Product of exponentials and spectral radius of random <span class="mathjax-formula">$k$</span>-circulants

www.imstat.org/aihp 2012, Vol. 48, No. 2, 424–443

DOI:10.1214/10-AIHP404

© Association des Publications de l’Institut Henri Poincaré, 2012

Product of exponentials and spectral radius of random k -circulants

Arup Bose

, Rajat Subhra Hazra

and Koushik Saha

aStatistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.

E-mail:bosearu@gmail.com;rajat_r@isical.ac.in

bDepartment of Mathematics, Bidhannagar Govt. College, Salt Lake City, Kolkata 700064, India. E-mail:koushiksaha877@gmail.com Received 17 March 2010; revised 6 November 2010; accepted 16 November 2010

Abstract. We considern×nrandomk-circulant matrices withn→ ∞andk=k(n)whose input sequence{a_l}l≥0is independent and identically distributed (i.i.d.) random variables with finite(2+δ)moment. We study the asymptotic distribution of the spectral radius, whenn=k^g+1. For this, we first derive the tail behaviour of thegfold product of i.i.d. exponential random variables.

Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.

Résumé. Nous considérons des matrices aléatoiresk-circulantes de taillen×navecn→ ∞etk=k(n), dont les entrées{al}l≥0 sont des variables aléatoires, de moment(2+δ)fini, indépendantes et identiquement distribuées. Nous étudions la distribution asymptotique du rayon spectral, lorsquen=k^g+1. Pour établir cette distribution asymptotique, nous calculons d’abord le com- portement de la queue du produit degvariables aléatoires exponentielles i.i.d. Ensuite, en utilisant un résultat sur le comportement des queues et les techniques appropriées d’approximation normale, nous montrons que, après renormalisation et recentrage, la dis- tribution limite est une distribution de Gumbel. Nous identifions explicitement les constantes de recentrage et de remise à l’échelle.

MSC:Primary 60B20; secondary 60B10; 60F05; 62E20; 62G32; 15A52; 60F99; 60F05

Keywords:Eigenvalues; Gumbel distribution;k-circulant matrix; Laplace asymptotics; Large dimensional random matrix; Linear process; Normal approximation; Spectral radius; Spectral density; Tail of product

1. Introduction

Random matrices are matrices whose elements are random variables. In random matrix theory, one of the most im- portant areas of research is the behaviour of the eigenvalues. In particular the spectral radius and spectral norm have been important objects of study.

Thespectral radiussp(A)of any matrixAis defined as sp(A):=max

|μ|: μis an eigenvalue ofA , where|z|denotes the modulus ofz∈C.

1Research supported by J. C. Bose National Fellowship, Department of Science and Technology, Government of India.

Thespectral normAof any matrixAwith complex entries is the square root of the largest eigenvalue of the positive semidefinite matrixA^∗A:

A = λ_max

A^∗A ,

whereA^∗ denotes the conjugate transpose ofA. Therefore ifAis ann×nnormal matrix (e.g. ifAis symmetric), with eigenvaluesλ1, λ2, . . . , λn, then

A =sp(A)= max

1≤i≤n|λ_i|.

Here is a brief description of the existing results on these and related quantities for random matrices. Geman [20]

considered ann×nrandom matrixM_n=((m_ij))where{m_ij}are i.i.d. random variables with mean zero and showed that lim sup_n√¹

nsp(Mn)is bounded almost surely under suitable assumptions. Later the same conclusion was obtained by Bai and Yin [4] under weaker assumptions. Silverstein [31] considered the case where the mean is nonzero and the fourth moment is finite, and showed that the spectral radius converges almost surely and also converges weakly to a normal distribution after proper scaling and centering.

Kostlan [22] gave a simple characterization of the moduli of the eigenvalues ofMnwith complex Gaussian entries:

the squared moduli of its eigenvalues are distributed as independentχ_2i², 1≤i≤n. Rider [27], using joint probability distribution of eigenvalues, showed that the spectral radius ofM_n when the entries are i.i.d. complex Gausian with mean zero and variance 1/n, converges to a Gumbel distribution with appropriate scaling and centering.

For work on spectral norm of (symmetric) Toeplitz matrices, see Bose and Sen [11], Meckes [25] and Adamczak [1]. Bryc, Dembo and Jiang [12] studied the almost sure convergence rate of spectral norm of Markov matrices.

