THIELE CENTRE. On spectral distribution of high dimensional covariation matrices. Claudio Heinrich and Mark Podolskij

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1 THIELE CENTRE for appied mathematics in natura science On spectra distribution of high dimensiona covariation matrices Caudio Heinrich and Mark Podoskij Research Report No. 02 December 2014

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3 On spectra distribution of high dimensiona covariation matrices Caudio Heinrich and Mark Podoskij Department of Mathematics, Aarhus University Abstract In this paper we present the asymptotic theory for spectra distributions of high dimensiona covariation matrices of Brownian diffusions. More specificay, we consider N-dimensiona Itô integras with time varying matrix-vaued integrands. We observe n equidistant high frequency data points of the underying Brownian diffusion and we assume that N/n c (0, ). We show that under a certain mixed spectra moment condition the spectra distribution of the empirica covariation matrix converges in distribution amost surey. Our proof reies on method of moments and appications of graph theory. Keywords: diffusion processes, graphs, high frequency data, random matrices. AMS 2010 Subject Cassification: 62M07, 60F05, 62E20, 60F17. 1 Introduction Last decades have witnessed an immense progress in the theory of random matrices and their appications to probabiity, statistica physics and number theory. Since the semina work [16], and increasingy so since [9], the asymptotic behaviour of the spectrum of random matrices received a great dea of attention. We refer to the monographs [1, 4, 10] for a detaied exposition of recent resuts and techniques. This paper is devoted to the study of spectra distribution of empirica covariation matrices of Brownian integras. On a fitered probabiity space (Ω, F, (F t ) t [0,1], P) we consider a diffusion process (X t ) t [0,1] that is defined as X t = X 0 + t 0 f s dw s, (1.1) where W denotes an N-dimensiona Brownian motion and f is a R N N -vaued step function given as m f t = T 1 [t 1,t )(t) (1.2) 1

4 where 0 = t 0 < < t m = 1 is a fixed partition of the interva [0, 1] and the matrices T j, 1 j m, are either deterministic or independent of the driving Brownian motion W. In mathematica finance one of the most centra objects is the empirica covariation of X, which is defined via [X] N n := n i=1 ( X i n )( X i 1 X i n n ). X i 1 (1.3) n Here and throughout the paper A denotes the transpose of a matrix A. For a fixed dimension N it is we known that [X] N n converges to the covariation matrix [X] N = 1 f 0 sf s ds as n whenever the Itô integra at (1.1) is we defined. When N converges to infinity at the same rate as n the situation becomes much more deicate. In the foowing we briefy review some recent work on spectra distribution of arge covariance/covariation matrices. Reca that for a given matrix A R N N with rea eigenvaues λ 1,..., λ N the spectra distribution of A is defined via F A (x) := 1 N N j=1 1 {λj x}. In [8] the author studies the spectra distribution of the empirica high dimensiona covariance matrix based on i.i.d. data, which corresponds to our mode (1.1) with f being constant. In this framework the spectra distribution of the empirica covariance matrix converges and, more importanty, there is a one-to-one connection between the imit of the Stietjes transform of F [X]N n and the imit of F [X] N (given the atter exists). It is exacty this reationship, caed Marčenko-Pastur equation, which makes the estimation of the spectra distribution of the covariation matrix [X] N possibe. In another paper [18] the authors consider the mode (1.1), where the time variation of f comes soey from a scaar function. In other words, they study processes of the type f s = a s T, where a : R R is a scaar function and T R N N. In this situation the methods of [8] can not be directy appied to infer F [X]N, but a certain modification of the functiona [X] N n, which separates the scaar function a and the matrix T, sti eads to a feasibe procedure. Unfortunatey, both methods do not work when the function f has the form (1.2). More precisey, the Stietjes transform method is hardy appicabe in our setting uness a matrices T 1,..., T m have the same eigenspaces for a N. In this work we foow the route of method of moments, which has been originay proposed by [17] in the context of random matrices. The basic idea is to show the amost sure convergence of a moments of the random probabiity measure F [X]N n. Then, under Careman s condition, the imiting distribution is uniquey determined by the imits of moments. The idea of the proof is heaviy based on combinatorics of coored graphs. The main resut of the paper is the foowing theorem. Theorem 1.1. Assume that N/n c (0, ) and the foowing conditions hod: (i) There exists a constant τ 0 > 0 such that T op τ 0 for a 1 m and uniformy in N. 2

