Extremum of Circulant type matrices: a survey Arup Bose

∗

Rajat Subhra Hazra

Koushik Saha

March 15, 2008

Abstract We survey some recent results on the behaviour of maximum, minimum, spectral norm and point process convergence of the eigenvalues of circulant type random matrices with both light tailed and heavy tailed input sequences. We also point out some recent results on other related matrices such as Toeplitz and Hankel matrices. We end the article with a description of a few open problems on these matrices.

Keywords Circulant matrix, eigenvalues, Gumbel distribution, k-circulant matrix, large dimensional random matrix, reverse circulant matrix, Hankel matrix, spectral norm, spectral radius, stable law, symmetric circulant matrix, Toeplitz matrix, weak convergence. AMS 2000 Subject Classification 60B20, 60G70, 60G55.

1

Introduction

Study of the properties of the eigenvalues of random matrices emerged first from data analysis and then from statistical models for heavy nuclei atoms. The research on limiting spectral analysis (LSA) of large dimensional random matrix (LDRM) has attracted considerable interest among mathematicians, probabilists and statisticians. Three of the matrices that have received the most attention are the Wigner matrix W which is a symmetric matrix with i.i.d. entries, the sample variance covariance matrix S = n−1 Xn XnT where Xn is a n × p matrix with i.i.d. entries and the square i.i.d. matrix with all entries being i.i.d. The Marˇcenko-Pastur limit law for S, the semicircular limit law for W , the circular limit law for the i.i.d. matrix and many other limit laws have been established over the last six decades. As these limit laws were discovered and established in increasing generality for the “bulk” of the spectrum, another aspect that became the focus of research was the the limiting behaviour near the “edge”: of the extreme eigenvalues, spectral norm and spectral radius. This behaviour of the extreme eigenvalues and related quantities is very nontrivial for most random matrices. The focus of this article is to review the distributional behaviour of extreme eigenvalues of a specific type of patterned random matrices, namely, circulant, symmetric circulant, reverse circulant, k-circulant and the related Toeplitz and Hankel matrices. In a later section, we ∗

Research supported by J.C.Bose Fellowship, Dept. of Science and Technology, Govt. of India.

1

have provided a quick overview of the literature and the history of the results for other matrices which are not discussed in any detail in this article. We have also listed a mixed bag of interesting open questions for future research. We first describe the matrices that we shall deal with in this article. The sequence of variables {xi ; i ≥ 0} which will be used to construct the matrices is called the input sequence. The Circulant matrix is defined as  x0 x1  xn−1 x0   Cn =  xn−2 xn−1   x1 x2

x2 . . . xn−2 xn−1 x1 . . . xn−3 xn−2 x0 . . . xn−4 xn−3 .. . x3 . . . xn−1

x0

      

.

n×n

The (j + 1)-th row is obtained by giving the j-th row a right circular shift by one position. Its corresponding symmetric version is  x0  x1   SCn =  x2   x1 The Reverse circulant matrix is a  x0  x1   RCn =  x2   xn−1

the Symmetric circulant matrix defined as  x1 x2 . . . x2 x1 x0 x1 . . . x3 x2   x1 x0 . . . x2 x3  .   ..  . x2 x3 . . . x1 x0 n×n

symmetric matrix defined as  x1 x2 . . . xn−2 xn−1 x2 x3 . . . xn−1 x0   x3 x4 . . . x0 x1  .   ..  . x0 x1 . . . xn−3 xn−2 n×n

The k-circulant matrix is defined as  x0 x1 x2  xn−k xn−k+1 xn−k+2   Ak,n =  xn−2k xn−2k+1 xn−2k+2   xk xk+1 xk+2

. . . xn−2 xn−1 . . . xn−k−2 xn−k−1 . . . xn−2k−2 xn−2k−1 .. . ...

xk−2

xk−1

      

.

n×n

All subscripts appearing above are calculated modulo n. The (j + 1)-th row is obtained by giving the j-th row a right circular shift by k positions (equivalently, k mod n positions). The circulant and reverse circulant are special cases of the k-circulant when we let k = 1 and k = n − 1 respectively.

2

The Toeplitz matrix is a symmetric matrix  x0 x1 x2  x1 x0 x1   x2 x1 x0 Tn =    xn−1 xn−2 xn−3 The Hankel matrix is given by  x0  x1   Hn =  x2   xn−1

x1 x2 x3

x2 x3 x4

given by . . . xn−2 xn−1 . . . xn−3 xn−2 . . . xn−4 xn−3 .. . ...

... ... ... .. .

x1

xn−2 xn−1 xn

x0

xn−1 xn xn+1

xn xn+1 . . . x2n−2 x2n−1

      

.

n×n

      

.

n×n

All of these matrices arise as important objects in statistics and mathematics. Nonrandom Toeplitz matrices when treated as operators are very classical objects in mathematics. See for example the book by Grenander and Szeg˝o (1984). Recent information on this matrix may be found in B¨ottcher and Silbermann (1999). The Hankel matrix is closely related to the Toeplitz matrix. For detailed properties of Hankel matrices see the above references. Matrices with the Toeplitz structure appear elsewhere too–as the covariance matrix of stationary processes, in shift-invariant linear filtering and in many aspects of combinatorics, and in harmonic analysis. The circulant matrices play a crucial role in the study of large dimensional Toeplitz matrices with nonrandom input. For details see, Grenander and Szeg˝o (1984) and Gray (2006). The classic book by Davis (1979) has a wealth of information on circulant matrices. Pn−1 2πij/n xl e , 1 ≤ j ≤ n, are the eigenvalues of the It is interesting to note that n−1/2 l=0 −1/2 circulant matrix n Cn . This is also the scaled version of the discrete Fourier transform. They arise in time series analysis in a natural way. The periodogram is fundamental in the Pn−1 −1 c ≤ j ≤ b n−1 c. spectral analysis of time series. It is defined as n | l=0 xl e2πij/n |2 , −b n−1 2 2 The maximum of the periodogram has been studied in Davis and Mikosch (1999), Mikosch et al. (2000) and Lin and Liu (2009). The reverse circulant matrix has gained importance recently due to its connection with the concept of half independence (see for example Bose et al. (2010c) and Banica et al. (2010)). We shall see later how the periodogram and the eigenvalues of the reverse circulant are intimately connected. Quite amusingly on page 68 of Rao (1973) the description of a circulant matrix that is given, is inadvertently that of the reverse circulant matrix. The k-circulant matrices and their block versions appear in the multi-level supersaturated design of experiment (see Georgiou and Koukouvinos (2006)) and time series analysis (see Pollock (2002)). If A is an n × n matrix with entries 0 or 1, then the digraph of A is defined as the graph with vertices {1, · · · , n} and there is an edge between i and j if the (i, j)th entry of A is 1. Now suppose A is a k-circulant matrix of order n with entries 0 or 1. An important question one studies here is what are the conditions on k and n which ensures that there exists an integer m such that Am = Jn where Jn is matrix whose all elements are 1. 3

