvalue of X A, and let mλ max X A be any median of λ max X A, then P λ max X A mλ max X A Cα ασs2 1 α/2 α, 1.15 whenever > [ 1/α, 2σS Cα] 2 1 and where Cα = 4 α 2 α + eα/α2 α. Remark 1.8. Let M be a Wigner matrix whose entries M i,i,1 i,m R i,j,1 i<j, and MI i,j, 1 i<j, are iid random variables, such that the distribution of M 1,1 belongs to the domain of attraction of an α-stable distribution, i.e., for any >, P M 1,1 > = L α, for some slowly varying positive function L such that lim Lt/L = 1, for all t >. Soshnikov [3] showed that, for any >, lim P λ max b 1 M = 1 exp α, where b is a normalizing factor such that lim 2 Lb /b α = 2 and where λ maxb 1 M is the largest eigenvalue of b 1 M. In fact lim 2 α ǫ /b = and lim b / 2 α +ǫ =, for any ǫ >. As stated in [13], when the random vector X is in the domain of attraction of an α-stable distribution, concentration inequalities similar to 1.14 or 1.15 can be obtained for general Lipschitz function. In particular, if the Lévy measure of X is given by + νb = σdξ 1 B rξ Lrdr, 1.16 r1+α S 2 1 for some slowly varying function L on [, +, and if we still choose the normalizing factor b such that lim σs 2 1 Lb /b α is constant, then, 115

10 P λ max b 1 M mλ maxb 1 M CασS2 1 2 α/2 b α L b 2 α, 1.17 whenever b α 2 1+α/2 CασS 2 1 L b / 2. ow, recall that for an 2 dimensional vector with iid entries, σs 2 1 = 2 ˆσ1 + ˆσ 1, where ˆσ1 is short for σ1,,..., and similarly for ˆσ 1. Thus, for fixed, our result gives the correct order of the upper bound for large values of, since for > 1, e 1 e α 1 e α 1 α. Moreover, in the stable case, L becomes constant, and b = 2/α. Since λ max 2/α M is a Lipschitz function of the entries of the matrix M with Lipschitz constant at most 2 2/α, for any median mλ max 2/α M of λ max 2/α M, we have, P λ max 2 α M mλmax ˆσ1+ˆσ α M Cα 2 α/2 α, 1.18 whenever [ 2Cα ˆσ1 + ˆσ 1 ] 1/α. Furthermore, using Theorem 1 in [14], it is not difficult to see that mλ max 2/α M can be upper and lower bounded independently of. Finally, an argument as in Remark 1.15 below will give a lower bound on λ max 2/α M of the same order as The following proposition will give an estimate on any median of a Lipschitz function of X, where X is a stable vector. It is the version of Proposition 1.1 for α-stable vectors. Proposition 1.9. Let < α < 2, and let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be an α-stable random vector in R 2, with Lévy measure ν given by Let f Lip1, then i any median mfx of fx satisfies mfx f J 1 α := σs 2 1 4α 1/α α 42 α + 3k α 42 α 1/4 +E, 1.19 ii the mean E [fx] of fx, if it exists, satisfies E [fx] f J 2 α := σs 2 1 4α 1/α α 42 α + k α 42 α 1/4 +E,

11 where k α/42 α x, x >, is the solution, in y, of the equation y y + α ln α 42 αy = lnx, α and where E = 2 k=1 e k, β 4σS 2 1 1/α< y 1 e k, y νdy α + 1/2 2 1< y 4σS 2 1 1/α e k, y νdy, 1.21 α with e 1, e 2,...,e 2 being the canonical basis of R 2. Remark 1.1. When the components of X are independent, a direct computation shows that, 1/α up to a constant, E in both J 1 α and J 2 α is dominated by, as. σs 2 1 4α In complete similarity to the finite exponential moments case, we can obtain concentration results for the spectral measure of matrices with α-stable entries. Theorem Let < α < 2, and let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be an α-stable random vector in R 2, with Lévy measure ν given by i Then, where P C, b, α = sup f Lip b 1 tr fx A E [tr fx A ] C, b, α aα σs 2 1 α α 1, α J1 α+1 1+α C 1 α + b + C 2 α, with C 1 α and C 2 α constants depending only on α, and with J 1 α as in Proposition 1.9. ii For any probability measure µ, P d W ˆµ A,µ md W ˆµ A,µ Cα ασs2 1 α α, 1.23 whenever [ 1/α 2σS Cα] 2 1 and where Cα = 4 α 2 α + eα/α2 α. 117