5 (ii) For any k 1 and any muti-index {1,..., m} k the mixed spectra moment condition hods: M k 1 ( k ) := im N N tr T i T i (1.4) i=1 exists in the amost sure sense and is non-random. Then F [X]N n converges in distribution to a non-random probabiity measure F amost surey. The k-th moment m k of F is given via m k = k r=1 c r 1 ν 1 + +ν r=k {1,...,m} k c r,ν, r a=1 M νa (a) m (t t 1 ) s,ν,, (1.5) where (a) = ( (a) 1,..., ν (a) a ) {1,..., m} νa are such that = ( (1),..., (r) ). The power s,ν, is defined as s,ν, = r where n (a) = a=1 n(a) { #{j : (1) j = }, if a = 1, #{j 1 : (a) j = }, ese. The definition of c r,ν, is given in section 2.4. The paper is structured as foows. In section 2 we present an overview about reated probems and give some remarks on the conditions of Theorem 1.1. At the end of this section we aso give the definition of the constant c r,ν,. Section 3 is devoted to the proof of Theorem Reated probems and remarks In this section we review some reated studies and comment on conditions of Theorem Limit theory for a fixed dimension N As we mentioned in the introduction, the definition of a covariation matrix impies the convergence in probabiity [X] N n P [X] N as n when the dimension N is fixed. Furthermore, the asymptotic resuts of [5, Theorem 2.5] impy the foowing theorem. Theorem 2.1. Assume that the process f is càdàg (not necessariy of the form (1.2)). Then we obtain the stabe convergence ( n [X] N n [X] N) 1 d st 0 A 1/2 s dw s, (2.1) where W is a N 2 -dimensiona Brownian motion independent of the σ-agebra F and the N 2 N 2 -dimensiona matrix A s is given as A jk,j k s = Cs jj Cs kk + Cs jk Cs kj with C s = f s fs. 3

6 Quite surprisingy, Theorem 2.1 hods for genera càdàg stochastic processes f. We remark that Theorem 2.1 can be transformed into a feasibe standard centra imit theorem (cf. [11, Exampe 3.5]), thus making statistica inference for components of [X] N possibe. Such genera resuts do not hod anymore when N and one requires much stronger conditions on the process f. 2.2 Limit theory in the setting N/n c (0, ) In this subsection we shorty review the resuts of [8, 18]. In [8] the author considers empirica covariance matrices of i.i.d. vectors. In the setting of our mode (1.1) it means that the function f is deterministic and constant over the interva [0, 1]. In order to state the main resut we introduce the Stietjes transform, which is defined via m µ (z) = R 1 x z µ(dx), z C+, (2.2) where µ is a measure on R and C + := {z C : Im z > 0}. Since the matrix f is constant, we can write (in distribution) X t = X 0 + Σ 1/2 W t with Σ = [X] N. The foowing path breaking resut, caed Marčenko-Pastur equation, has been shown in [9] for the case of a diagona matrix Σ and extended ater to genera covariance matrices Σ in [12]. Theorem 2.2. Assume that the spectra distribution F Σ of Σ converges in distribution to F as N. When N/n c (0, ) the foowing resuts hod: (i) Define the function v [X] N n (z) := z 1 (1 N/n) + Nm F [X] N n (z)/n for z C+. Then there exists a deterministic function v such that v [X] N n (z) v(z) amost surey. (ii) The function v from (i) satisfies the Marčenko-Pastur equation 1 v(z) = z c 0 x F (dz). (2.3) 1 + xv(z) (iii) The equation (2.3) has a unique soution, which is the Stietjes transform of a measure. In practice it is of course impossibe to check whether the spectra distribution F Σ converges as N. A pragmatic soution to this probem is to assume that N is arge enough, so that F Σ can be identified with its theoretica imit F. In the next step, as proposed in [8], discretization and convex optimization can be appied to construct a numerica agorithm to compute the function F from Marčenko-Pastur equation (2.3). At this step the approximation v [X] N n (z) v(z) can be used. Finay, since we have identified F Σ with F, the spectra density of the covariance matrix Σ can be recovered from F. This procedure shows the importance of Marčenko-Pastur equation for statistica inference. 4