The study of such solutions are generally achieved by considering the digraphs of A. These are also closely related in graph theory to spectra of De Bruijn digraphs. In fact, adjacency matrix of a De Bruijn graph forms a k-circulant matrix. For results of this type we refer to Strok (1992) and Wu et al. (2002). The plan of this paper is as follows. We describe the eigenvalues of all the circulant type matrices in Section 2. In Section 3 we describe the distributional behaviour of the extreme when we have i.i.d. Gaussian inputs. In Subsection 3.3 we briefly describe how and why the same limit (universality) holds for the spectral norm when we have i.i.d. inputs with E|ai |s < ∞ for some s > 2. In Section 4 we describe the distributional behaviour of the point process based on the eigenvalues and show how it helps to determine the joint distributional behaviour of upper ordered eigenvalues and their scaled spacings. In Section 5 we describe the behaviour of the spectral norm when the input sequence is “heavy tailed”. In Section 6 we point out some known results on Toeplitz and Hankel matrices. In Section 7, we give a short review of the results known for other matrices without going into any details. Finally in Section 8 we state some open problems in this area.

2

Description of eigenvalues

For most of the matrices in LDRM literature, explicit expression of eigenvalues are not known. However, a formula solution is known for the eigenvalues of the k-circulant. This description is based on Zhou (1996). Define ωk =

2πk for 0 ≤ k ≤ n − 1. n

(2.1)

Let 2

ν = νn := cos(2π/n) + i sin(2π/n), i = −1 and λk =

n−1 X

xl ν kl , 0 ≤ k < n.

(2.2)

l=0

Let p1 < p2 < . . . < pc be all the common prime factors of k, n, so that, n = n0

c Y

pβq q and k = k 0

q=1

c Y

pαq q .

q=1

Here αq , βq ≥ 1 and n0 , k 0 , pq are pairwise relatively prime. For any positive integer s, let Zs = {0, 1, 2, . . . , s − 1}. Define  S(x) = xk b mod n0 : b ≥ 0 , 0 ≤ x < n0 and gx = |S(x)|. (2.3) Define υk,n0 := |{x ∈ Zn0 : gx < g1 }| . We observe the following about the sets S(x).  (i) S(x) = xk b mod n0 : 0 ≤ b < |S(x)| . (ii) For x 6= u, either S(x) = S(u) or S(x) ∩ S(u) = φ. 4

(2.4)

As a consequence, the distinct sets from the collection {S(x) : 0 ≤ x < n0 } forms a partition of Zn0 . We shall call {S(x)} the eigenvalue partition of {0, 1, 2, . . . , n − 1} and we will denote the partitioning sets and their sizes by {P0 , P1 , . . . , Pl−1 } , and ni = |Pi |, 0 ≤ i < l.

(2.5)

Define yj :=

Y

λty , j = 0, 1, . . . , l − 1 where y = n/n0 .

(2.6)

t∈Pj

Then the characteristic polynomial of Ak,n (whence its eigenvalues follow) is given by χ (Ak,n ) = λ

n−n0

`−1 Y

(λnj − yj ) .

(2.7)

j=0

As special cases of the above formula, we obtain the following: Circulant matrix. Its eigenvalues {λk , 0 ≤ k ≤ n − 1} are given by λk =

n−1 X

n−1 X

xj exp(iωk j) =

j=0

xj cos(ωk j) + i

j=0

n−1 X

xj sin(ωk j), 0 ≤ k ≤ n − 1.

j=0

Symmetric circulant matrix. The eigenvalues {λk , 0 ≤ k ≤ n − 1} of SCn are given by: (a) for n odd: λn−k = λk for 1 ≤ k ≤ b n2 c where, n

λ0 = x0 + 2

b2c X

xj

j=1 n

λ k = x0 + 2

b2c X

n xj cos(ωk j), 1 ≤ k ≤ b c. 2 j=1

(b) for n even: λn−k = λk for 1 ≤ k ≤ n2 , where n

λ0 = x0 + 2

−1 2 X

xj + xn/2

j=1 n

λ k = x0 + 2

−1 2 X

xj cos(ωk j) + (−1)k xn/2 , 1 ≤ k ≤

j=1

n . 2

Reverse circulant matrix. The eigenvalues {λk , 0 ≤ k ≤ n − 1} are: λ0 =

n−1 X

xj

j=0

λ n2 =

n−1 X

(−1)j xj , if n is even

j=0

λk = −λn−k

n−1 X n−1 = xj exp(iωk j) , 1 ≤ k ≤ b c. 2 j=0

5

3

Behaviour of extreme eigenvalues

For any matrix A, its spectral radius sp(A) is defined as n o sp(A) := max |λ| : λ is an eigenvalue of A , where |z| denotes the modulus of z ∈ C. The spectral norm kAk is the square root of the largest eigenvalue of the positive semidefinite matrix A∗ A: p kAk = λmax (A∗ A) where A∗ denotes the conjugate transpose of A. Therefore if A is an n×n real symmetric matrix or A is a normal matrix, with eigenvalues λ1 , λ2 , . . . , λn , then kAk = sp(A) = max |λi |. 1≤i≤n

It appears difficult to establish distributional convergence of spectral norm or spectral radius for k-circulant matrix for all possible values of (k, n). A special case (n = k 2 + 1) was tackled in Bose et al. (2010a). After that Bose et al. (2010b) generalized the result for n = k g + 1. First in the next section, we deal with Gaussian inputs which is rather straightforward and helps to visualize the general non-Gaussian case.