Another related aspect to study is the behaviour of maximum and minimum eigenvalues. The literature in this area is extensive. Geman [19] proved that the largest eigenvalue of a sample covariance matrix converges almost surely under growth conditions on the moments of the underlying distribution. Yin, Bai and Krishnaiah [41] proved the same result under the finiteness of the fourth moment, and Bai, Silverstein and Yin [3] proved that the finitness of the fourth moment is also necessary for the existence of the limit. Bai and Yin [5] found necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of the Wigner matrix. Yin, Bai and Krishnaiah [40], Silverstein [29] and Bai and Yin [6] considered the almost sure limit of the smallest eigenvalue of a sample covariance matrix. Silverstein [30] found a necessary and sufficient condition for weak convergence of the largest eigenvalue of a sample covariance matrix to a nonrandom limit. The distributional convergence of the largest eigenvalue of Gaussian orthogonal, unitary and symplectic ensembles were studied by Tracy and Widom in a series of articles. See Tracy and Widom [37] for a brief survey of such results.

There are a few results in the literature for limiting behaviour of maximum eigenvalues of matrices with heavy tailed entries. See Soshnikov [32,33]; Auffinger, Ben Arous and Peche [2]. For results on spectral norm of circulant matrices and Toeplitz matrices with heavy tailed entries, see Bose, Hazra and Saha [8].

In general the eigenvalues of a random matrix are dependent and hence the behaviour of the spectral norm or the maximum eigenvalue is not apriori comparable to the behaviour of the maximum of i.i.d. random variables. However, for one class of patterned matrices the classical theory of extreme values of i.i.d. random variables can be applied after approximating the eigenvalues with i.i.d. random variables or their suitable functions. This is the class ofk-circulant matrices for suitable values ofk. In this article we study the limiting behaviour of spectral radius of a specific type of k-circulant matrices.

Suppose a= {a_l}l≥0 is a sequence of real numbers or random variables. For positive integers k andn, thek- circulant matrix withinputsequence{a_l}is defined as

A_k,n(a)=

⎡

⎢⎢

⎣

a0 a1 · · · an−1

a_n−k a_n−k+1 · · · a_n−k−1 a_n−2k a_n−2k+1 · · · a_n−2k−1

...

⎤

⎥⎥

⎦

n×n

.

We writeAk,n(a)=Ak,n. The subscripts appearing in the matrix entries above are calculated modulon. The(j+1)th row ofAk,nis a right-circular shift of thejth row bykpositions (equivalently,k modnpositions). The value ofk

may change with the increasing dimension of the matrix. Note thatA_1,nis the well-known circulant matrix(C_n)and An−1,nis thereverse circulant matrix(RCn). Without loss of generality,kmay always be reduced modulon.

These matrices and their block versions arise in many different areas of Mathematics and Statistics such as, multi- level supersaturated design of experiment (Georgiou and Koukouvinos [21], Chen and Liu [14]), spectra of De Bruijn graphs (Strok [35]) and,(0,1)-matrix solutions toA^m=J_n(Wu, Jia and Li [39]).

The initial advantage in dealing with this matrix is that a formula solution is known for its eigenvalues (see Zhou [42]). However, this formula depends heavily on the number theoretic relation betweenkandn. This makes it hard to study any type of spectral properties for arbitrary choice of these values.

The spectral norm and spectral radius of reverse circulant random matrices were considered in Bose, Hazra and Saha [7] and results on symmetric circulant (the usual circulant with symmetry imposed) matrices are available in Bose, Hazra and Saha [7] and Bryc and Sethuraman [13]. Bose, Mitra and Sen [9] considered thek-circulants with n=k²+1 with i.i.d. entries having mean zero variance one and E[|a₁|^2+δ]<∞for someδ >0 and showed that the spectral radius, appropriately scaled and centered, converges to the (standard) Gumbel distribution

Λ(x)=exp

−exp(−x)

, x∈R.

When the entries are i.i.d. Gaussian, this result is relatively easy to establish using some properties of the eigenvalues and the tail behaviour ofH₂(x)=P(E1E₂> x)whereE₁andE₂are i.i.d. standard exponential variables. When the entries are not necessarily Gaussian, appropriate normal approximation arguments were used.

We provide a significant extension of this result to the casen=k^g+1 wheregis any positive integer. In particular we show that asn→ ∞,

sp(n^−1/2A_k,n)−d_q c_q

→D Λ, (1.1)

where{q=q(n), cn, dn}are constants defined later in Theorem4.