12 It is also possible to obtain concentration results for smaller values of. Indeed, the lower and intermediate range for the stable deviation obtained in [5] provide the appropriate tools to achieve such results. We refer to [5] for complete arguments, and only provide below a sample result. Theorem Let < α < 2, and let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be an α-stable random vector in R 2, with Lévy measure ν given by For any ǫ >, there exists ηǫ, and constants D 1 = D 1 α, a,, σs 2 1 and D 2 = D 2 α, a,, σs 2 1, such that for all < < ηǫ, P sup tr fx A E [tr fx A ] f Lip b ǫ D 1 α+1 α exp D 2 2α+1 α Remark i In 1.14, 1.15 or 1.23, the constant Cα is not of the right order as α 2. It is, however, a simple matter to adapt Theorem 2 of [13] to obtain, at the price of worsening the range of validity of the concentration inequalities, the right order in the constants as α 2. ii Let us now provide some estimation of D 1 and D 2, which are needed for comparison with the GUE{ results of [1] see } iii below. Let Cα = 2 α eα + 2 α/22 α, Kα = max 2 α /α 1, Cα, Lα = α 1/α α/α 1 2 α/1 and let 2α 1 D =2 α 12 Cα Kα 1 α 1 α J2 αb 1 α CασS 2 1 As shown in the proof of the theorem, D 1 = 24D, while 1 α α 12 Cα Kα 1 α b α+1 α b 1 α D 2 = Lα σs α 1 α α D α α 1 Thus, as +, D 1 is of order 1/2 σs 2 1 1/α, while D 2 is of order 3α/2α 2 σs 2 1 2/1 α. iii Guionnet and Zeitouni [1], obtained concentration results for the spectral measure of matrices with independent entries, which are either compactly supported or satisfy a logarithmic Sobolev inequality. In particular for the elements of the GUE, their upper bound of concentration for the spectral measure is C 1 + b 3/2 exp 3/2 { C 2 8ca 22 5 C 1 + b 3/ }, 1.26

13 where C 1 and C 2 are universal constants. In Theorem 1.12, the order, in b, of D 1 is at most b α+1/α, while that of D 2 is at least b α+1/α 1. For α close to 2, this order is thus consistent with the one in Taking into account part ii above, the order of the constants in 1.24 are correct when α 2. Following [5] see also Remark 4 in [21], we can recover a suboptimal Gaussian result by considering a particular stable random vector X α and letting α 2. Toward this end, let X α be the stable random vector whose Lévy measure has for spherical component σ, the uniform measure with total mass σs 2 1 = 2 2 α. As α converges to 2, X α converges in distribution to a standard normal random vector. Also, as α 2, the range of in Theorem 1.12 becomes, + while the constants in the concentration bound do converge. Thus, the right hand side of 1.24 becomes { } D 1 exp D 3/2 2 5, which is of the same order, in, as However our order in is suboptimal. iv In the proof of Theorem 1.12, the desired estimate in 2.56 is achieved through a truncation of order 1/α, which, when α 2, is of the same order as the one used in obtaining However, for the GUE result, using Gaussian concentration, a truncation of order ln12b/ gives a slightly better bound, namely, C 1 ln 12b { exp C } 8ca 2 ln 12b, where C 1 and C 2 are absolute constants different from those of Wishart matrices are of interest in many contexts, in particular as sample covariance matrices in statistics. Recall that M = Y Y is a complex Wishart matrix if Y is a K matrix, K >, with entries Y i,j = Y R i,j + 1Y I i,j a real Wishart matrix is defined similarly with YI i,j = and M = Y t Y. Recall also that if the entries of Y are iid centered random variables with finite variance σ 2, the empirical distribution of the eigenvalues of Y Y/ converges as K,, and K/ γ, + to the Marčenko-Pastur law [4], [23] with density p γ x = 1 2πxγσ 2 c2 xx c 1, c 1 x c 2, where c 1 = σ 2 1 γ 1/2 2 and c 2 = σ γ 1/2 2. When the entries of Y are iid normal, Johansson [16] and Johnstone [17] showed, in the complex and real case respectively, that the properly normalized largest eigenvalue converges in distribution to the Tracy-Widom law [33], [34]. Soshnikov [29] extended the results of Johansson and Johnstone to Wishart matrices with symmetric subgaussian entries under the condition that K = O 1/3. As kindly indicated to us by a referee, this last condition has recently been removed by Péché. Soshnikov and Fyodorov [32] studied the distribution of the largest eigenvalue of the Wishart matrix Y Y, when the entries of Y are iid Cauchy random variables. We are interested here in concentration for the linear statistics of the spectral measure and for the largest eigenvalue of the Wishart matrix Y Y, where the entries of Y form an infinitely divisible vector and, in particular, a 119