3.1

Spectral radius and norm with Gaussian input

Suppose {xl }l≥0 are independent, mean zero and variance one random variables. Fix n. For 1 ≤ t < n, let us split λt into real and complex parts as λt = at,n + ibt,n , that is, at,n =

n−1 X

 xl cos

l=0

2πtl n

 , bt,n =

n−1 X

 xl sin

l=0

2πtl n

 .

(3.1)

Recall the identities:       X   n−1 n−1 n−1 X X 2πtl 2πt0 l 2πtl 2πtl 0 2 2 sin = 0, ∀t, t , cos = sin = n/2 ∀0 < t < n. cos n n n n l=0 l=0 l=0 (3.2) n−1 X l=0

 cos

2πtl n



 cos

2πt0 l n

 = 0,

n−1 X

 sin

l=0

2πtl n



 sin

2πt0 l n



= 0 ∀t 6= t0 (

mod n). (3.3)

For z ∈ C, by z¯ we mean, as usual, the complex conjugate of z. For all 0 < t, t0 < n, the following identities can easily be verified using the above orthogonality relations E(at,n bt,n ) = 0, and E(a2t,n ) = E(b2t,n ) = n/2, ¯ t = λn−t , E(λt λt0 ) = nI(t + t0 = n), E(|λt |2 ) = n. λ Recall the notations Pj and yj introduced in (2.5) and (2.6). 6

Lemma 3.1. (Bose et al. (2010a)) Fix k and n. Suppose that {xl }0≤l
p n−1 1 ], dq = log q and cq = √ , 2 2 log q

(3.5)

(ii) Bn = SCn , q = q(n) = n, cn = (2 log n)−1/2 and dn = (2 log n)1/2 +

log 2 log log n + log 4π − , (2 log n)1/2 2(2 log n)1/2 (3.6)

(iii) Bn = Ak,n with n = k g + 1, q = q(n) =

n 1 , cn = 1/2 , 2g 2g (log n)1/2

 1/2   log Cg − g−1 log g g−1 log n (g − 1) log log n 1 2 dn = + 1+ , Cg = √ (2π) 2 . (3.7) 1/2 1/2 2g (log n) g 4 log n g

7

Remark 3.1. (i) Since RCn , Cn and SCn are normal matrices, similar conclusion as (3.4) holds for spectral norm of RCn , Cn and SCn also. The k-circulant matrices for k 6= 1, n − 1 are not normal. So we cannot conclude the distributional convergence of the spectral norm of those matrices from the above result on spectral radius. (ii) Though Bose et al. (2010b) consider only the case n = k g + 1, a result similar to Theorem 3.1(iii) can be obtained also when n = k g − 1. This is achieved by establishing the analogue of the most crucial Lemma 3 of Bose et al. (2010b) in the present case. The limiting result remains exactly the same as in this case. In Section 3.3 we shall outline how the above result continues to hold for non Gaussian input.

3.2

Minimum of modulus of eigenvalues with Gaussian input

It is not at all obvious how the minimum of the absolute values of the eigenvalues behave and the answer is not known in general for circulant type matrices. Here we shed some light when the input sequence is Gaussian and for a specific subclass of k-circulant matrices. Consider a k-circulant matrix with n = k g + 1. Then from the previous section it is clear that now we have to deal with behaviour of the g-fold product of exponentials around zero. Now we know that P[E1 ≤ x] ∼ x as x → 0 where E1 is a standard exponential random variable. This translates into P[ E11 > x] ∼ x1 as x → ∞. Hence if E1 , E2 , · · · , Em are i.i.d. exponentials then it follows from Lemma 4.1 of Jessen and Mikosch (2006) that P[E1 E2 · · · Em ≤ x] ∼ x(− log x)m−1

as x → 0.

So we have for m ≥ 1,   Hm (x) := P (E1 E2 · · · Em )1/2m ≤ x ∼ (2m)m−1 x2m (− log x)m−1

(3.8)

as x → 0.

(3.9)

Now it follows from Theorem 2.1.5 of Galambos (1987), that if we have i.i.d. random variables {Yi } with common distribution function Hm , then, lim P[ min Yi ≤ xcn ] → 1 − exp(−x2m ),

n→∞

1≤i≤n

for x > 0,

where cn = sup{x : Hm (x) ≤ n1 }. Using the description of cn , it can be shown that cn ∼ n−1/2m (log n)−

m−1 2m

.