To prove (1.1), first assume that the entries are Gaussian. Once the result is established for Gaussian entries, the case of general entries is tackled by using appropriate normal approximation results. When the entries are Gaussian, we show that the result is easy to establish once we can derive the tail behaviour of

H_g(x)=P[E₁E₂· · ·E_g> x], (1.2) where{E_i}are i.i.d. standard exponentials. The tail behaviour ofH₂(·)is available in the literature. This has been obtained through different approaches. Springer and Thompson [34] and Lomnicki [24] used Mellin transform, Tang [36] used a clever substitution and Bose, Mitra and Sen [9] set up a second order differential equation forH2(·)and observed that the solution is a modified Bessel function.

Forn >2, it is not hard to set up a higher order differential equation forH_n(·). However, this does not seem to provide the precise information that we need on the tail. We show in Theorem1that for anyn≥1,

Hn(x)=Cnx^αne^−nx1/ngn(x), n≥1, (1.3)

for some suitable sequence{α_n, C_n}and a sequence of functions{g_n(x)→1}for everyx. The proof is based on the so called Laplace asymptotics (see Erdélyi [17]) after an appropriate substitution similar to that used in Tang [36], in conjunction with induction onn. This result may be of independent interest. The tail behaviour ofH_n when{E_i} are not necessarily exponential is an open question. For information on the properties of product of general random variables, see Galambos and Simonelli [18].

The outline of the rest of the paper is as follows. In Theorem1of Section2we establish (1.3). This immediately implies that(E₁· · ·E_g)^1/2gbelongs to the max domain of attraction of the Gumbel distribution. In Section3we give a brief description of the known results on spectral radius of circulant matrices and state our main theorem (Theorem4) on thek-circulant.

In Sections4.1–4.5we develop the proof of the main result. In particular, we present the known formula for the eigenvalues of thek-circulant in Section4.1and more detailed properties forn=k^g+1 in Section4.2. Section4.3 contains the properties of the eigenvalues when the input sequence is i.i.d. Gaussian. Section4.4has two preparatory lemmas on truncation and normal approximation. Section4.5has the final steps of the proof.

In Section 5.1, we remark about the case sn=k^g+1. In Section5.2, we deal with the case where the input sequence is an infinite order linear process and derive the limit for the maxima of the eigenvalues when scaled by the spectral density.

2. Tail of product and extreme values

It does not seem to be known when the product of i.i.d. random variables belongs to the max domain of attraction of the Gumbel distribution. However, for our purposes, we need only the exponential case that is given in the following theorem.

Theorem 1. There exists constants{C_n, α_n}such that

H_n(x)=C_nx^αne^−nx1/ng_n(x), n≥1, (2.1)

where forn≥1, C_n= 1

√n(2π)^(n−1)/2, α_n=n−1

2n and g_n(x)→1 asx→ ∞.

Proof. SinceH₁(x)=P[E₁> x] =e^−x, we haveC₁=1, α1=0 andg₁(x)=1 for allx. Now H₂(x)=

_∞

0

e^−ye^−x/ydy

=x^1/2 _∞

0

e^{−x1/2(t+1/t )}dt (substitutingy=t x^1/2)

=x^1/2 _∞

0

f (t )e^−x1/2g(t)dt,

wheref (t )=1 andg(t)=t+ ¹_t. Note that gassumes a strict minimum at t=1 andf (1)=1=0. So applying Laplace asymptotics (see Section 2.4 of Erdélyi [17]) we have

H2(x)=x^1/2e^−x1/2g(1)f (1)

2π

x^1/2g(1)g2(x)=√

πx^1/4e^−2x1/2g2(x),

whereg2(x)→1 asx→ ∞. HenceC2=√

π=^√1₂(2π)^1/2, αk=1 and the result is true forn=2. Now suppose (2.1) is true forn=k. We shall prove it forn=k+1.

H_k+1(x)= _∞

0

e^−yH_k x

y

dy

=C_{k ∞}

0

e^−y x

y α_k

e^−k(x/y)1/kg_k x

y

dy

=xkC_{k ∞}

0

e^−(ks+x/sk)s^kαk−k−1g_k s^k

ds (substitutingx/y=s^k)

=x^{(kαk+1)/(k+1)}kC_{k ∞}

0

e^{−(kt+1/tk)x1/(k+1)}t^kαk−k−1g_k

t^kx^k/(k+1)

dt (substitutings=x^1/(k+1)t )

=x^{(kαk+1)/(k+1)}kC_{k ∞}

0

f (t )e^{−x1/(k+1)g(t)}dt,

where

f (t )=t^kαk−k−1g_k

t^kx^k/(k+1)

and g(t)=kt+ 1 t^k.