14 stable one. We restrict our work to the complex framework, the real framework being essentially the same. It is not difficult to see that if Y has iid Gaussian entries, Y Y has infinitely divisible entries, each with a Lévy measure without a known explicit form. However the dependence structure among the entries of Y Y prevents the vector of entries to be, itself, infinitely divisible this is a well known fact originating with Lévy, see [27]. Thus the methodology, we used till this point, cannot be directly applied to deal with functions of eigenvalues of Y Y. However, concentration results can be obtained when we consider the following facts, due to Guionnet and Zeitouni [1] and already used for that purpose in their paper. Let for 1 i K, 1 j K for + 1 i K +, K + 1 j K + A i,j = 1 for 1 i K, K + 1 j K + 1 for + 1 i K +, 1 j K, and for 1 i K, 1 j K for + 1 i K +, K + 1 j K + ω i,j = Ȳ i,j for 1 i K, K + 1 j K + Y i,j for + 1 i K +, 1 j K, Y then X A = M Y K+ K+ C, and X 2 Y A = Y YY Moreover, since the spectrum of Y Y differs from that of YY only by the multiplicity of the zero eigenvalue, for any function f, one has and trfx 2 A = 2trfY Y + K f, λ max M 1/2 = max 1 i λ ix A, where M 1/2 is the unique positive semi-definite square root of M = Y Y. ext let P K, be the joint law of Y R i,j,yi i,j 1 i K,1 j on R 2K, and let E K, be the corresponding expectation. We present below, in the infinitely divisible case, a concentration result for the largest eigenvalue λ max M, of the Wishart matrices M = Y Y. The concentration for the linear statistic tr fm could also be obtained using the above observations. Corollary Let M = Y Y, with Y i,j = Y R i,j + 1Y I i,j. i Let X = Yi,j R,YI i,j 1 i,1 j K be a random vector with joint law P K, IDβ,, ν such that E K, [e t X ] < +, for some t >. Let T = sup{t > : E K, [e t X ] < + } and let h 1 be the inverse of hs = u e s u 1νdu, < s < T. R 2K 12