We omit the details of the algebra. Using this, we have the following theorem. Theorem 3.2. Suppose {xi }i≥0 are i.i.d. standard normal random variables. Consider any one of the circulant type matrices {Bn } with the input {xi }. Then as n → ∞, min1≤i≤n |λi | D → F, cq where {λi , 1 ≤ i ≤ n} are the eigenvalues of n−1/2 Bn and for 8

(3.10)

(i) Bn = RCn or Bn = Cn , q = q(n) = [

n−1 ], cq = q −1/2 and F (x) = 1 − exp(−x2 ), 2

(ii) Bn = SCn , n q = q(n) = [ ], cq = 2

r

2 −1 q and F (x) = 1 − exp(−x), π

(iii) Bn = Ak,n with n = k g + 1, g−1 n q = q(n) = , cq = q −1/2g (log q)− 2g and F (x) = 1 − exp(−x2g ). 2g

3.3

Non-Gaussian input

When we drop the Gaussian assumption on the input sequence, we cannot use Lemma 3.1 anymore. We outline now how Theorem 3.1 continues to hold in this case. The idea is to then use a sufficiently strong normal approximation to reduce the general case to the Gaussian case. There is also an appropriate truncation involved. To make this work, we now assume that E |x1 |s < ∞ for some s > 2. We briefly sketch the main steps involved. It may be noted that when the input belongs to the domain of attraction of a stable law with index α, 0 < α < 1, then a completely different behaviour is obtained (see Section 5). The behaviour is not known when the moment condition is violated but the distribution is not so heavy tailed as above. Note that the eigenvalues are functions of the following type of random variables: qn

Yn,k,X

1 X θ(t, k, n)xt , =√ n t=1

(3.11)

P where |θ(t, k, n)| < C where C is independent of t, k and n. Also, nt=1 θ(t, k, n) = 0 and P P n 1 2 k, n) = 1 and nt=1 θ(t, k, n)θ(t, k 0 , n) = 0 for k 6= k 0 . For example, one can take t=1 θ (t,√ n θ(t, k, n) = 2 cos( 2πtk ) and they satisfy above mentioned properties (see (3.2)). n So let {xi } be i.i.d. with mean zero, variance 1 and E[|x1 |s ] < ∞ for some s > 2. et = Xt I(|Xt | < n1/s ) and X t = X et − E[X et ]. Since Pn θ(t, k, n) = 1. Truncation: Let X t=1 0 we have Yn,k,X = Yn,k,Xe . Now it follows from the Borel-Cantelli lemma that there P et | = 0 for all n ≥ max{N (ω), |X1 |s , · · · , |XN (ω) |s }. exists N (ω) such that, nt=1 |Xt − X Hence Yn,k,X = Yn,k,Xe almost surely for all sufficiently large n. This implies that we may restrict attention to a truncated input sequence. 2. Smoothing: After truncation, we smoothen the Yn,k,X by adding an independent sequence of Gaussian random variables to the sequence {X i } with variance σn → 0 with appropriate rate. Now observe that | max Yn,k,X − max Yn,k,X+σn N | ≤ max |Yn,k,X − max Yn,k,X+σn N | ≤ σn max |Yn,k,N |, k

k

k

k

k

where Yn,k,N is the sequence formed by i.i.d. Gaussian random variables {Ni }. Now we use the order of the maximum of Yn,k,N ’s with Gaussian entries from the previous section to show that the last quantity in the above inequality is negligible in probability. 9

3. Bonferroni inequalities: The next crucial step is to use the Bonferroni inequality to reduce the problem to a problem about the behaviour of sum. This also allows the use of normal approximation in the next step. Let xn = cn x + dn . Note that     P max Yn,k,X+σn N > xn = P ∪nk=1 {Yn,k,X+σn N >xn } . 1≤k≤n

Now by Bonferroni inequalities we have, 2k X

j−1

(−1)

Sj ≤ P



∪nk=1 {Yn,k,X+σn N }

j=1



≤

2k−1 X

(−1)j−1 Sj ,

j=1

  P where Sj = 1≤i1 <···id ≤n P Yn,i1 ,X+σn N > xn , · · · , Yn,id ,X+σn N > xn . Now if one defines vd (t) = (θ(t, i1 , n), · · · , θ(t, id , n)) and An = {(y1 , · · · , yd ) : yi > xn , for i = 1, · · · , d} then we have, # " n X 1 X (X t + σn Nt )vd (t) ∈ An . Sj = P √ n t=1 1≤i <···i ≤n 1

d

4. Normal approximation: Now we may use Normal approximations results (cf. Davis and Mikosch, 1999, Lemma 3.4) for sums to show that, # Z " n 1 X (X t + σn Nt )vd (t) ∈ An ∼ ϕ1+σn2 (x)dx, P √ (3.12) n t=1 An where ϕ1+σn2 is the Gaussian density with mean zero, variance 1 + σn2 and the convergence holds uniformly over the tuples (i1 , · · · , id ). Now to complete the argument, use the fact that σn2 → 0 to show that as n → ∞, Z ϕ1+σn2 (x)dx → e−x . An

Following the above outline, it was shown in Bose et al. (2009, 2010b) that Theorem 3.1 remains valid when {xi } are i.i.d., E(xi ) = 0 and E |x1 |s < ∞ for some s > 2.

4

Upper order eigenvalues and Poisson convergence

The next natural question to study is the convergence of the joint distribution of ordered eigenvalues. Again, such studies appear to have been only limited. See Section 7 for a brief description of the existing results for other matrices. For reverse circulant and symmetric −1/2 circulant matrices we consider the point process based on the points (ωk , n cλq k −dq ) where λk is the k-th eigenvalue and cq , dq are appropriate scaling and centering constants appearing in the weak convergence of the maximum. We show that the limit measure is Poisson. For the case n = k 2 + 1 in k-circulant matrices we show the limit measure of an appropriate 10

point process based on the eigenvalues is Poisson. As a consequence, this yields the distributional convergence of any p-upper ordered eigenvalues and also yields the joint distributional convergence of any p spacings of the upper ordered eigenvalues. Let E be a subset of a compactified Euclidean space. A measure µ is said to be a point measure on E if µ is of the following form: µ(·) =

∞ X

xi (·) where {xi , i ≥ 1} ⊂ E.

i=1

Here x (·) denotes the measure which gives unit mass to any set containing x. Let Mp (E) be the set of all point measures on E. For {µn }, µ ∈ Mp (E), we say µn converges vaguely to µ if µn (B) → µ(B) as n → ∞ for all relatively compact sets B ⊂ E with µ(∂B) = 0. Endow Mp (E) with the topology of vague convergence. See Resnick (1987) for further details and properties of Mp (E) and vague convergence. A point process on E is a measurable map N : (Ω, F, P) → (Mp (E), Mp (E)). A sequence of point processes Nn on E is said to converge weakly to N , if for all f : Mp (E) → R, bounded continuous Nn (f ) = E[f (Nn )] → E[f (N )] as n → ∞. We use the D notation Nn → N .

