Asymptotic results for empirical measures of weighted sums of independent random variables

Transcription

1 Asymptotic results for empirical measures of weighted sums of independent random variables B. Bercu and W. Bryc University Bordeaux 1, France Seminario di Probabilità e Statistica Matematica Sapienza Università di Roma, 2009 B. Bercu and W. Bryc Empirical measures of weighted sums 1 / 24

2 Outline 1 Motivation 2 B. Bercu and W. Bryc Empirical measures of weighted sums 2 / 24

3 Outline Motivation 1 Motivation 2 B. Bercu and W. Bryc Empirical measures of weighted sums 3 / 24

4 Let (X n ) be a sequence of random variables and consider the symmetric circulant random matrix A n = 1 n X 1 X 2 X 3 X n 1 X n X 2 X 3 X 4 X n X 1 X 3 X 4 X 5 X 1 X 2 X n X 1 X 2 X n 2 X n 1. Goal. Asymptotic behavior of the empirical spectral distribution F n (x) = 1 n n 1 1 {λk x}. k=0 B. Bercu and W. Bryc Empirical measures of weighted sums 4 / 24

5 Trigonometric weighted sums We shall make use of [ ] n 1 r n = 2 and of the trigonometric weighted sums S n,k = 1 n T n,k = 1 n ( 2πkt X t cos n ( 2πkt X t sin n ), ). B. Bercu and W. Bryc Empirical measures of weighted sums 5 / 24

6 Eigenvalues Motivation Lemma (Bose-Mitra) The eigenvalues of A n are given by if n is even and for all 1 k r n λ 0 = 1 n λ n/2 = 1 n λ k = λ n k = X t, ( 1) t 1 X t, S 2 n,k + T 2 n,k. B. Bercu and W. Bryc Empirical measures of weighted sums 6 / 24

7 Eigenvalues Motivation Lemma (Bose-Mitra) The eigenvalues of A n are given by if n is even and for all 1 k r n λ 0 = 1 n λ n/2 = 1 n λ k = λ n k = X t, ( 1) t 1 X t, S 2 n,k + T 2 n,k. B. Bercu and W. Bryc Empirical measures of weighted sums 6 / 24

8 Eigenvalues Motivation Lemma (Bose-Mitra) The eigenvalues of A n are given by if n is even and for all 1 k r n λ 0 = 1 n λ n/2 = 1 n λ k = λ n k = X t, ( 1) t 1 X t, S 2 n,k + T 2 n,k. B. Bercu and W. Bryc Empirical measures of weighted sums 6 / 24

9 Theorem (Bose-Mitra, 2002) Assume that (X n ) is a sequence of iid random variables such that E[X 1 ] = 0, E[X 2 1 ] = 1, E[ X 1 3 ] <. Then, for each x R, lim E[(F n(x) F (x)) 2 ] = 0 n where F is given by F(x) = 1 exp( x 2 ) if x 0, 2 2 exp( x 2 ) if x 0. Remark. F is the symmetric Rayleigh CDF. B. Bercu and W. Bryc Empirical measures of weighted sums 7 / 24

10 Let (X n ) be a sequence of random variables and consider the empirical periodogram defined, for all λ [ π, π[, by I n (λ) = 1 2 e itλ X n t. At the Fourier frequencies λ k = 2πk, we clearly have n I n (λ k ) = S 2 n,k + T 2 n,k. Goal. Asymptotic behavior of the empirical distribution F n (x) = 1 r n r n 1 {In(λk ) x}. k=1 B. Bercu and W. Bryc Empirical measures of weighted sums 8 / 24

11 Theorem (Kokoszka-Mikosch, 2000) Assume that (X n ) is a sequence of iid random variables such that E[X 1 ] = 0 and E[X1 2 ] = 1. Then, for each x R, lim E[(F n(x) F (x)) 2 ] = 0 n where F is the exponential CDF. Remark. (F n ) also converges uniformly in probability to F. B. Bercu and W. Bryc Empirical measures of weighted sums 9 / 24