15 Then, P K, λ max M 1/2 E K, [λ max M 1/2 ] e R / 2 h 1 sds, 1.29 for all < < ht. ii Let X = Yi,j R,YI i,j 1 i K,1 j be an α-stable random vector with Lévy measure ν given by νb = S σdξ + 2K 1 1 B rξdr/r 1+α. Then, P K, λ max M 1/2 mλ max M 1/2 Cα 2 ασs2k 1 α, whenever > [ 2σS 2K 1 Cα ] 1/α and where Cα = 4 α 2 α + eα/α2 α. Remark i As already mentioned, Soshnikov and Fyodorov [32] obtained the asymptotic behavior of the largest eigenvalue of the Wishart matrix Y Y when the entries of, the K matrix, Y are iid Cauchy random variables. They further argue that although the typical eigenvalues of Y Y are of order K, the correct order for the largest one is K 2 2. The above corollary combined with Remark 1.1 and the estimate 1.19, shows that when the entries of Y form an α-stable random vector, the largest eigenvalue of Y Y is of order at most σs 2K 1 2/α. There is also a lower concentration result, described next, which leads to a lower bound on the order of this largest eigenvalue. Thus, from these two estimates, if the entries of Y are iid α-stable, the largest eigenvalue of Y Y is of order K 2/α 2/α. ii Let X IDβ,, ν in R d, then see Lemma 5.4 in [6] for any x >, and any norm on R d, P X x 1 { 1 exp ν { u R d : u 2x }}. 4 But, λ max M 1/2 is a norm of the vector X = Y R i,j,yi i,j, which we denote by X λ, if X is a stable vector in R 2K. P K, λ max M 1/2 mλ max M 1/2 = P K, λ max M 1/2 + mλ max M 1/2 1 { 1 exp ν { λ max M 1/2 2 + mλ max M 1/2 }} 4 1 { 1 exp ν { X λ 2 + mλ max M 1/2 }} 4 = 1 { σ S 2K 1 } 1 exp λ 4 α + mλ max M 1/2 α, 1.3 where S 2K 1 λ is the unit sphere relative to the norm λ and where σ is the spherical part of the Lévy measure corresponding to this norm. Moreover, if the components of X are independent, in which case the Lévy measure is supported on the axes of R 2K, σ S 2K 1 λ is of order K, and so, as above, the largest eigenvalue of M 1/2 is of order K 1/α 1/α. 121

16 iii For any function f such that gx = fx 2 is Lipschitz with Lipschitz constant g Lip := f L, trgx A = trfx 2 A is a Lipschitz function of the entries of Y with Lipschitz constant at most 2 f L K +. Hence, under the assumptions of part i of Corollary 1.14, P K, tr fm E K, [tr fm] K + { / 2K+ f L } exp h 1 sds, 1.31 for all < < f L ht / 2K +. iv Under the assumptions of part ii of Corollary 1.14, for any function f such that gx = fx 2 is Lipschitz with g Lip = f L, and any median mtr fm of tr fm we have: P K, tr fm mtr fm K + f α L σs 2K 1 Cα 2 α K + α α, 1.32 whenever > f L [ 2σS 2K 1 Cα ] 1/α / 2K +, and where Cα = 4 α 2 α + eα/α2 α. Remark In the absence of finite exponential moments, the methods described in the present paper extend beyond the heavy tail case and apply to any random matrix whose entries on and above the main diagonal form an infinitely divisible vector X. However, to obtain explicit concentration estimates, we do need explicit bounds on V 2 and on ν. Such bounds are not always available when further knowledge on the Lévy measure of X is lacking. 2 Proofs: We start with a proposition, which is a direct consequence of the concentration inequalities obtained in [12] for general Lipschitz function of infinitely divisible random vectors with finite exponential moment. Proposition 2.1. Let X = ωi,i R, ωr i,j, ωi i,j 1 i<j be a random vector with joint law P IDβ,, ν such that E [ e t X ] < +, for some t > and let T = sup{t > : E [ e t X ] < + }. Let h 1 be the inverse of hs = u e s u 1 νdu, < s < T. R 2 i For any Lipschitz function f, P tr fx A E [tr fx A ] exp for all < < f Lip ht /. { f Lip h 1 sds }, 122

17 ii Let λ max X A be the largest eigenvalue of the matrix X A. Then, P λ max X A E [λ max X A ] exp for all < < ht /. { } h 1 sds, Proof of Theorem 1.2: For part i, following the proof of Theorem 1.3 of [1], without loss of generality, by shift invariance, assume that min{x : x K} =. ext, for any v >, let if x g v x = x if < x < v 2.33 v if x v. Clearly g v Lip1 with g v = v. ext for any function f Lip K 1, any >, define recursively f x = for x, and for j 1 x j, j = 1,..., x, let x f x = g j, where g j := 21 {fj >f j 1 } 1g x j 1. Then f f and the 1-Lipschitz function f is the sum of at most K / functions g j Lip1, regardless of the function f. ow, for > 2, P tr fx A E [tr fx A ] P sup f Lip K 1 sup f Lip K 1 j=1 { tr f X A E tr f X A + tr fx A tr f X A + E [tr fx A ] E [tr f X A ] } P sup tr f X A E tr f X A > 2 f K 8 K sup P tr g j g j Lip1 exp X A E [tr g j X A] 2 K { 2 } 8 K h 1 sds, K ht /, and where the last inequality follows from part i of whenever < < the previous proposition by taking also = /4. 123