4.1

Reverse circulant (RCn )

Since the eigenvalues occur in pairs with opposite signs (except one eigenvalue when n is odd), −1/2 it suffices for our purposes to define our point process based on the points (ωk , n cλq k −dq ) √ 1 for k = 0, 1, 2, . . . , [n/2], where ωk = 2πk . With q = [ n2 ], cq = 2√log and dq = log q, define n q ηn (·) =

q X j=0

 ωj ,

n−1/2 λj −dq cq


(·).

(4.1)

We then have the following theorem. Theorem 4.1 (Bose et al. (2010e)). Let {xt } be i.i.d random variables with E[x0 ] = 0, D E[x0 ]2 = 1 and E |x0 |s < ∞ for some s > 2. Then ηn → η, where ηn is as in (4.1) and η is a Poisson process on E = [0, π] × (−∞, ∞] with intensity measure π −1 dt × e−x dx. Let xp < · · · < x1 be any real numbers, and write Ni,n = ηn ([0, π] × (xi , ∞)) for the n−1/2 λj −dq number of exceedances of xi by , j = 1, . . . , q. Then cq  n−1/2 λn,(1) − dq n−1/2 λn,(p) − dq ≤ x1 , . . . , ≤ xp = {N1,n = 0, N2,n ≤ 1, . . . , Np,n ≤ p − 1}. cq cq From this it is easy to derive the distributional convergence of the p upper order eigenvalues using Theorem 4.1. For every n let the ordered version of the sample λj , j = 0, 1, . . . , n − 1, λn,(q) ≤ · · · ≤ λn,(2) ≤ λn,(1) . Then we have the following corollary of Theorem 4.1. 11

Corollary 4.1 (Bose et al. (2010e)). Under the assumption of Theorem 4.1, as n → ∞, (i) for any real numbers xp < · · · < x2 < x1 ,  −1/2  n λn,(1) − dq n−1/2 λn,(p) − dq P ≤ x1 , · · · , ≤ xp → P(Y(1) ≤ x1 , · · · , Y(p) ≤ xp ), cq cq where (Y(1) , · · · , Y(p) ) has the density exp(− exp(−xp ) − (x1 + · · · + xp−1 )).   −1/2 n λn,(i) −n−1/2 λn,(i−1) D −→ (i−1 Ei )i=1,...,p where {Ei } is i.i.d (ii) As a consequence, cq i=1,...,p

standard exponential.

4.2

Symmetric circulant (SCn ) −1/2

Now define a sequence of point processes based on the points (ωk , n cλq k −dq ) for k = 0, 1, . . . , q(= [ n2 ]), where ωk = 2πk . Note that we have not considered the eigenvalues λn−k n for k = 1, . . . , [ n2 ] to define the point process since λn−k = λk for k = 1, . . . , [ n2 ] and it does not affect our goal of finding the limit distribution of upper order eigenvalues. Define ηn (·) =

q X



j=0

ωj ,

n−1/2 λj −dq cq


(·)

(4.2)

where cq , dq are as in (3.6). Theorem 4.2 (Bose et al. (2010e)). Let {xt } be i.i.d random variables with E[x0 ] = 0, D E[x0 ]2 = 1 and E[x0 ]s < ∞ for some s > 2. Then ηn → η, where ηn is as in (4.2) and η is a Poisson process on [0, π] × (−∞, ∞] with intensity measure π −1 dt × e−x dx. The analogue of Corollary 4.1 holds with {λn,(i) }1≤i≤p as the largest p ordered eigenvalues of the symmetric circulant matrix.

4.3

k-circulant with n = k 2 + 1

For simplicity, here we consider k-circulant matrices only for n = k 2 + 1. One can consider point process based on eigenvalues of k-circulant matrices for n = k g + 1 where g > 2 and can prove result similar to Theorem 4.3. But for general g > 2 the algebraic details will be much more complicated. In the present case, that is, when n = k 2 + 1, clearly n0 = n and k 0 = k. From Section 4.1 of Bose et al. (2010a), g1 = 4 and the eigenvalue partition of {0, 1, 2, . . . , n − 1} contains exactly q = [ n4 ] sets of size 4 and each set is self-conjugate. Moreover, if k is even then there is only one more partition set containing only 0, and if k is odd then there are two more partition sets containing only 0 and only n/2 respectively. For the development of the point process we need a clear picture of the eigenvalue partition of {0, 1, 2, . . . , n − 1}. For this we represent the set Zn = {0, 1, 2, . . . , n − 1} in the following form Zn = {ak + b; 0 ≤ a ≤ k − 1, 1 ≤ b ≤ k} ∪ {0}. (4.3) 12

Then we can write S(x) defined in (2.3) as follows: S(ak + b) = {ak + b, bk − a, n − ak − b, n − bk + a}; 0 ≤ a ≤ k − 1, 1 ≤ b ≤ k. For n = k 2 + 1, [

Zn =

S(ak + b)

[

S(0), if k is even

(4.4)

[

(4.5)

0≤a≤[ k−2 ],a+1≤b≤k−a−1 2

and [

Zn =

S(ak + b)

[

S(0)

S(n/2), if k is odd

],a+1≤b≤k−a−1 0≤a≤[ k−2 2

where all {S(ak + b)} are mutually disjoint and hence form the eigenvalue partition of Zn . Now we are ready to define our point process based on the eigenvalues of the k-circulant matrix. For our purpose we neglect the eigenvalues λ0 , λn/2 if n is even and the eigenvalue λ0 if n is odd. Denote Tn = {(a, b) : 0 ≤ a ≤ [