Note that g assumes a strict minimum at t =1 and f (1)=gk(x^k/(k+1))=0, g(1)=k(k+1). Again applying Laplace asymptotics we have

Hk+1(x)=x^{(kαk+1)/(k+1)}kCke^{−x1/(k+1)g(1)}f (1)

2π

x^1/(k+1)g(1)h(x)

=x^{(kαk+1)(k+1)}kCke^{−(k+1)x1/(k+1)}gk

x^k/(k+1)

2π

x^1/(k+1)k(k+1)h(x), whereh(x)→1 asx→ ∞. Substituting the values ofαkandCkwe get

H_k+1(x)=x^k/(2(k+1)) 1

√k+1(2π)^k/2e^{−(k+1)x1/(k+1)}g_k

x^k/(k+1) h(x)

=C_k+1x^αk+1e^{−(k+1)x1/(k+1)}g_k+1(x), where

α_k+1= k

2(k+1), C_k+1= 1

√k+1(2π)^k/2, g_k+1=g_k

x^k/(k+1) h(x)

andg_k+1(x)→1 asx→ ∞. Hence the result is true forn=k+1 and this completes the proof.

The next two results shall be needed in the study of the spectral radius. The first is an easy consequence of The- orem1and standard calculations in extreme value theory as found in Rootzèn [28], Embrechts et al. [16] and Exer- cise 1.1.4 of Resnick [26]. We omit the details.

Lemma 1. Let{X_n}be a sequence of i.i.d.random variables whereX_i=^D(E₁E₂· · ·E_k)^1/2k and{E_i}1≤i≤k are i.i.d.

Exp(1)random variables.Then max1≤i≤nX_i−d_n

c_n

→D Λ,

where

c_n= 1

2k^1/2(logn)^1/2, d_n=logC_k−((k−1)/2)logk 2k^1/2(logn)^1/2 +

logn k

1/2

1+(k−1)log logn 4 logn

,

Ck= 1

√k(2π)^(k−1)/2.

The next result is immediate from Lemma1and will be useful in the proof of Lemma7.

Lemma 2. Let{Ei},cn anddn be as in Lemma1.Letσ_n²=n^−c,c >0. Then there exists some positive constant K=K(x),such that for all largenwe have

P

(E1E2· · ·Ek)^1/2k>

1+σ_n²₋1/2

(cnx+dn)

≤K

n, x∈R.

3. Spectral radius of randomk-circulant matrices

To put our main result (Theorem4) in perspective, we now briefly describe the known results on various circulant type matrices when the entries do not have heavy tails. The following result can be derived following the argument given for symmetric Toeplitz matrix in Bose and Sen [11].

Theorem 2. Consider the reverse circulant matrix RCn with the input sequence{a_i}which is i.i.d.withE(a0)=μ andVar(a0)=1.Let RC⁰_n=RCn−μnu_nu^T_n withu_n=(1, . . . ,1)^T.Ifμ >0,then

RCn

n →μ almost surely and RC⁰_n

RCn

→0 almost surely.

Similar results hold for circulant matrix(C_n)also.

Theorem 3 (Bose, Hazra and Saha [7]). Consider the reverse circulant matrix RCn and the circulant matrixC_n both with the input sequence{a_i}which is i.i.d.with meanμandE[|a_i|^2+δ]<∞for someδ >0.

(a) Ifμ=0then, RCn − |μ|n

√n

→D N (0,1).

(b) Ifμ=0then, (1/√

n)RCn −d_q cq

→D Λ,

where

q=q(n)= n−1

2

, d_q=

lnq and c_q= 1 2√

lnq. The above conclusions continue to hold forC_nalso.

We now state the following significant generalisation of a result of Bose, Mitra and Sen [9] who proved the result for the caseg=2. See Sections5.1and5.2for extensions to the casesn=k^g+1 and to the case of dependent entries.

Theorem 4. Suppose{a_l}l≥0is an i.i.d.sequence of random variables with mean zero and variance1andE[|a_l|^γ]<

∞for someγ >2.Ifn=k^g+1for some fixed positive integerg,then asn→ ∞, sp(n^−1/2A_k,n)−d_q

c_q

→D Λ,

whereq=q(n)=_2gⁿ and the normalizing constantscnanddncan be taken as follows

c_n= 1

2g^1/2(logn)^1/2, d_n=logCg−((g−1)/2)logg 2g^1/2(logn)^1/2 +

logn g

1/2

1+(g−1)log logn 4 logn

,

C_g= 1

√g(2π)^(g−1)/2.