18 In order to prove part ii, for any f Lip b 1, i.e, such that f Lip 1, f b, and any τ >, let f τ be given via: fx if x < τ fτ signfτx τ if τ x < τ + fτ f τ x = f τ + signf τx + τ if τ f τ < x τ otherwise. Clearly f τ Lip1 and suppf τ [ τ f τ, τ + fτ ]. Moreover, sup tr fx A E tr fx A f Lip b 1 sup f Lip b 1 sup f Lip b 1 tr f τ X A E tr f τ X A + sup f Lip b 1 tr fx A f τ X A E [tr fx A f τ X A ] tr f τ X A E tr f τ X A tr g b X A τ +2E [tr g b X A τ], 2.36 with g b given as in ow, P sup tr fx A E tr fx A f Lip b 1 P sup f τ X A E f Lip b 1 tr tr f τ X A 3 + P 2tr g b X A τ + 2E [tr g b X A τ] 2 3 P sup f τ X A E f Lip b 1 tr tr f τ X A 3 +P tr g b X A τ E [tr g b X A τ] 3 2E [tr g b X A τ] P sup f τ X A E f Lip b 1 tr tr f τ X A 3 +P tr g b X A τ E [tr g b X A τ] 3 2bE [tr 1 { XA τ}] Let us first bound the second probability in Recall that the spectral radius ρx A = max λ i is a Lipschitz function of X with Lipschitz constant at most a 2/. Hence, for any 1 i 124

19 < t T, and γ > such that νp γ 1/4, E [tr 1 { XA τ}] = 1 P ρx A τ i=1 P ρx A P λ i X A τ E [ ρx A ] { } exp Ht τ G 2 γ t τ G 2 γ 2.38 where, above, we have used Proposition 1.1 in the next to last inequality and where the last inequality follows from Theorem 1 in [12] p with t Ht = hsds = e t u t u 1 νdu. R 2 We want to choose τ, such that E [tr 1 { XA τ}] /12b. This can be achieved if 12b ln + Ht τ G 2 γ t Since and d 2 ln 12b dt 2 d ln 12b + Ht = dt t + Ht t tht ln 12b Ht t 2, = t3 H t 2ttht ln 12b Ht t 4, it is clear that the right hand side of 2.39 is minimized when t = t, where t is the solution of and the minimum is then ht. Thus, if tht Ht ln 12b τ = C, b := =, G 2 γ + ht, 2.4 then and so, E [tr 1 { XA τ}] 12b, P tr g b X A τ E [tr g b X A τ] 3 2bE [tr 1 { XA τ}] P tr g b X A τ E [tr g b X A τ] 6 { } 6 exp h 1 sds,

20 for all < < 6 ht /, where Proposition 2.1 is used in the last inequality. For τ chosen as in 2.4, letting K = [ τ b, τ + b], it follows that for any f Lip b 1, f τ Lip K 1. By part i, the first term in 2.37 is such that P sup f τ X A E f Lip b 1 tr tr f τ X A 3 P sup f τ X A E f τ Lip K 1 tr [tr f τ X A ] 3 { 48C }, b + b C,b+b exp h 1 sds, 2.42 for all < C, b + b ht /. Hence, returning to 2.37, using 2.41 and 2.42 and for { < min 6 h T /, 144 C, b + b } ht /, we have P sup f Lip b C, b+b tr fx A E tr fx A exp { C, b + b C,b+b h 1 sds } +exp { 6 h 1 sds { } C,b+b exp h 1 sds, 2.43 } since only the case 2b presents some interest otherwise the probability in the statement of the theorem is zero. Part ii is then proved. Proof of Proposition 1.4: As a function of x R 2, d W ˆµ A, µx is Lipschitz with Lipschitz constant at most /. Indeed, for x, y R 2, d W ˆµ A,µx = sup tr fxa x fdµ f Lip b 1 R sup tr fx A x tr fx A y f Lip b 1 + sup tr fx A y fdµ f Lip b 1 R x y + d Wˆµ A,µy