λt =

n−1 X

xl e

l=0

i2πtl n

k−2 ], a + 1 ≤ b ≤ k − (a + 1)}, 2 Y

, β(a, b) =

λt and λ(a, b) = (β(a, b))1/4 .

t∈S(ak+b)

Note that {λ(a, b); (a, b) ∈ Tn } are the eigenvalues of the k-circulant matrix for n = k 2 + 1. −1/2 q ) : (a, b) ∈ Tn } Define the sequence of point processes based on points {( √an , √bn , n λ(a,b)−d cq as X (4.6) ηn (·) =  a b n−1/2 λ(a,b)−dq
(·) √ ,√ , n n

(a,b)∈Tn

cq

where q = q(n) = b n4 c and −1/2

cn = (8 log n)

(log n)1/2 √ and dn = 2

  1 log log n 1 π log . 1+ + 1/2 4 log n 2(8 log n) 2

(4.7)

Theorem 4.3 (Bose et al. (2010e)). Let {xt } be i.i.d random variables with E[x0 ] = 0, D E[x0 ]2 = 1 and E |x0 |s < ∞ for some s > 2. Then ηn → η, where ηn is as in (4.6) and η is a Poisson process on [0, 1/2] × [0, 1] × [0, ∞] with intensity measure 4I{s≤t≤1−s} e−x dsdtdx. As before, let xp < · · · < x1 be real numbers, and let Ni,n = ηn ([0, 12 ] × [0, 1] × (xi , ∞)) −1/2

q be the number of exceedances of xi by n λ(a,b)−d . Then the joint distribution of the p cq upper order eigenvalues can be written in terms of {Ni,n }1≤i≤p . From this it is easy to derive the distributional convergence of the p upper order eigenvalues. Hence a result similar to Corollary 4.1 holds with {λn,(i) }1≤i≤p representing the highest ordered eigenvalues of the k-circulant matrix.

13

5

Behaviour of eigenvalues with heavy tail entries

So far all results have been obtained assuming E |x1 |s < ∞ for some s > 2. It is not known what happens for all the rest of the cases. In this section we deal with a subclass of the remaining cases, namely the heavy tailed case. The idea of Poisson convergence of the point process based on the eigenvalues helps here too. Let E = [0, 1] × ([−∞, ∞]\{0}) and {Zt , t ∈ Z} be a sequence of i.i.d. random variables with common distribution F where F is in the domain of attraction of an α-stable random variable with 0 < α < 1. Thus, there exist p, q ≥ 0 with p + q = 1 and a slowly varying function L(x), such that P(Z1 ≤ −x) P(Z1 > x) = p, lim = q and P(|Z1 | > x) ∼ x−α L(x) as x → ∞. (5.1) x→∞ P(|Z1 | > x) x→∞ P(|Z1 | > x) lim

Now let {Γj }, {Uj } and {Bj } be three independent sequences defined on the same probability space where {Γj } is the arrival sequence of a unit rate Poisson process on R, Uj are i.i.d U (0, 1) and Bj are i.i.d. satisfying P(B1 = 1) = p and P(B1 = −1) = q,

(5.2)

where p and q are as defined in (5.1). The following convergence result follows from Proposition 3.21 of Resnick (1987): Nn :=

n X

D

(k/n,Zk /bn ) → N :=

∞ X

(Uj ,Bj Γ−1/α ) in Mp (E).

j=1

k=1

(5.3)

j

Suppose f is a bounded continuous complex valued function defined on R and without loss of generality assume |f (x)| ≤ 1 for all x ∈ R. Now pick η > 0 and define Tη : Mp (E) −→ C[0, ∞) as follows: X (Tη m)(x) = vj 1{|vj |>η} f (2πxtj ) j

P

if m = j (tj ,vj ) ∈ Mp (E) and vj ’s are finite. Elsewhere, set (Tη m)(x) = 0. Interestingly the point process based on the eigenvalues can be viewed as a function of the above point process in (5.3). Using this idea we can shed some light on the behaviour of the spectral norm of certain circulant type matrices. A random variable Yα is said to have a stable distribution Sα (σ, β, µ) if there are parameters 0 < α ≤ 2, σ ≥ 0, −1 ≤ β ≤ 1 and µ real such that its characteristic function has the form  exp{iµt − σ α |t|α (1 − iβ sgn(t) tan(πα/2))}, if α 6= 1, E[exp(itYα )] = exp{iµt − σ|t|(1 + (2iβ/π) sgn(t) log |t|)}, if α = 1. If β = µ = 0, then Yα is symmetric α-stable SαS. We also define Yα =

∞ X

−1/α Γj

∼

−1 Sα (Cα α , 1, 0)

Z where Cα =

x 0

j=1

14

−1

∞ −α

sin xdx

.

(5.4)

For a nondecreasing function f on R, let f ← (y) = inf{s : f (s) > y}. Then the scaling sequence {bn } is defined as ←  1 (n) ∼ n1/α L0 (n) for some slowly varying function L0 . bn = P[|Z1 | > ·] Theorem 5.1 (Bose et al. (2010d)). Suppose {Zi } are i.i.d. random variables satisfying (5.1). Consider the circulant type matrices with input {Zi }. Then for α ∈ (0, 1), D

D

−1 (i) kb−1 n Cn k → Yα and kbn RCn k → Yα , D

1−1/α Yα , (ii) kb−1 n SCn k → 2

where Yα is as in (5.4). Remark 5.1. Since RCn , Cn and SCn are normal matrices, conclusions (i), (ii) in Theorem 5.1 hold for the spectral radius of RCn , Cn and SCn also.