Remark. Bose and Sen[11]showed that thelimiting spectral distribution (LSD)of the symmetric circulant and the reverse circulant exist if E(a₁²) <∞. Bose,Mitra and Sen[10] showed that the LSD of thek-circulant exists for suitable values of k under the slightly stronger assumptionE[|a1|^2+δ]<∞.This assumption helps to use normal approximation for non-Gaussian entries.

For i.i.d.variables,it is well known that the finiteness of any moment is not necessary for its maximum to have a limit distribution.However,in our case,due to the approximation of the non-Gaussian case by the Gaussian case,we are compelled to use a moment condition.It is an open question if Theorem4remains true under weaker conditions and in particular if convergence to some extreme value distribution holds solely under appropriate tail behaviour ofa₁.

4. Proof of Theorem4

The proof is long and is developed in the following sections. In Section4.1we describe the eigenvalues of a general k-circulant matrix and in Section4.2we provide more detailed description of the eigenvalues of thek-circulant matrix forn=k^g+1. In Section4.3we discuss some distributional properties of the eigenvalues ofk-circulant matrix when the input sequence is i.i.d. Gaussian. Section4.4has two preparatory lemmas on truncation and normal approximation.

Drawing on the developments of Sections2,4.2and4.4, we finish the proof in Section4.5.

4.1. Description of eigenvalues of ak-circulant

The formula solution by Zhou [42], given below in Theorem5, for the eigenvalues of ak-circulant is our starting point. A proof is also provided in Bose, Mitra and Sen [10]. Let

ω=ω_n:=cos(2π/n)+i sin(2π/n), i²= −1 and λ_t=

n−1

l=0

a_lω^{t l}, 0≤t < n. (4.1) Note that{λ_t,0≤t < n}are eigenvalues of the usual circulant matrixA_1,n. Letp₁< p₂<· · ·< p_cbe all the common prime factors ofnandk. Then we may write,

n=n c q=1

p^β_q^q and k=k c q=1

p_q^αq. (4.2)

Hereα_q, β_q≥1 andn,k,p_qare pairwise relatively prime. For any positive integerm, let Zm= {0,1,2, . . . , m−1}.

We introduce the following family of sets S(x):=

xk^bmodn: b≥0

, x∈Zn. (4.3)

Note thatx∈S(x)for everyx. SupposeS(x)∩S(y)=∅. Then,xk^b1=yk^b2modn for some integersb₁, b₂≥1.

Multiplying both sides byk^gx−b1we see that,x∈S(y)so that,S(x)⊆S(y). Hence, reversing the roles,S(x)=S(y).

Thus, the distinct sets in{S(x)}x∈Zn forms a partition, called theeigenvalue partition, ofZn. Denote the partitioning sets and their sizes by

P0= {0}, P1, . . . ,P−1 and k_j=#Pj, 0≤j < . (4.4) Define

Π_j:=

t∈Pj

λ_{t n/n}, j=0,1, . . . , −1. (4.5)

Theorem 5 (Zhou [42]). The characteristic polynomial ofA_k,nis given by

χ (A_k,n)(λ)=λⁿ⁻ⁿ

−1 j=0

λ^kj−Π_j

. (4.6)

4.2. Additional description of eigenvalues whenn=k^g+1

Letg_x=#S(x). We callg_xtheorderofx. Note thatg₀=1. It is easy to see that g_x=min

b >0:bis an integer andxk^b=xmodn

(4.7) and

S(x)=

xk^bmodn: 0≤b < gx

.

Define

J_k:= {Pi: #Pi=k}, n_k:=#Jk, X(k):= {x: x∈Znandxhas orderk},

(4.8) v_k,n:=1

n#{x: x∈Znandg_x< g₁}.

Lemma 3. The eigenvalues{η_i}of thek-circulant withn=k^g+1,g≥2,satisfy the following:

(a) η₀=_n−1

t=0a_t,is always an eigenvalue and ifnis even,thenη_n/2=_n−1

t=0(−1)^ta_t,is also an eigenvalue and both have multiplicity one.

(b) Forx∈Zn\ {0,ⁿ₂},g_x=g₁or ^g_b¹ for someb≥2and ^g_2b¹ is an integer.

(c) For all largen,g₁=2g.Hence from(b),forx∈Zn\ {0,ⁿ₂},g_x=2gor ^2g_b.The total number of eigenvalues corresponding toJ_2gis

2g×#J2g=#X(2g)∼n.