7 In the work [18] the authors propose an extension of this procedure to timevarying matrices f s, where the time variation is described by a scaar function. More precisey, they consider modes of the type (1.1) with [X] N n := tr([x]n n ) n f s = a s Σ 1/2, where a : [0, 1] R is a scaar function and Σ is a positive definite matrix with tr(σ) = N (possiby random, but independent of W ). In this setting the Marčenko- Pastur aw for [X] N n can not be expected to hod in genera as it has been demonstrated in [18, Proposition 3]. The functiona [X] N n requires a modification to satisfy the Marčenko-Pastur equation (2.3). Such a modification is given as ( )( ) n X i X i 1 X i X i 1 n n n n i=1 X i n X i 1 n 2, (2.4) where denotes the Eucidean norm. Intuitivey speaking, the proposed transformation of the origina statistic [X] N n eiminates the scaar variation a s and the methods of [8] become appicabe. Indeed, under certain conditions, the spectra distribution F [X] N n is connected to F Σ through the Marčenko-Pastur equation (2.3). We refer to [18, Theorem 2] for a detaied exposition of the asymptotic theory. 2.3 Remarks on conditions of Theorem 1.1 In this subsection we provide a discussion of conditions of Theorem 1.1. First of a, we remark that the mixed spectra moment condition at (1.4) is a rather strong condition, which however seems to be necessary according to our proofs. Nevertheess, in some specia cases this assumption can be repaced by an easier condition. For instance, in the setting of a constant function f, i.e. T 1 = = T m = T, a necessary condition for Theorem 1.1 to hod becomes F T T F, (2.5) where F T T is the spectra distribution of T T and the convergence is in distribution amost surey towards a non-random distribution function F. This assumption is used in cassica works [13, 17]. In this framework the boundedness of the operator norm at (i) of Theorem 1.1 is not required as this condition can be overcome by a truncation argument. More precisey, defining F T T τ assumption (2.5) impies the convergence F T T τ F τ, (x) := N 1 N i=1 1 {λ i x τ}, where F τ is a non-random distribution function, for a τ > 0. The convergence of moments resut simiar to (1.5) is then proved by showing the corresponding assertion for a fixed τ and etting τ. We refer to e.g. [4] for a detaied exposition. Aso the condition (1.4) of Theorem 1.1 foows directy from (2.5) and boundedness of T T op due to the obvious reation 1 N tr(t T ) k = 5 x k F T T (dx).

8 However, in the genera framework of (1.2) the convergence of, say, joint spectra distribution of matrices T 1 T1,..., T m Tm is not sufficient to concude convergence of mixed spectra moments at (1.4). The reason is that the behaviour of the expression at (1.4) is not soey determined by the eigenvaues of the invoved matrices, but cruciay depends on their eigenspaces. For the very same reason the truncation argument of [17] does not work, and spectra boundedness at (i) of Theorem 1.1 has to be assumed expicity. Therefore it seems hard to avoid imposing condition (1.4). Let us remark however that when matrices T 1 T1,..., T m Tm have the same eigenspaces for a N, i.e. there exist eigenvectors v 1,... v N such that T T v i = λ () i v i, then conditions (i) and (ii) of Theorem 1.1 can be repaced by assuming the amost sure weak convergence of the joint spectra distribution N F (T 1,...,T m) (x 1,..., x m ) = 1 N i=1 1 (1) {λ i x 1,...,λ (m) i x m} towards a non-random distribution function F. It is worth noticing that in the framework of free probabiity the mixed moment condition is referred to as the convergence of the joint distribution of the noncommutative random variabes T 1 T1,..., T m Tm, as N. See [3, 15] for an overview of this theory and its appications to random matrix theory. In particuar, asymptotic freeness of T 1 T1,..., T m Tm aows to weaken the mixed moment condition. Denoting for N N random matrices τ N (A) = 1 E[tr(A)], the matrices T N 1T1,..., T m Tm are asymptoticay free if for a i 1 i 2 i k and a p 1,..., p k > 0 im τ [( N (Ti1 Ti N 1 ) p 1 τ N ((T i1 Ti 1 ) p 1 ) ) ((T ik Ti k ) p k τ N ((T i1 Ti 1 ) p 1 ) )] = 0. By inearity of τ N it is then obvious that a mixed imiting moments exist if and ony if the spectra distributions F T iti converge to nonrandom imiting distributions F i with finite moments of a orders for i = 1,..., m, amost surey. 2.4 Definition of c r,ν, In this subsection we give the definition of the constant c r,ν, that appears in Theorem 1.1. Given {1,..., m} k and ν 1,..., ν r with ν ν r = k, we et (a) = ( (a) 1,..., ν (a) a ) {1,..., m} νa such that = ( (1),..., (r) ). We reca the definition { n (a) = #{j : (1) j = }, if a = 1, #{j 1 : (a) j = }, ese. Given a tree, i.e. a connected graph without cyces, G with r vertices H 1,..., H r, we define for {1,..., m} and a {1,..., r} numbers n (a),g in the foowing way: Let H a1,..., H ap be the vertices adjacent to H a in G (i.e. the vertices connected to H a by a path of ength 1), where we eave out the vertex that ies on the path from H a to H 1, if a > 1. We set n (a),g = #{j {1,..., p} : (a j) 1 = }. 6

