STAT C206A / MATH C223A : Stein s method and applications 1. Lecture 31

Transcription

1 STAT C26A / MATH C223A : Stein s method and applications Lecture 3 Lecture date: Nov. 7, 27 Scribe: Anand Sarwate Gaussian concentration recap If W, T ) is a pair of random variables such that for all Lipschitz ϕ, then for any σ 2 > E[W ϕw )] = E[T ϕ W )] ) LW ), N, σ 2 ) ) 2E T σ2 σ 2, 2) where LW ) is the law of W. In particular, if σ 2 = VarW ), it is easy to see that E[T ] = σ 2, and hence LW ), N, σ 2 ) ) 2 VarT ) σ 2. 3) So how do we plan to use this? We have the following canonical construction: let X = X, X 2,..., X n ) be a vector of iid N, ) random variables and f : R n R be an absolutely continuous function. Then for W = fx) with E[W ] =, E[W 2 ] < we have for all Lipschitz ϕ, where E[W ϕw )] = E[T ϕ W )], 4) T = 2 t i= f X) f X t )dt 5) X i X i where X t = t X + t Y and Y = Y, Y 2,..., Y n ) are also iid N, ). With this tool we proved the Gaussian Poincaré inequality and the Gaussian concentration inequality. Today we will start a method for obtaining normal approximations for quite complicated functions. For example, we will look at linear statistics of the eigenvalues of random matrices. 3-

2 2 A CLT for functions of Gaussians To get a CLT we first need to prove the concentration of T given by 5). Clearly, we can replace T by the conditional expectation T x) = E[T X = x]. This requires some ugly but straightforward calculation. We begin by writing T x): [ T x, x 2,..., x n ) = 2 t E f x) f t x + ] t Y) dt 6) x i x i Letting σ 2 = VarW ), we have By the Poincaré inequality, i= LW ), N, σ 2 ) ) 2 σ 2 VarT x)) 7) VarT X)) E T X) 2 8) These give what one might call the 2nd order Poincaré inequalities. Continuing with the computation: T x) = x i 2 t E 2 f x) f t x + t Y) x j= i x j x j }{{} A i t) 9) + f 2 f t x) t x + t Y) dt. ) x j= j x i x j }{{} B i t) What we really want to bound is the sum of squares of this expression. Using the inequality a + b) 2 2a 2 + 2b 2 and Jensen s inequality, we get ) T 2 ) 2 x) 2 x i i= 2 t E[A it)]dt + 2 2E 2 A i t) 2 dt + 2E t i= You should be used to this by now! i= i= ) 2 2 E[B it)]dt ) B i t) 2 dt. 2) 4 i= 3-2

3 Turning to the first term, A i t) 2 = 2 f x) f t x + 2 t Y) 3) x i= j= i x j x j = Hess fx) f t x + t Y) 2 4) Hess fx) 2 f t x + t Y) 2. 5) i= We can get a similar bound for B i t). Note that in computing E T X) we will encounter terms that can be bounded using the Cauchy-Schwarz inequality and the fact that X d = X t. Thus [ E Hess fx) 2 fx t ) 2] E Hess fx) 4) /2 E fx t ) 4) /2. 6) VarT X)) E T X) 7) ) 2 2 t dt dt E Hess fx) 4 E fx t ) 4) /2 8) = 5 E Hess fx) 4 E fx t ) 4) /2. 9) 2 We then have L, N, σ 2 ) ) 2 5 E σ 2 Hess fx) 4 E fx t ) 4) /4 2) 2 = σ 2 E Hess fx) 4 E fx t ) 4) /4. 2) We have proved the following Theorem If W = fx, X 2,..., X n ) where X = X, X 2,..., X n ) is a vector of iid N, ) random variables with E[W ] =, E[W 2 ] = σ 2, and f C 2 R n ), then LW ), N, σ 2 ) ) σ 2 E Hess fx) 4 E fx t ) 4) /4. 22) Exercise 2 Improve this theorem so that it doesn t have any 4th powers. 3-3

4 3 Looking forward : eigenvalues of random matrices What sort of problems can we tackle with this machinery? Suppose are iid N, ) random variables and let X ij ) i,j< 23) A n = n X ij ) i,j<. 24) This is sometimes called the real Ginibré ensemble. The eigenvalues λ, λ 2,..., λ n of A n are approximately uniformly distributed on the unit disc, in the following sense: n i= We can look at sums of the form δ λi a.s. Uniformunit disc in C). 25) fλ i ), 26) i= for some function f : C C. It turns out that under very general conditions on f, this is asymptotically Gaussian, meaning [ ] fλ i ) E fλ i ) 27) i= converges in law. For symmetric random matrices, Sinai and Soshnikov proved this in 998. We will conclude with a brief chronology of the relevant results. i=. Jonsson, D. Some limit theorems for the eigenvalues of a sample covariance matrix J. Multivariate Anal ). Discusses sample covariance or Wishart matrices, which are of the form A T A, where A is a matrix whose rows are sample data points. 2. Ya. Sinaĭ, A. Soshnikov, Central limit theorem for traces of large random symmetric matrices, Bol. Soc. Brasil. Mat., 29, No., ). 3. Ya. Sinaĭ, A. Soshnikov, A refinement of Wigner s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Functional Anal. Appl. 32, No. 2, 998). These papers prove a refinement and CLT for Wigner matrices, which are symmetric random matrices. 3-4

5 4. Johansson, K. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J ). This paper studies matrices whose entries have joint density proportional to exp nt rv A))), where V is a polynomial. 5. Diaconis, P. and Evans, S.N. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc ). This paper studies random unitary matrices and uses connections to symmetric functions. 6. Chatterjee, S. Fluctuations of eigenvalues and second order Poincaré inequalities. arxiv:75.224v2 [math.pr]. This will be our plan for the next few lectures. 3-5