12 Let (X n ) be a sequence of iid random variables such that E[X n ] = 0 and E(X 2 n ) = 1 and denote S n = 1 n Theorem (Lacey-Phillip, 1990) X t. The sequence (X n ) satisfies an ASCLT which means that for any x R 1 1 lim n log n t 1 {S t x} = Φ(x) a.s. where Φ stands for the standard normal CDF. B. Bercu and W. Bryc Empirical measures of weighted sums 10 / 24

13 Outline Motivation 1 Motivation 2 B. Bercu and W. Bryc Empirical measures of weighted sums 11 / 24

14 Assumptions Motivation Let (U (n) ) be a family of real rectangular r n n matrices with 1 r n n, satisfying for some constants C, δ > 0 (A 1 ) max 1 k r n, 1 t n u(n) k,t (A 2 ) max 1 k, l r n u (n) k,t u(n) l,t C (log(1 + r n )) 1+δ, δ k,l C (log(1 + r n )) 1+δ. Let (V (n) ) be such a family and assume that (U (n), V (n) ) satisfies (A 3 ) max u (n) k,t v (n) l,t C (log(1 + r n )) 1+δ. 1 k, l r n B. Bercu and W. Bryc Empirical measures of weighted sums 12 / 24

15 Trigonometric weights Motivation (A 1 ) to (A 3 ) are fulfilled in many situations. For example, if [ ] n 1 r n 2 and for the trigonometric weights ( u (n) 2 2πkt k,t = n cos n ( v (n) 2 2πkt k,t = n sin n ), ). We shall make use of the sequences of weighted sums S n,k = u (n) k,t X t and T n,k = v (n) k,t X t. B. Bercu and W. Bryc Empirical measures of weighted sums 13 / 24

16 Motivation Theorem (Bercu-Bryc, 2007) Assume that (X n ) is a sequence of independent random variables such that E[X n ] = 0, E[Xn 2 ] = 1, sup E[ X n 3 ] <. If (A 1 ) and (A 2 ) hold, we have lim sup 1 r n 1 n {Sn,k x} Φ(x) = 0 a.s. x R r n k=1 In addition, under (A 3 ), we also have for all (x, y) R 2, lim n 1 r n r n k=1 1 {Sn,k x,t n,k y} = Φ(x)Φ(y) a.s. B. Bercu and W. Bryc Empirical measures of weighted sums 14 / 24

17 Sketch of proof Motivation For all s, t R, let Φ n (s, t) = 1 r n Lemma (Bercu-Bryc, 2007) r n k=1 exp(iss n,k + itt n,k ). Under (A 1 ) to (A 3 ), one can find some constant C(s, t) > 0 such that for n large enough [ E Φ n (s, t) Φ(s, t) 2] where Φ(s, t) = exp( (s 2 + t 2 )/2). C(s, t) (log(1 + r n )) 1+δ B. Bercu and W. Bryc Empirical measures of weighted sums 15 / 24

18 Lemma (Lyons, 1988) Let (Y n,k ) be a sequence of uniformly bounded C-valued random variables and denote Z n = 1 r n r n k=1 Y n,k. Assume that for some constant C > 0, E[ Z n 2 ] Then, (Z n ) converges to zero a.s. C (log(1 + r n )) 1+δ. By the two lemmas, we have for all s, t R, lim Φ n(s, t) = Φ(s, t) n a.s. B. Bercu and W. Bryc Empirical measures of weighted sums 16 / 24

19 Corollary The result of Bose and Mitra on empirical spectral distributions of symmetric circulant random matrices holds with probability one Corollary lim sup F n (x) F(x) = 0 a.s. n x R The result of Kokoszka and Mikosch on empirical distributions of periodograms at Fourier frequencies holds with probability one if sup E[ X n 3 ] < lim sup F n (x) (1 exp( x)) = 0 a.s. n x 0 B. Bercu and W. Bryc Empirical measures of weighted sums 17 / 24