9 Then, we have c r,ν, = G 1 S,G m r a=1 (n (a) n (a)! n (a),g )! 1 {n (a),g, n (a) } where the summation runs for a trees G on {H 1,..., H r }. Here, S,G is the set of a permutations π on the {2,..., r} for which at east one of the foowing hods: (i) (π(p)) (p) for some p {2,..., r} (ii) G π G, where G π is the graph obtained from G by permuting the vertices H 2,..., H r according to π. 3 Proof For the proof of Theorem 1.1 we rey on the we known moment convergence theorem. Theorem 3.1. Let (F n ) be a sequence of p.d.f.s with finite moments of a orders m k,n = x k df n (x). Assume m k,n m k for n for k = 1,... where (a) m k < for a k and (b) k=1 [m 2k(F )] 1 2k =. Then, F n converges weaky to the uniquey determined probabiity distribution function F with moment sequence (m k ). Condition (b) is known as Careman s condition. For the proof we refer to [7, Theorem ]. The strategy for proving Theorem 1.1 is the foowing: In the next subsection we introduce coored Q + -graphs. In the two subsections thereafter, these graphs take a key roe in showing that E[m k (F [X]N n )] mk (3.1) hods for a k, where m k is defined as in Theorem 1.1. Then, in subsection 3.4 we argue that ) 4 ] E [(m k (F [X]N n ) E[mk (F [X]N n )] = O(N 2 ), (3.2) which yieds m k (F [X]N n ) mk, amost surey, by virtue of the Bore-Cantei Lemma. Finay, verifying that the sequence (m k ) satisfies Careman s condition and appying Theorem 3.1 competes the proof. Our proof extends the proof given in [17] (see aso [4] and [2]) for the case of constant function f. In order to dea with our more genera setting we introduce coored graphs and use new combinatorica arguments. Throughout the proof, we denote the entries of the matrices T by (T ) ij = t () ij, and ikewise for other matrices, in order to maintain readabiity. 7

10 i 1 = i 19 i 2 = i 18 i 3 = i 17 i 4 i 5 = i 15 i 16 i 6 = i 14 i 7 i 8 = i 9 i 10 i 11 = i 12 i 13 i j 1 = j 6 j 2 = j 5 j 3 j 4 j (1) j (2) Figure 1: A coored Q + -graph for k = 6 and m = 2. Here, = (1, 2, 2, 1, 2, 1) where 1 = green and 2 = red. 3.1 Coored Q + -graphs For = 1,..., m et Y be N [n(t t 1 )] matrices containing i.i.d. standard norma variabes independent of T, where [n(t t 1 )] denotes the integer part of n(t t 1 ). Set [X] N n := 1 m T Y Y T. n By virtue of the we known fact F A F B 1 N rank(a B) for N N symmetric matrices A and B, it is easy to see that F [X]N n F [X] N n 0, (3.3) as n, N. Therefore, we can repace [X] N n by [X] N n for the proof of Theorem 1.1. Conditioning on a T as given aows us, moreover, to restrict ourseves to nonrandom T for the proof. In order to show the convergence of the expected k-th spectra moment E[m k (F [X] N n )] we are faced with the equation E[m k (F [X] 1 1 [ N n )] = N n E tr k ( m [ = N 1 n k E,i,j ) k ] T Y Y T t ( 1) i 1 i 2 y ( 1) i 2 j 1 y ( 1) j 1 i 3 t ( 1) i 3 i 4 t ( k) i 3k 2 i 3k 1 y ( k) i 3k 1 j k y ( k) j k i 3k t ( k) i 3k i 1 ], (3.4) Here, the summation runs over a = ( 1,..., k ) {1,..., m} k and i = (i 1,..., i 3k ) {1,..., N} 3k. For a a, the index j a varies over {1,..., [n(t a t a 1)]}. In order to carry out the summation we introduce coored Q + -graphs which correspond to the summands in the above equation. These graphs are reated to Q-graphs as used by the authors of [17]. 8

