Spectral Measure of Large Random Toeplitz Matrices

Transcription

1 Spectral Measure of Large Random Toeplitz Matrices Yongwhan Lim June 5, 2012

2 Definition (Toepliz Matrix) The symmetric Toeplitz matrix is defined to be [X i j ] where 1 i, j n; that is, X 0 X 1 X 2 X n 2 X n 1 X 1 X 0 X 1 X n 2. T n = X 2 X 1 X X2 X n 2 X 0 X 1 X n 1 X n 2 X 2 X 1 X 0 We will further assume that E[X ] = 0, E[ X 2 ] = 1, and E[ X r ] < for all r > 2.

3 Theorem (1.1) Let {X k : k = 0, 1, 2, } be a sequence of i.i.d. real-valued random variables with Var(X 1 ) = 1. Then with probability 1, ˆµ(T n / n) converges weakly as n to a non-random symmetric probability measure γ T which does not depend on the distribution of X 1, and has unbounded support. This theorem falls short of establishing that the limiting distributions have smooth densities. What happens to the maximal eigenvalue of T n? What is limit empirical measure in non-symmetrc case, where the moment method no longer is applicable?

4 Definition (Moments) For a probability measure γ on (R, B), its moments are defined to be m k (γ) = x k γ(dx). Example (γ T ) We will see that the odd moments are zero (symmetric) and the even moments are sums of numbers labeled by the pair partitions of {1,, 2k}.

5 Definition (Partition Words) The partition words are words of length 2k with k pairs of letters such that the first occurrences of each of the k letters are in alphabetical order. Example (k = 2) In this case, we have 1 3 = 3 partition words: Their pair partitions are: aabb, abba, abab. {1, 2} {3, 4}, {1, 4} {2, 3}, {1, 3} {2, 4}.

6 Definition Let w[j] be the letter in position j of word w. Example (w = abab) w[1] = a, w[2] = b, w[3] = a, and w[4] = b.

7 To every partition word w of length 2k, we associate the following system of equations in unknowns x 0,, x 2k : x 1 x 0 + x m1 x m1 1 = 0 if there is m 1 > 1 such that w[1] = w[m 1 ]. x 2 x 1 + x m2 x m2 1 = 0 if there is m 2 > 2 such that w[2] = w[m 2 ].. x i x i 1 + x mi x mi 1 = 0 if there is m i > i such that w[i] = w[m i ].. x 2k 1 x 2k 2 + x 2k x 2k 1 = 0 if w[2k 1] = w[2k]. We write p T (w) to mean the volume of a cross section of the cube I k+1, after forcing the k + 1 dependent variables to lie in the interval I = [0, 1].

8 Example (w = abab) We have x 3 = x 0 x 1 + x 2 and x 4 = x 0. This defines the solid {x 0 x 1 + x 2 I } {x 0 I } I 3, which has the (Euclidean) volume, p T (abab) = 4/3! = 2/3. We will show in the second talk that the measure is symmetric (i.e., the odd moments are 0) and that: m 2k (γ T ) = p T (w) w: w =2k γ T is uniquely determined since m 2k is at most the number (2k 1)!! of words of length 2k.

9 For each partition word w of length 2k with nonzero volume p(w), we may write: k M p(w) = P n i,j U j [0, 1] i=1 where n i,j are integers, M = k, and U j s are the independent uniform U[0, 1] random variables. Using this idea, we can prove the following proposition: Proposition (A.1) j=0 A symmetric measure γ T has unbounded support.

10 Proof. It suffices to show that (m 2k ) 1/k. Let w be a partition word of length 2k. Writing S i = j n i,ju j 1/2, where i = 1,, k, we have: ( k ) p T (w) = P { S i < 1/2}. i=1 Now, since the coefficients n i,j take values 0, ±1 only and j n i,j = 1, we can rewrite S i as: S = (U α 1/2) + L (U β(j) U γ(j) ) j=1 where α, β(j), γ(j) with j = 1,, L are all different.

11 Proof. Fix ɛ > 0. After standard calculation, we conclude that: p T (w) 1 2 P(A) = 1 (ɛ(k + 1)) (k+1) 2 Since there are more than k! partition words of length 2k, for sufficiently large k, we have: So, as desired. m 2k 1 2 k!(ɛ(k + 1)) (k+1) (3ɛ) k. lim sup m 1/k 2k 1 k 3ɛ

12 Definition (Circulant Matrix) The symmetric circulant matrix is the symmetric Toeplitz matrix with an additional constraint that X i = X n i for all 0 < i < n; that is, X 0 X 1 X 2 X 2 X 1 X 1 X 0 X 1 X 2. C n = X 2 X 1 X X2 X 2 X 0 X 1 X 1 X 2 X 2 X 1 X 0

13 Lemma The limiting spectral measure is standard normal for the circulant matrix of i.i.d. first row entries. Proof. From linear algebra, we know that eigenvalues are real and given by the discrete fourier transform of the first row of matrix; hence, Gaussian. Standard calculation shows that entries of this DFT are then uncorrelated. So, joint law of eigenvalue is that of standard normal i.i.d. Hence, by Glivenko-Cantelli Theorem, their empirical measure converges to standard normal.

14 Theorem If the sum of square differences of entries of the first row of two families of Toeplitz matrices goes to zero as matrix size goes to infinity, the spectral measure of both matrices have the same limit. Since circulant matrices are also Toeplitz, in the context of independent entries, if Var(X i ) goes to zero fast enough in i then the limit spectral measure for Toeplitz will also be standard normal, with a proper scaling; this is, of course, not the case for i.i.d. X i s.

15 1. Bryc, W., Dembo, A. and Jiang, T. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), pp H. Zessin, The method of moments for random measures, Z. Wahrsch. Verw. Gebiete 62, No. 3 (1983), Grenander, U., Szego G. Toeplitz forms and their applications. Berkeley, University of California Press (1958). 4. Bose, A. and Mitra, J. Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 (2002),