20 Sketch of proof Motivation The empirical measure ν n = 1 r n r n k=1 δ (Sn,k,T n,k ) ν a.s. where ν stands for the product of two independent normal probability measure. Let h be the continuous mapping h(x, y) = 1 2 (x 2 + y 2 ). F n is the CDF of µ n = ν n (h). Consequently, as µ n µ a.s. where µ = ν(h) is the exponential probability measure lim F n(x) = 1 exp( x) n a.s. B. Bercu and W. Bryc Empirical measures of weighted sums 18 / 24

21 In all the sequel, we only deal with trigonometric weights. We shall make use of Sakhanenko s strong approximation lemma. Definition. A sequence (X n ) of independent random variables satisfies Sakhanenko s condition if E[X n ] = 0, E[X 2 n ] = 1 and for some constant a > 0, (S) sup a E[ X n 3 exp(a X n )] 1. n 1 Remark. Sakhanenko s condition is stronger than Cramér s condition as it implies for all t a/3 sup E[exp(tX n )] exp(t 2 ). n 1 B. Bercu and W. Bryc Empirical measures of weighted sums 19 / 24

22 A keystone lemma Motivation Lemma (Sakhanenko, 1984) Assume that (X n ) is a sequence of independent random variables satisfying (S). Then, one can construct a sequence (Y n ) of iid N (0, 1) random variables such that, if S n = X t and T n = then, for some constant c > 0, [ ( )] E exp ac max S t T t 1 t n Y t 1 + na. B. Bercu and W. Bryc Empirical measures of weighted sums 20 / 24

23 Theorem (Bercu-Bryc, 2007) Assume that (X n ) is a sequence of independent random variables satisfying (S). Suppose that (log n) 2 r 3 n =o(n). Then, for all x R, 1 rn r n k=1 (1 {Sn,k x} Φ(x)) In addition, we also have 1 r n rn sup 1 x R {Sn,k x} Φ(x) r n k=1 L N (0, Φ(x)(1 Φ(x))). L n + K where K stands for the Kolmogorov-Smirnov distribution. B. Bercu and W. Bryc Empirical measures of weighted sums 21 / 24

24 Remark. K is the distribution of the supremum of the absolute value of the Brownian bridge. For all x > 0, P(K x) = ( 1) k exp( 2k 2 x 2 ). k=1 Remark. One can observe that the CLT also holds if (S) is replaced by the assumption that for some p > 0 sup E[ X n 2+p ] <, n 1 as soon as r 3 n = o(n p/(2+p) ). B. Bercu and W. Bryc Empirical measures of weighted sums 22 / 24

25 Relative entropy Motivation We are interested in the large deviation principle for the random empirical measure µ n = 1 r n δ Sn,k. r n k=1 The relative entropy with respect to the standard normal law with density φ is given, for all ν M 1 (R), by I(ν) = R log f (x) φ(x) f (x)dx if ν is absolutely continuous with density f and the integral exists and I(ν) = + otherwise. B. Bercu and W. Bryc Empirical measures of weighted sums 23 / 24

26 Theorem (Bercu-Bryc, 2007) Assume that (X n ) is a sequence of independent random variables satisfying (S). Suppose that log n =o(r n ), r 4 n =o(n). Then, (µ n ) satisfies an LDP with speed (r n ) and good rate function I, Upper bound: for any closed set F M 1 (R), lim sup n 1 r n log P(µ n F) inf ν F I(ν). Lower bound: for any open set G M 1 (R), lim inf n 1 r n log P(µ n G) inf ν G I(ν). B. Bercu and W. Bryc Empirical measures of weighted sums 24 / 24

27 Theorem (Bercu-Bryc, 2007) Assume that (X n ) is a sequence of independent random variables satisfying (S). Suppose that log n =o(r n ), r 4 n =o(n). Then, (µ n ) satisfies an LDP with speed (r n ) and good rate function I, Upper bound: for any closed set F M 1 (R), lim sup n 1 r n log P(µ n F) inf ν F I(ν). Lower bound: for any open set G M 1 (R), lim inf n 1 r n log P(µ n G) inf ν G I(ν). B. Bercu and W. Bryc Empirical measures of weighted sums 24 / 24