11 i i 1 = i 13 i 2 = i 12 i 3 = i 11 i 4 i 5 = i 6 i 7 i 8 = i 9 = i 10 j (1) j (2) j 1 = j 4 j 2 = j 3 Figure 2: A coored Q + -graph in C 3 and its piar. Definition 3.2. Let k > 0. Given muti-indices {1,..., m} k, i {1,..., N} 3k, and j = (j 1,..., j k ) where j a {1,..., [n(t a t a 1)]}, we define the coored Q + - graph Q,i,j in the foowing way. Choose m arbitrary coors c 1,..., c m. For brevity we wi usuay not distinguish between {1,..., m} and its associated coor c. Draw m+1 horizonta ines, the i-, j (1) -,..., j (m) -ine. Mark the numbers {1,..., N} on the i-ine and, for a, the numbers {1,..., [n(t t 1 )]} on the j () -ine. For s = 1,..., k, draw horizonta edges coored in s from i 3s 2 to i 3s 1 and from i 3s to i 3s+1 with the convention that i 3k+1 = i 1. For s = 1,..., k, draw a vertica (down) edge from i 3s 1 to j s on the j (s) -ine and a vertica (up) edge from j s to i 3s, both edges aso coored in s. The resut is a connected directed graph forming a cyce. It consists of 4k edges and aways 4 subsequent edges are of the same coor. Figure 1 provides an exampe of a coored Q + -graph. There is a one to one correspondence between coored Q + -graphs and the summands of (3.4). Highighting this correspondence we introduce the notation (ty) Q,i,j = E [ t ( 1) i 1 i 2 y ( 1) i 2 j 1 y ( 1) j 1 i 3 t ( 1) i 3 i 4 t ( k) i 3k 2 i 3k 1 y ( k) i 3k 1 j k y ( k) j k i 3k t ( k) i 3k i 1 ]. (3.5) We wi organize the coored Q + -graphs in three categories and then derive the imit for (3.4) if the summation runs ony for graphs from one of these categories. To this end, the foowing definitions are required. Definition 3.3. The head H(Q,i,j ) of a coored Q + -graph Q,i,j is the subgraph of a vertices on the i-ine and a horizonta edges. Definition 3.4. The piar of a coored Q + -graph Q,i,j is the Graph obtained from Q,i,j by first guing together coincident vertica edges, then guing a vertices on the i-ine that are connected in the head of Q,i,j, and then deeting a horizonta edges. The piar is undirected and cooress. See Figure 2 for an exampe. We divide the coored Q + -graphs in the foowing three categories. Category C 1 contains a graphs Q,i,j such that every down edge of Q,i,j coincides with exacty one up edge, and such that the piar of Q,i,j is a tree. An exampe of a coored Q + - graph in this category is the graph in Figure 1. Note that coincident vertica edges are aways of the same coor. Category C 2 contains a coored Q + -graphs that have at east one singe vertica edge. Category C 3 contains a other coored Q + -graphs. The graph in Figure 2 is in this category since its piar contains a cyce. 9

12 Now we can spit the sum (3.4) into [ E[m k (F [X] ] N n )] = N 1 n k (ty) Q + (ty) Q + (ty) Q. (3.6) Q C 1 Q C 2 Q C 3 The second sum vanishes since a vertica edge in Q which is singe corresponds to a factor y () ij in (3.5) that occurs ony once. Hence, the expectation is 0 due to independence. In the foowing section we argue that the third sum is negigibe for n, N. In the section thereafter, the imit for the first sum is derived. 3.2 The imit for the sum of C 3 graphs We make the foowing conventions on notation: For a Q + -graph Q we denote by r the number of connected components of the head. By s 1,..., s m we denote the numbers of noncoincident j (1),..., j (m) -vertices, respectivey, and s = s s m. Denote further by p the number of noncoincident vertica edges of Q. Lemma 3.5. For a category C 3 coored Q + -graph Q,i,j it hods that p + s 1 < k. Furthermore, the degree of each vertex of H(Q,i,j ) is at east 2. Proof. The piar of Q,i,j has r + s vertices and p edges and is connected. Therefore, r + s 1 p where equaity impies that the piar is a tree. We distinguish two different cases. Case 1. If every vertica edge of Q,i,j has coincidence mutipicity 2, it hods that p = k, since Q,i,j contains 2k vertica edges. If, in this case, the piar woud be a tree, Q,i,j woud be in C 1. Therefore, we have r + s 1 < p = k. Case 2. One vertica edge of Q,i,j has coincidence mutipicity arger 2. We have p < k and thus r + s 1 p < k. Every i-vertex of Q,i,j connects either with at east two horizonta edges or with one horizonta and one vertica edge, which is then singe. Therefore, if some vertex of H(Q,i,j ) has degree one, we have Q,i,j C 2. In order to show that the sum corresponding to C 3 in (3.6) is negigibe for N, we introduce the concept of isomorphic Q + -graphs. Definition 3.6. Two coored Q + -graphs Q,i,j and Q,i,j are isomorphic, or Q,i,j Q,i,j, if we can obtain Q,i,j from Q,i,j by permuting the numbers on the ines i, j (1),..., j (m). In particuar, Q,i,j Q,i,j impies =. Lemma 3.7. It hods that E 3 := N 1 n k Q C 3 (ty) Q 0 (3.7) for N, n with N/n c (0, ). 10

