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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Remark. K is the distribution of the supremum of the absolute value of the Brownian bridge. For all x > 0, P(K x) = 1 + 2 ( 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

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

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