(d) X(^2g_b)=∅for2≤b < g,b even.Ifg is even thenX(^2g_g)=X(2)is either empty or contains exactly two elements with eigenvalues

η_l= |λ_l|, η_n−l= −|λ_l| for some1≤l≤n 2.

(e) Supposebis odd, 3≤b≤gand ^g_b is an integer.For eachPj ∈J_2g/b there are ^2g_b eigenvalues given by the

2g

bth roots ofΠj.Total number of eigenvalues corresponding to the setJ2g/bis 2g

b ×#J2g/b=#X 2g

b

∼

k^g/b+1

1+n^−a

for somea >0.

There are no other eigenvalues.

Proof. Sincen=k^g+1,nandkare relatively prime and hencen=n.

(a) P0=S(0)= {0}and the corresponding eigenvalue is η₀=_n−1

t=0a_t with multiplicity one. Similarly ifn is even thenkis odd and henceS(n/2)= {ⁿ₂}, and the corresponding eigenvalue isηn/2=n−1

t=0(−1)^tat of multiplicity one.

(b) From (4.7) it is easy to see thatg_x dividesg₁and henceg_x=g₁org_x=^g_b¹ for someb≥2. Also for every integert≥0,t k^g=(−1+n)t= −tmodn. Henceλ_t andλ_n−tbelong to same partition blockS(t)=S(n−t ). Thus eachS(t)contains even number of elements, except fort=0,ⁿ₂. Hence^g_b¹ must be even, that is,_2b^g1 must be an integer.

(c) From Lemma 5(i) of Bose, Mitra and Sen [10],g₁=2gfor all but finitely manynandv_k,n→0 asn→ ∞.

For each Pj∈J_2g we have 2gmany eigenvalues and which are 2gth roots ofΠ_j. Now the result follows from the fact that

n=2g#J2g+nυ_k,n.

(d) Suppose b=2 and x ∈X(^g₂¹)=X(^2g₂). Then xk^g1/2 =xk^g =x modn. But k^g = −1 modn and so, xk^g= −xmodn. Therefore 2x=0 modnandx can be either 0 orn/2. But we have already seen in part (a) that g₀=g_n/2=1. HenceX(^2g₂)=∅.

Now supposeb >2, even. From Lemma 3(ii) of Bose, Mitra and Sen [10], #X(^2g_b)≤gcd(k^2g/b−1, k^g+1)for b≥3. Now observe that forbeven,

gcd

k^2g/b−1, k^g+1

=1 ifkeven, 2 ifkodd.

So we have #X(^2g_b)≤2 forb >2 andbeven.

Suppose if possible, there existx∈Znsuch thatg_x=^2g_b. Then #S(x)=^2g_b and for ally∈S(x), g_y=^2g_b. Hence

#

y: g_y=2g b

≥2g

b >2 forg > b >2, beven.

This contradicts the fact that #X(^2g_b)≤2 forg > b >2,beven. HenceX(^2g_b)=∅forbeven andg > b >2.

Ifb=gand it is even, then from previous discussion #X(^2g_g)=0 or 2. In the later case there are exactly two ele- ments inZnwhose order is 2 and there will be only one partitioning set containing them. So corresponding eigenvalues will be

η_l= |λ_l|, η_n−l= −|λ_l| for some 1≤l≤n 2. (e) We first show that forbodd,

k^g/b+1

−

bi>b,biodd, g/biinteger

k^g/bi+1

≤#X 2g

b

≤k^g/b+1.

Note that (e) is a simple consequence of this. Let Z_n,b=

x: x∈Znandxk^2g/b=xmod

k^g+1 . Then it is easy to see that

X 2g

b

⊆Z_n,b. (4.9)

Letx∈Zn,band ^g_b=m. Then k^g+1|x

k^2g/b−1

⇒ k^bm+1|x

k^2m−1

⇒ k^(b−1)m−k^(b−2)m+k^(b−3)m− · · · −k+1|x k^m−1

.

But gcd(k^m−1, k^(b−1)m−k^(b−2)m+k^(b−3)m− · · · −k+1)=1, and thereforexis a multiple of(k^(b−1)m−k^(b−2)m+ k^(b−3)m− · · · −k+1). Hence

#Zn,b=

k^bm+1

(k^(b−1)m−k^(b−2)m+k^(b−3)m− · · · −k+1)

=k^m+1=k^g/b+1 and combining with (4.9),

#X 2g

b

≤#Zn,b=k^g/b+1.