13 Proof. Observe the identity E 3 = N 1 n k Q 3 Q [Q 3 ] (ty) Q, (3.8) where the first summation is taken for a representative system of pairwise not isomorphic graphs in category C 3 and the second summation for a Q + -graphs isomorphic to Q 3. It is sufficient to show that for arbitrary Q 3 C 3 we have N 1 n k (ty) Q 0. (3.9) Q [Q 3 ] Gue coincident vertica edges of Q 3 into cooress down edges. Let every vertica edge that connects with the j () -ine correspond to the matrix Y (µ) = {(µ 1)!!} N [n(t t 1 )], where µ denotes the coincidence mutipicity of the edge. Appying Theorem A 35. of [4] and Lemma 3.5 yieds that there is a constant C k such that N 1 n k (ty) Q C k N 1 n k N r+s = O(N 1 ), (3.10) and the proof is compete. Q [Q 3 ] 3.3 Limit of the Expected k-th Spectra Moment In this subsection we derive the imit of the first sum in (3.6). For a coored Q + - graph Q C 1, the expectation factor E[y ( 1) i 2 j 1 y ( 1) i 3 j 1 y ( k) i 3k 1 j k y ( k) i 3k j k ] of (ty) Q equas 1. Therefore, N 1 n k (ty) Q = N 1 n k (t) H(Q) Q C 1 Q C 1 depends on the heads of the graphs ony. Using the notations introduced in the ast subsection, there are m / m [n(t t 1 )]! ([n(t t 1 )] s )! coored Q + -graphs with the same head as Q. Every graph Q 1 C 1 has k noncoincident vertica edges and its piar is a tree with r + s vertices and k edges where s = s s m. Consequenty, we have k = r + s 1. Therefore, it hods that N 1 n k Q C 1 (t) H(Q) = N 1 H(Q) H(C 1 ) n r+1 (t) H(Q) m (t t 1 ) s + o(1) (3.11) where H(C 1 ) denotes the set of coored heads for graphs in C 1. We first derive the imit for this term if the summation runs for a cass of simiar heads. 11

14 Definition 3.8. A Q + -graph Q induces a partition of the set {1,..., 3k}, where a and b are in the same partition set if and ony if i a and i b are connected in H(Q). Let Q and Q be coored Q + -graphs with the same cooring vector. The heads H(Q) and H(Q ) are simiar (sometimes we aso say Q and Q are simiar) if they induce the same partition. The equivaence cass of heads simiar to H(Q) wi be denoted by [[H(Q)]]. See Figure 3 for an exampe. At this point it is convenient to introduce the notion of component cooring muti-indices (CCMIs). For a head of a coored Q + -graph we denote the connected components by H 1,..., H r and their sizes (i.e. the number of edges they contain) by 2ν 1,..., 2ν r. For some component H a of the head, the CCMI (a) = ( (a) 1,..., ν (a) a ) {1,..., m} νa is defined in the foowing way. We obtain a natura order for the edges of the Q + -graph by the order of indices in (3.5), i.e. the first edge connects i 1 and i 2, the second i 2 and j 1 and so on. We set (a) b = where is the coor of the b-th up edge that connects to H a. Remark. Note that for a given Q + -graph Q the muti-index ( (1),..., (r) ) is not uniquey determined since it depends on the abeing of the head components H 1,..., H r. We foow the convention that H 1 contains the index i 1. The abeing of the components H 2,..., H r, however, remains arbitrary, for reasons that wi be expained ater. Let us now anayze the summation of (t) H(Q) for a simiarity cass [[H(Q)]]. Lemma 3.9. Introducing the notation (T T ) = q i=1 T i T i for = ( 1,..., q ) {1,..., m} q, we have for a coored Q + graphs Q 1 C 1 with CCMIs (1),..., (r) Proof. We write H(Q) [[H(Q 1 )]] H(Q) [[H(Q 1 )]] (t) H(Q) = r (tr(t T ) (a) + O(1)). a=1 (t) H(Q) = (t)h1 (t) Hr, (3.12) where (t) H1,..., (t) Hr are products of entries of T 1,..., T m, T1,..., Tm associated with the distinct cyces H 1,..., H r of the head in the same manner as coored Q + - graphs are associated to products of the form (3.5). The summation Σ runs for a i i 1 = i 10 i 7 i 3 = i 5 i 4 i i 1 = i 10 i 2 = i 6 i 7 i 8 = i 9 i 3 = i 4 = i 5 Figure 3: Two simiar heads. 12