21 Theorem 1.4 then follows from Theorem 1 in [12]. Proof of Corollary 1.5: For Lévy measures with bounded support, E [ e t X ] < +, for all t, and moreover Hence and Thus, one can take { exp x Ht = e ht V 2 tr 1. R t hsds V 2 R 2 s tr 1 tr, } { x x h 1 sds exp R R + V 2 R 2 ln 1 + Rx } V 2. C, b = C G 2 γ + V 2 R e t R 1 + b, where t is the solution, in t, of Applying Theorem 1.2 ii yields the result. V 2 R 2 tre tr e tr + 1 = ln 12b. In order to prove Theorem 1.11, we first need the following lemma, whose proof is essentially as the proof of Theorem 1 in [13]. Lemma 2.2. Let X = ω R i,i, ωr i,j, ωi i,j 1 i<j be an α-stable vector, < α < 2, with Lévy measure ν given by For any x, x 1 >, let g x,x 1 x = g x1 x x, where g x1 x is defined as in Then, tr P g x,x 1 X A E [tr g x,x 1 X A ] Cα aα σs 2 1 α α, whenever 1+α > 2 1+α σs 2 1 x 1 /α 1+α and where Cα = 2 5α/2 2eα + 2 α/α2 α. Proof of Theorem 1.11 For part i, first consider f Lip K 1. Using the same approximation as in Theorem 1.2, any function f Lip K 1 can be approximated by f, which is the sum of at most K / functions g j Lip1, regardless of the function f. ow, and as before, for > 2, 127

22 whenever P sup tr fx A E tr fx A f Lip K 1 tr P g j X A E [tr g j X A] K sup g j Lip b1 j=1,, K 2 K 4 K 8 α a α C 2 ασs 2 1 K α α 2α, K > 2 σs α, α and where the last inequality follows from Lemma 2.2, taking also = /4. For any f Lip b 1, and any τ >, let f τ be given as in Then, f τ Lip K 1, where K = [ τ b, τ + b], and moreover, P sup f Lip b 1 tr fx A E tr fx A P tr g τ,b X A E [tr g τ,b X A ] 3 2bE [tr 1 { XA τ}] + P sup f τ X A E f τ Lip K 1 tr tr f τ X A The spectral radius ρx A is a Lipschitz function of X with Lipschitz constant at most /. Then by Theorem 1 in [13], E [tr 1 { XA τ}] = 1 P λ i X A τ i=1 P ρx A > τ P ρx A mρx A > τ J 1 α C 1α2 α/2 a α σs 2 1 α/2 τ J 1 α α, 2.48 whenever α τ J 1 α 2C 1α2 α/2 a α σs 2 1 α/2, 2.49 and where C 1 α = 4 α 2 α + eα/α2 α. ow, if τ is chosen such that C 1 α2 α/2 a α σs 2 1 α/2 τ J 1 α α 12b, 128

23 that is, if it then follows that α τ J 1 α 12bC 1α2 α/2 a α σs 2 1 α/2, 2.5 E [tr 1 { XA τ}] 12b. Since g τ,b X A is the sum of two functions of the type studied in Lemma 2.2 with x 1 = b, we have, P tr g τ,b X A E [tr g τ,b X A ] 3 2bE [tr 1 { XA τ}] 2P tr g τ,b X A E [tr g τ,b X A ] 12 2C 2 α 12α a α σs 2 1 α α, 2.51 whenever 2 1+α12 1+α 1+α σs 2 1 b >, 2.52 α and where C 2 α = 2 5α/2 2eα+2 α/α2 α. The respective ranges 2.5 and 2.52 suggest that one can choose, for example, τ = J 1 α +. Then, there exists α, a,, ν such that for > α, a,, ν, P sup tr fx A E [tr fx A ] f Lip b 1 P sup f τ X A E f τlip K 1 tr [tr f τ X A ] 3 + P tr g τ,b X A E [tr g τ,b X A ] 3 2bE [tr 1 { XA τ}] C 3 αa α σs α J 1 α + b + α 1+2α + C 4αa α σs 2 1 α α, where C 3 α = 2 4+2α 12 α C 2 α, C 4 α = 212 α C 2 α and α, a,, ν is such that 2.46 and 2.52 hold. Part ii is a direct consequence of Theorem 1 of [13], since d W ˆµ A, µ Lip / as shown in the proof of Proposition 1.4. Proof of Theorem For any f Lip1, Theorem 1 in [13] gives a concentration inequality for fx, when it deviates from one of its medians. For 1 < α < 2, a completely similar even simpler argument gives the following result, P fx E [fx] x CασS2 1 x α,