On the other hand, ifx∈Z_n,bthen eitherg_x=^2g_b org_x<^2g_b. For the second caseg_x=^2g_bi for someb_i > b,b_i odd and thereforex∈Zn,b_i. Hence

#X 2g

b

≥#Zn,b−

bi>b,biodd, g/biinteger

#Zn,bi≥

k^g/b+1

−

bi>b,biodd, g/biinteger

k^g/bi+1 .

4.3. Properties of eigenvalues of Gaussian circulant matrices

Suppose{a_l}l≥0are independent, mean zero and variance one random variables. Fixn. For 1≤t < n, let us splitλ_t into real and complex parts asλt=at,n+ibt,n, that is,

a_t,n=

n−1

l=0

a_lcos 2πt l

n

, b_t,n=

n−1

l=0

a_lsin 2πt l

n

. (4.10)

Forz∈C,z¯denotes its complex conjugate. For all 0< t, t< n, the following identities can easily be verified using the orthogonality relations of sine and cosine functions.

E(at,nb_t,n)=0 and E a_t,n²

=E b²_t,n

=n/2, λ¯_t=λ_n−t, E(λtλ_t)=nI

t+t=n

, E

|λ_t|²

=n.

The following lemma is due to Bose, Mitra and Sen [10].

Lemma 4. Fixkandn.Suppose that{a_l}0≤l<nare i.i.d.standard normal random variables.

(a) For every n,n^−1/2a_t,n, n^−1/2b_t,n, 0≤t ≤n/2, are i.i.d.normal with mean zero and variance1/2.Conse- quently,any subcollection{Πj₁, Πj₂, . . .}of{Πj}0≤j <,so that no member of the corresponding partition blocks {Pj₁,Pj₂, . . .}is a conjugate of any other,are mutually independent.

(b) Suppose1≤j < andPj =n−Pj andn/2∈/Pj.Thenn^−nj/2Π_j are distributed as(n_j/2)-fold product of i.i.d.exponential random variables with mean one.

4.4. Truncation and normal approximation lemmata 4.4.1. Truncation

From Section4.2,n=nandS(t)=S(n−t )except fort=0, n/2. So forPj=S(0), S(n/2), we can defineAjsuch that

Pj= {x: x∈Aj orn−x∈Aj} and #Aj=1

2#Pj. (4.11)

For any sequence of random variablesb= {b_l}l≥0, define forPk∈J_2j β_b,j(k)=

t∈Aj

√1 n

n−1

l=0

b_lω^{t l}

2

, whereω=exp 2πi

n

. (4.12)

For eachn≥1, define a triangular array of centered random variables{¯a_l⁽ⁿ⁾}0≤l<nby

¯

al= ¯a⁽ⁿ⁾_l =alI_|al|≤n1/γ−EalI_|al|≤n1/γ.

Lemma 5. AssumeE[|al|^γ]<∞for someγ >2.Then,almost surely,

1max≤j≤q

β_a,g(j )1/2g

− max

1≤j≤q

β_a,g¯ (j )1/2g

=o(1).

Proof. Since_n−1

l=0ω^{t l}= 0 for 0< t < n, it follows thatβ_a,n¯ (j )=β_a,n˜ (j )where

˜

a_l= ˜a_l⁽ⁿ⁾= ¯a_l+EalI_|a

l|≤n^1/γ=a_lI_|a

l|≤n^1/γ. By Borel–Cantelli lemma,_∞

t=0|a_l|I_|a

l|>l^1/γ is finite a.s. and has only finitely many nonzero terms. Thus there exists a positive integerN (ω)such that

n t=0

|at− ˜at| = n t=0

|at|I_|at|>n1/γ≤ ∞ t=0

|at|I_|at|>t1/γ =

N (ω)

t=0

|at|I_|at|>t1/γ. (4.13)

It follows that forn≥ {N (ω),|a₁|^γ, . . . ,|a_{N (ω)}|^γ}the left-hand side of (4.13) is zero. Consequently, for allnsuffi- ciently large,

β_a,g(j )=β_a,g¯ (j ) a.s. for allj (4.14)

and the assertion follows immediately.

4.4.2. Normal approximation

Ford≥1, and any distinct integersi₁, i₂, . . . , i_d, from{1,2, . . . ,ⁿ⁻₂¹}, define v2d(l)=

cos

2πi_jl n

,sin

2πi_jl n

: 1≤j≤d T

, l∈Zn.

Letϕ_Σ(·)denote the density of the 2d-dimensional Gaussian vector having mean zero and covariance matrixΣ and letI_2dbe the identity matrix of order 2d.