15 indices of these entries over {1,..., N}, with the restriction that entries of different cyces have distinct indices. Consider first (t) Ha for some a {1,..., r}. By the definition of the CCMI (a) and recaing that the piar of Q is a tree it is not difficut to verify that (t)ha = t ( (a) 1 ) i 1 i 2 t ((a) 2 ) i 2 i 3 t ((a) 2 ) i 3 i 4 i {1,...,N} 2νa... t ((a) νa ) i 2νa 1 i 2νa t ((a) 1 ) i 2νa i 1 = tr(t T ) (a). Then, appying the incusion-excusion principe, and recaing T op τ 0 for a, yieds t ((a) 1 ) t ((a) 2 )... t ((a) 1 ) i 2νa i 1 = tr(t T ) (a) + O(1), i {1,...,N} 2νa {i 1,...,i 2νa } M= i 1 i 2 i 2 i 3 for any finite set M Z +. The statement foows now by induction over the distinct cyces of the head. Appying this Lemma we find N 1 H(Q) [[H(Q 1 )]] n r+1 (t) H(Q) m (t t 1 ) s i m c r 1 (t t 1 ) s i r a=1 M νa (a). (3.13) Now, in order to derive the imit of E[m k (F [X]N n )], it is sufficient to determine, for given r, ν 1,..., ν r, s 1,..., s m and (1),..., (r), the number of simiarity casses with this specific parameters. Definition Two components H a and H b of the head of a coored Q + -graph in C 1 are verticay connected if there is a down edge starting at some vertex in H a, which is foowed by an up edge that ends at some vertex in H b. Note that if H a and H b are verticay connected, then there is exacty one down edge eaving H a that is foowed by an up edge connecting to H b and exacty one down edge eaving H b that is foowed by an up edge connecting to H a. These four edges form two pairs of coincident edges and are of the same coor. Therefore, we may understand the vertica connections as coored as we. Definition For a coored Q + -graph Q C 1 with components H 1,..., H r we define the connectivity tree G Q to be the graph with vertex set {H 1,..., H r } where (H a, H b ) is an edge in G Q if and ony if H a and H b are verticay connected in Q. Note that due to the arbitrary abeing of H 2,..., H r most Q + -graphs have more than one possibe connectivity tree. Lemma It hods that E[m k (F [X] N n )] mk 13

16 where m k = k c r 1 r=1 ν 1 + +ν r=k {1,...,m} k c r,ν, r a=1 M νa (a) m (t t 1 ) s,ν,. Here, = ( (1),..., (r) ) where (a) has ength ν a. For the definition of s,ν, see Theorem 1.1, for the definition of c r,ν, see section 2.4. Proof. Recaing (3.6) and Lemma 3.7 it is sufficient to derive that N 1 n k Q C 1 (ty) Q m k. Thus, by virtue of (3.11) and (3.13) there are two things eft to show: (1) For a Q + -graph Q with CCMIs (1),..., (r), the number of noncoincident vertices on the j () -ine is s = r a=1 n(a). (2) There are c r,ν, simiarity casses of Q + -graphs with CCMIs (1),..., (r). For (1) note that every vertex on the j () -ine has either degree 2 or 4 and its degree is 4 if and ony if it ies on a vertica connection of coor. Therefore, s is the number of up edges coored in minus the number of vertica connections of coor. The number of up edges coored in is the number of -s in the CCMIs (1),..., (m). Let H a and H b be two verticay connected components where in the connectivity tree G Q H a ies on the path from H b to H 1. Then, the coor of the vertica connection (H a, H b ) is (b) 1. Therefore, the entries (b) 1 for b > 1 correspond one to one to the coors of the vertica connections of Q. This proves caim (1). For (2) we first show that there are m r a=1 (n (a) n (a)! n (a),g )! 1 {n (a),g n (a) } simiarity casses of Q + -graphs with connectivity tree G and CCMIs (1),..., (r). Within a component H a a vertica connection (H a, H b ) is at a certain position p {1,..., ν a }, meaning that the p-th down edge eaving H a is foowed by an up edge connecting to H b. It is straightforward to verify that two Q + -graphs with the same connectivity tree G and the same CCMIs (1),..., (r) are simiar if and ony if within a components a vertica connections are at the same positions. Consider component H 1, and et H a1,..., H ap be the components adjacent to it in G. A Q + -graph Q with connectivity tree G contains the corresponding vertica connections (H 1, H a1 ),..., (H 1, H ap ), n (1),G of which are coored in. Since H 1 has eaving down edges of coor we have n (1) m (n (1) n (1)! n (1),G )! 1 {n (1),G n (1) } possibiities of positioning the vertica connections among the vertica edges eaving H 1. Now turn to some component H a H 1. There is one component H a0 verticay connected to H a that ies on the path from H a to H 1 in G. By construction, the 14