24 whenever x α KασS 2 1, where Cα = 2 α eα + 2 α/α2 α and Kα = max { 2 α /α 1, Cα }. ext, following the proof of Theorem 1.2, approximate any function f Lip b 1 by f τ Lip [ τ b,τ+b] 1 defined via Hence, P sup tr fx A E tr fx A f Lip b 1 P sup f τ X A E f τ Lip K 1 tr tr f τ X A 3 +P tr g τ,b X A E [tr g τ,b X A ] 3 2bE [tr 1 { XA τ}] For ρx A the spectral radius of the matrix X A, and for any τ, such that τ E [ρx A ] KασS 2 1 1/α, E tr 1 { XA >τ} P ρx A E [ρx A ] τ E [ρx A ] αcασs 2 1 τ E [ρx A ] α, 2.55 where we have used, in the last inequality, 2.53 and the fact that ρx A Lip /. For Q >, let τ = E [ρx A ] + Q 1/α. With this choice, we then have: E tr 1 { XA >τ} αcασs 2 1 τ E [ρx A ] α provided Q α / > KασS 2 1 /, and αcασs 2 1 Q α 12b, 2.56 αcασs 2 1 /Q α 1/12b. ow, taking Q = 12bCασS 2 1 1/α /, and recalling, for 1 < α < 2, the lower range concentration result for stable vectors Theorem 1 and Remark 3 in [5]: For any ǫ >, there exists η ǫ, such that for all < < f Lip η ǫ/, P tr fx A E tr fx A { 2 α α 1 α α 1 1 α 1 + ǫexp σs 2 1 1/α 1 α f Lip α 1 α α 1 }

25 With arguments as in the proof of Theorem 1.2, if < ηǫ := 72 1/α 1/2 J 2 α + b + KασS 2 1 η ǫ, there exist constants D 1 α, a,, σs 2 1 and D 2 α, a,, σs 2 1, such that the first term in 2.54 is bounded above by 1 + ǫ D 1 α, a,, σs 2 1 exp α+1 α D 2 α, a,, σs 2 1 2α+1 α Indeed, with the choice of τ above and D as in 1.25, 2τ + b D / 1/α. Moreover, as in obtaining 2.34, D 1 can be chosen to be 24D, while D 2 can be chosen to be 2 α 1 α 1 α α 1 α σs α 1 α α D α α 1 ext, as already mentioned, J 2 α can be replaced by E [ X ]. In fact, according to 1.7 and an estimation in [14], if E [X] =, then 1 42 α 1/ασS2 1 1/α E [ X ] αα 1 1/α σs2 1 1/α. Finally, note that, as in the proof of Theorem 1.2 ii, the second term in 2.54 is dominated by the first term. The theorem is then proved, with the constant D 1 a,, σs 2 1 magnified by 2. Proof of Corollary As a function of Y R i,j,yi i,j 1 i K,1 j, with the choice of A made in 1.27, λ max X A Lip 2. Hence parti is a direct application of Theorem 1 in [12], while partii can be obtained by applying Theorem 1.7. Acknowledgements Both authors would like to thank the organizers of the Special Program on High-Dimensional Inference and Random Matrices at SAMSI. Their hospitality and support, through the grant DMS-11269, greatly facilitated the completion of this paper. 131

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