6

Toeplitz and Hankel matrices

Toeplitz and Hankel matrices are two very important matrices in random matrix literature and not many spectral properties are known about these matrices. For results related to the LSD of these and related matrices see Hammond and Miller (2005), Bryc et al. (2006), Massey et al. (2007) and Bose and Sen (2008). There are some close realtives of the Toeplitz and circulant type matrices which have palindromic structure in the first row of the matrices. The limiting spectral distribution of these type of matrices were dealt in Kologlu et al. (2010) and Jackson et al. (2010). For LSD results on Toeplitz matrices with band structure one may refer to Kargin (2009), Basak and Bose (2010) and Liu and Wang (2010). Here we describe some of the results known on the spectral norm of Toeplitz and Hankel matrices. It was shown in Bose and Sen (2007) that if the entries {xi } are i.i.d. with E(x0 ) = µ > 0 and V ar(x0 ) = 1 and if Tn0 = Tn − µnun uTn with un = n−1/2 (1, 1, . . . , 1)T , then (i)

kTn k n

0

→ µ almost surely and k kTTnn k k → 0 almost surely.

(ii) If E(x40 ) < ∞, then for Mn = kTn k or Mn = λn (Tn ), the maximum eigenvalue of Tn , Mn − µn √ → N (0, 4/3) in distribution. n (iii) If Tn and Tn0 are replaced by the corresponding symmetric Hankel matrices Hn and Hn0 , then (i) holds. Further, (ii) holds with the limiting variance being changed from 4/3 to 2/3. Meckes (2007) showed that if xi ’s are independent and centered uniformly subgaussian then √ EkTn k ≤ C n ln n. He also showed that if for all j and for some constant A, |xj | ≤ A or, if {xj } satisfy logarithmic Sobolev inequality with constant A, that is,     E f 2 (xj ) log f 2 (xj ) ≤ 2A E f 0 (xj )2 for every smooth f such that E f 2 (xj ) = 1, 15

then with probability 1 kTn k ≤ C, lim sup √ n n ln n

(6.1)

where C depends only on A. These results of Meckes were further improved in Adamczak (2010), where it was shown that for {xi } i.i.d. mean zero and finite variance, lim

n→∞

kTn k = 1 a.s. EkTn k

(6.2)

Further, kTn k (6.3) lim sup √ < ∞ a.s. if and only if Ex0 = 0 and Ex20 < ∞. n ln n The spectral norm of Toeplitz matrix with heavy tailed entries was studied in Bose et al. (2010d) and using point process arguments and convergence (5.3) they showed if the input sequence is i.i.d. {Zt } satisfying (5.1). Then for α ∈ (0, 1) and γ > 0, ∞ X

−1/α P 2 (1 − Uj )Γj > γ ≤ lim inf P b−1 n kTn k > γ j=1

n

≤ lim sup P n

b−1 n kTn k




>γ ≤P 2

∞ X

−1/α

Γj


>γ .

j=1

(6.4) It is not known if there is equality in any of the inequalities above.

7

Extremes of other matrices

We give a very brief account on the extreme eigenvalue literature for other matrices. Two of the most important matrices in random matrix literature are the sample covariance matrix (also known as the S matrix and, if further, the entries are complex or real Gaussian then they are referred to as complex or real Wishart respectively) and the Wigner matrix W . Historically, one of the first successes in the study of the extreme eigenvalues was by Geman (1980), who proved that the largest eigenvalue of S converges almost surely to a limit under certain growth conditions on the moments of the entries. Yin et al. (1988) proved the same result under the existence of the fourth moment, and Bai et al. (1988) proved that the existence of the fourth moment is also necessary for the existence of the limit. Silverstein (1989) found a necessary and sufficient condition for the weak convergence of the largest eigenvalue of S to a nonrandom limit. It was much harder to study the convergence of the smallest eigenvalue of S. The first breakthrough was obtained in Silverstein (1985), who established its almost sure convergence when the entries are i.i.d. standard normal. Bai and Yin (1993) proved the almost sure convergence of the smallest eigenvalue under finiteness of fourth moment of the underlying distribution. As a byproduct, they also established the almost sure limit of the largest eigenvalue of the S matrix. 16

Johansson (2000) proved that the properly scaled largest eigenvalue of S converges weakly to the Tracy-Widom law as n, p (dimension of Xn ) tends to ∞, n/p → γ > 0 and the entries are i.i.d. complex Gaussian. Johnstone (2001) proved a similar result when the entries are real Gaussian. Soshnikov (2002) generalized these results in two directions. He proved that the joint distribution of the upper ordered eigenvalues of Wishart matrices (after proper scaling) converge to the joint Tracy-Widom distribution and also extended the results to non-Gaussian entries provided n − p = O(p1/3 ). El Karoui (2003) extended the result of Johnstone to the case p/n → 0 or ∞. Onatski (2007) showed that the joint distribution of the centered and scaled several largest eigenvalues of p-dimensional complex Wishart matrix converges to the joint Tracy-Widom law when n and p tend to infinity so that n/p remains in a compact subset of (0, ∞). This result was the extension of results of Baik et al. (2005) and El Karoui (2007) who studied the asymptotic distribution of the largest eigenvalue of complex Wishart matrix as n and p go to infinity so that n/p remains in a compact subset of [1, ∞). P´ech´e (2009) generalized the result of Soshnikov (2002) when p/n → γ where γ ∈ (0, ∞]. For result on smallest singular values of n × n matrix with i.i.d. entries see Tao and Vu (2010). They showed that limiting distribution of smallest singular value is universal in the sense that it does not depend on the distribution of the entries. In particular, it converges to the same limiting distribution as in the special case when the entries are i.i.d. real gaussian, which was explicitly calculated by Edelman (1988). Juh´asz (1981) and F¨ uredi and Koml´os (1981) studied the asymptotic properties of the largest eigenvalue of W under the existence of moments of all order. Sometimes they assume the uniform boundedness of entries. Bai and Yin (1988) found necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of W . Some related work can be found in Geman (1986) and Bai and Yin (1986). Geman established an almost sure limit theorem for the operator norm of S with p = n and his result relates to studies on the spectrum of symmetric random matrices, and Bai and Yin considered the limiting behaviour of the operator norm of powers of random matrices (non symmetric) with i.i.d. entries. Another important class of matrices related to W are the Gaussian matrix ensembles, which are Gaussian measures on spaces of Hermitian matrices A, obtained by multiplying a translation-invariant measure by the Gaussian function exp(−Tr(A2 )). The three main examples are the Gaussian orthogonal ensemble on real Hermitian matrices, the Gaussian unitary ensemble on complex Hermitian matrices, and the Gaussian symplectic ensemble on quaternionic Hermitian matrices. The distributional convergence of the largest eigenvalue of Gaussian orthogonal, unitary and symplectic ensembles were studied by Tracy and Widom (1994, 1996) in a series of articles. See Tracy and Widom (2000) for a brief survey of such results. Soshnikov (1999) showed that after proper scaling, the first, second, third, etc. eigenvalues of Wigner random hermitian (respectively, real symmetric) matrix weakly converge to the distributions established by Tracy and Widom for Gaussian unitary (Gaussian orthogonal) cases. P´ech´e and Soshnikov (2007) established a probabilistic upper bound on the spectral radius of W with i.i.d. bounded centered but non symmetrically distributed entries. P´ech´e and Soshnikov (2008), established a probabilistic lower bound on the spectral radius of W with same type of entries and combining both the results, they established a rate of convergence result for the spectral radius of W . For some recent results on extreme gaps of the eigenvalues of Gaussian unitary ensembles see Ben Arous and Bourgade (2010). Soshnikov (2004) considered the point process based on the positive eigenvalues of ap17