17 vertica connection (H a, H a0 ) is at position ν a within H a and it is coored in (a) 1. For distributing a other vertica connections at H a on their possibe positions within H a, we are eft with m (n (a) n (a)! n (a),g possibiities. This eaves us, overa, with m r a=1 (n (a) n (a)! n (a),g )! 1 {n (a),g )! 1 {n (a),g n (a) } n (a) } possibiities for distributing a vertica connections of a components on their possibe positions. Most simiarity casses have more than one possibe connectivity tree and CCMIs since the components H 2,..., H r are arbitrariy abeed. By definition of the set S,G, introduced in section 2.4, a Q + -graph Q has S,G possibe connectivity trees and CCMIs where G is one possibe connectivity tree for Q. This proves (2). Remark. The arbitrary abeing of the components H 2,..., H r is necessary in order to appy the combinatorica arguments of the proof above. If we, for exampe, abe the components in order of appearance with respect to the natura order of edges, we impose subte restrictions on the CCMIs, eading to more invoved expressions. In the next subsection we compete the proof of Theorem Convergence of m k (F [X]N n ) The foowing Lemma ensures the a.s. convergence of m k (F [X] N n ). The proof reies on corresponding resuts for constant f. For more detais we refer to [4, Theorem 4.1]. Lemma It hods that E[m k (F [X] N n ) Emk (F [X] N n )] 4 = O(N 2 ). Proof. For a = 1,..., 4, given muti-indices a = ( (a) 1,..., (a) k ) {1,..., m}k, i a {1,..., N} 3k and j a = (j (a) 1,..., j (a) k ) with j(a) p {1,..., [n(t (a) t (a) p )]}, we denote p 1 by Q a the corresponding coored Q + -graph Q a,ia,j a. Then, we have [ E[m k (F [X] N n ) Emk (F [X] N n 1 [ 1 ] ] 4 )] 4 = E (ty) Nn k Q,i,j E (ty) Nn k Q,i,j,i,j = N 4 n 4k 1,...,j 4 E,i,j [ 4 ] ((ty) Qa E[(ty) Qa ]). (3.14) If, for some a, a vertica edges of Q a do not coincide with vertica edges of one of the other graphs, we obtain a=1 [ 4 ( E (ty)qa E[(ty) Qa ] )] = 0, a=1 15

18 from independence. Thus, Q = Q a consists of either one or two connected components. By expanding (3.14) we have E[m k (F [X] N n ) Emk (F [X] N n )] 4 = N 4 n ( [ 4 ] 4k E (ty) Qa ± + i 1,..., 4 a=1 4 a=1 ) E[(ty) Qa ] Appying Theorem 1A.35. of [4], in a simiar way as in the proof of Lemma 3.7, for each of the 16 summands within the brackets separatey, shows that this sum is O(N 2 ). Now, combining Lemma 3.12 and Lemma 3.13 we have, by virtue of the Bore- Cantei Lemma and (3.3), m k (F [X]N a.s. n ) m k, for a k, where m k is defined as in Theorem 1.1. Therefore, if the sequence (m k ) satisfies Careman s condition, appying Theorem 3.1 competes the proof of Theorem 1.1. Lemma The sequence of imiting spectra moments m k satisfies (m 2k ) 1 2k =. k=0 Proof. Consider the matrices S = 1 T n Y Y T for = 1,..., m. The spectra distribution of Y n(t t 1 ) Y is known to converge to the Marčenko-Pastur aw p y (x) with 1 support [(1 y ) 2, (1 + y ) 2 ] for = 1,..., m, where y = c(t t 1 ) 1. Thus, we have im N 1 N tr(sk ) im (t t 1 ) k T 2k 1 k N op n(t t 1 ) Y Y ( 1 + 2k max (y )), τ 2k 0 for = 1,..., m. Therefore, the resut foows from m k (F [X]N n ) 1 N mk 1 ( tr(s k 1 ) + + tr(s k m) ), which hods due to the convexity of the function A tr(a) k, see for exampe [6, Theorem 2.10]. op Acknowedgement We woud ike to thank Caros Vargas Obieta and Steen Thorbjørnsen for a fruitfu discussion. 16

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