Lemma 6 (Davis and Mikosch [15]). Fixd≥1, γ >2and letp˜_nbe the density function of 2^1/2n^−1/2

n−1

l=0

(a¯_l+σ_nN_l)v_2d(l),

where {Nl}l≥0 is a sequence of i.i.d. N (0,1) random variables, independent of {al}l≥0 and σ_n² =Var(a¯0)s_n². If n^−2clnn≤s_n²≤1withc=1/2−(1−δ)/γ for arbitrarily smallδ >0,then

˜

pn(x)=ϕ_(1+σ2

n)I2d(x)(1+εn) withεn→0 holds uniformly forx³=od(n^1/2−1/γ), x∈R^2d.

Corollary 1. Letγ >2andσ_n²=n^−cwherecis as in Lemma6.Then for an measurableB⊆R^2d,

Bp˜_n(x)dx−

B

ϕ_(1+σ2

n)I2d(x)dx ≤ε_n

B

ϕ_(1+σ2

n)I2d(x)dx+Od

exp

−n^η

for someη >0and uniformly over all thed-tuples of distinct integers1≤i₁< i₂<· · ·< i_d≤ ⁿ⁻₂¹. 4.5. Proof of Theorem4:Concluding arguments

To complete the proof we use the following lemmata whose proofs are given in the next section. Recall that {β_x,g(t)^1/2g}are the eigenvalues corresponding to the set of partitions having cardinality 2g. We derive the behaviour of their maximum in Lemma7. Lemma8 is technical and helps to conclude that the maximum of the remaining eigenvalues is negligible compared to the above.

Lemma 7.

max1≤t≤qβ_a,g(t)^1/2g−d_q c_q

→D Λ, (4.15)

whered_q, c_qare as in Lemma1,q=q_n=_2gⁿ −k_nand ^k_nⁿ→0asn→ ∞.As a consequence, max1≤t≤qβa,g(t)^1/2g−dn/2g

c_n/2g

→D Λ. (4.16)

Let

c_n(l)= 1

2l^1/2(logn)^1/2, d_n(l)=logC_l−(l−1)/2 logl 2l^1/2(logn)^1/2 +

logn l

1/2

1+(l−1)log logn 4 logn

,

Cl= 1

√l(2π)^(l−1)/2,

cn_2j=cn_2j(j ), dn_2j=dn_2j(j ), cn/2g=cn/2g(g) and dn/2g=dn/2g(g).

Lemma 8. Letn=k^g+1Ifj < gand for somea >0, 2j n2j =(k^j+1)(1+n^−a)∼n^j/g or is finite,then there exists a constantK=K(j, g)≥0such that,

cn/2g

c_n2j →K and d_n/2g−d_n2j

c_n2j → ∞ asn→ ∞.

Proof of Theorem 4. If #Pi =j, then the eigenvalues corresponding to Pi’s are the jth roots of Πi and hence these eigenvalues have the same modulus. From Lemma3, the possible values of #Pi are{1,2,2gand 2g/b,3≤b <

g, bodd,^g_b∈Z}. Recall from (4.12) thatβ_a,j(k)is the modulus of the eigenvalue associated with the partition setPk, where #Pk=2j.

In case of Gaussian entries it easily follows thatβ_a,j(k)is the product ofj exponential random variables and they are independent asktakesn_2jmany distinct values. So from Lemma1, ifn_2j→ ∞then the maximum ofβ_a,j(k)^1/2j has a Gumbel limit. For more general entries the method as in the proof of Lemma 7 can be adopted to get the following limit:

1≤maxk≤n_2j

β_a,j(k)^1/2j−d_n2j c_n2j

→D Λ asn_2j→ ∞, (4.17)

wherec_n2j andd_n2j are as above.

Letx_n=c_nx+d_n,q=q_n=_2gⁿ andB= {b: bodd, 3≤b < g,^g_b ∈Z}. Then P

sp

n^−1/2A_k,n

> x_q

≥P

max

j:Pj∈J2g

β_a,g(j )^1/2g> x_q

and P

sp

n^−1/2A_k,n

> x_q

≤ P

max

j:Pj∈J2g

β_a,g(j )^1/2g> x_q +

b∈B

P

max

j:Pj∈J2g/b

β_a,g/b(j )^b/2g> x_q

+P n^−1/2

n−1

l=0

al

> xq

!

+P n^−1/2

n−1

l=0

(−1)^lal

> xq

!

+P

j:Pmaxj∈J₂βa,2(j )^1/2> xq

=:A+B+C+D+E.