propriately scaled W with heavy tailed entries {xij } satisfying P(|xij | > x) = h(x)x−α where h is slowly varying function at infinity and 0 < α < 2. He showed that it converges to an inhomogeneous Poisson random point process and from there, he deduced the distributional convergence of the maximum eigenvalue of an appropriately scaled W with such heavy tailed entries. The limiting distribution was Φα (x) = exp(−x−α ). A similar result was proved for sample covariance matrices in Soshnikov (2006). These results were extended in Auffinger et al. (2009) to 2 ≤ α < 4.

8

Conclusion

The behaviour of the exterme eignevalues of large dimensional random matrices is a very nontrivial issue. The class of k-circulants admit a formula solution for its eigenvalues. This helps in the study of the extreme values but the issue of non Gaussianity of the entries can only be taken care of after considerable amount of approximation by the Gaussian case. Even then this requires the finiteness of a moment of order larger than two. Moreover, results are known only for certain subclasses of the k-circulants. There are some results known for the related Toeplitz and Hankel matrices but even there, a host of unanswered questions remain. Here we list some of the major problems for future study and hope that our article will generate sufficient activity in this interesting area of research. 1. For the Toeplitz matrices with mean zero entries nothing is known about the limiting distribution of the spectral norm (after centering and scaling). As seen in Section 6 only the almost sure and in probability convergence (see (6.2) and (6.3)) of spectral norm is known. It would be nice to find appropriate centering and scaling in such a case. Similar questions can be asked about the Hankel matrices. Moreover, even for the almost sure convergence results, the results are not completely sharp and the exact limits are now known. It would also be interesting to study the edge behaviour of the palindromic Toeplitz matrices and Toeplitz matrices with band structure. 2. In Subsection 3.2 we observed the distributional behaviour of the minimum of modulus of eigenvalues of circulant type matrices with Gaussian entries. A similar result for non-Gaussian entries is not known. The normal approximation results used for the maximums do not seem to be able to salvage the situation for the minimum. Similarly, the behaviour when the input sequence is heavy tailed is also not known. 3. Results on spectral radius and minimum of modulus of eigenvalues of k-circulant matrices are known for the case when n = k g ± 1 with g ≥ 1 . It would be interesting to find out what happens for other combinations of k and n. 4. Spectral norm for k-circulant matrix with k = 1, n − 1 can be derived from the results on spectral radius as remarked in Remark 3.1(i). The behaviour of the spectral norm for other cases is not known as the matrices in such cases are non-normal. 5. In Section 4 we saw that a detailed study of the eigenvalues is required to exhibit the point process convergence for k-circulant matrices with n = k 2 + 1. This explicit study for the eigenvalue partition is not known for n = k g + 1 when g > 2. If this study is accomplished then one can expect a point process convergence result in this case. 18

6. In Section 5 we saw the behaviour of the spectral norm of certain circulant type matrices when the input sequence is in the domain of attraction of α stable law with 0 < α < 1. Results for the case 1 ≤ α ≤ 2 are not known. 7. In the heavy tailed case the spectral norm of k-circulant matrices is not yet known even for the case when n = k 2 + 1. 8. It is interesting to study the extremes when one goes out of the independent regime. This is also required for example in the study of dependent processes such as stationary time series. Suppose the input sequence {xn } is an infinite order moving average P∞ P process, xn = i=−∞ ai n−i , where n |an | < ∞, are nonrandom and {i ; i ∈ Z} are i.i.d. with E(i ) = 0 and V (i ) = 1. It seems to be a nontrivial problem to derive properties of the spectral norm in this case. Some results can be obtained when one resorts to scaling each eigenvalue by the spectral density at the appropriate ordinate and then considering their maximum. This scaling has the effect of equalizing the variance of the eigenvalues. For results of such types one may see Bose et al. (2009, 2010e). However, it is not known what happens if we consider the maximum without such scaling. Acknowledgement. The authors are extremely grateful to the Referee for the lightning quick constructive comments on the manuscript.

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Koushik Saha Bidhannagar Government College, Sector I, Salt Lake, Kolkata 700064, India. [email protected].

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