Central Limit Theorem for Non-stationary Markov Chains

Transcription

1 Central Limit Theorem for Non-stationary Markov Chains Magda Peligrad University of Cincinnati April 2011 (Institute) April / 30

2 Plan of talk Markov processes with Nonhomogeneous transition probabilities -Contraction coe cient. Dobrushin central limit theorem for Markov chains. -Maximum coe cient of correlation. New central limit theorem for Markov chains. -Nonstationary martingale approximation. (Institute) April / 30

3 Dobrushin s contraction coe cient Dobrushin s contraction coe cient as a measure of degeneracy Let Q be a Markov transition probability on (X, B(X)) Dobrushin s contraction coe cient Q(x, A) = P(ξ 1 2 Ajξ 0 = x) Z (Qu)(x) = u(y)q(x, dy). X δ(q) = sup jq(x 1, A) x 1,x 2,A Q(x 2, A)j δ(q) = sup sup j(qu)(x 1 ) (Qu)(x 2 )j, u2u x 1,x 2 where U = fu, Osc(u) = sup x1,x 2 ju(x 1 ) u(x 2 )j 1g. Dobrushin s coe cient of independence α(q) = 1 δ(q) (Institute) April / 30

4 Properties of Dobrushin s coe cients Properties of Dobrushin s coe cients 0 δ(q) 1 δ(q) = 0 if and only if Q(x, A) does not depend on x Q is called nondegenerate if 0 δ(q) < 1 δ(q 1 Q 2 ) δ(q 1 )δ(q 2 ) For a matrix Q = (q i,j ) α(q) = 1 δ(q) = inf min(q i,k, q j,k ) i,j2i k2i (Institute) April / 30

5 Non-homogeneous transitions Markov chain with nonstationary transitions. - (ξ i ) 1in non-homogeneous Markov chain of length n with values in (X, B(X )) P(ξ i+1 2 Ajξ i = x) = Q i,i+1 (x, A) Q i,j = Q i,i+1 Q i+1,i+2...q j 1,j δ 1 = sup δ(q i,i+1 ) and δ k = 1i n 1 sup 1i n k δ(q i,i+k ). δ k δ k 1 (Institute) April / 30

6 Array of non-homogeneous Markov chains -Let (ξ n,i ) 1in be an array of non-homogeneous Markov chains with values in (X, B(X )) and X n,i = f n,i (ξ n,i ) and S n = EX n,i = 0, EX 2 n,i < n X n,i, i=1 and denote by σ 2 n = var S n and b 2 n = δ n,1, α n = 1 δ n,1. n var X n,i, i=1 (Institute) April / 30

7 Dobrushin s CLT Dobrushin s CLT Theorem Under the assumptions max jx n,i j C n a.s. and 1in C 2 n α 3 nb 2 n! 0 as n!, we have n i=1 X n,i σ n D! N(0, 1) as n!. Corollary When jx n,i j C < a.s. and var X n,i c > 0, then the CLT holds provided α 3 nn! as n!. (Institute) April / 30

8 Sharpness of Dobrushin s Theorem Sharpness of Dobrushin s Theorem Theorem There is a Markov chain with jx n,i j C < a.s. and var X n,i c > 0, and lim sup α 3 nn < which does not satisfy the CLT. Example 0 < p < 1 1 p p Q = p 1 p, π(1) = π(2) = 1, α(q) = 2p 2 T n = n 1 f1g (X i ) i=1 - For p = 1/n, T n /n ) G and lim n! n 2 var(t n ) = V 1 < - For p = 1 1/n, T n n/2 ) F and lim n! var(t n ) = V 2 < (Institute) April / 30

9 Sharpness of Dobrushin s Theorem continue Blocking argument: α n! 0, n 1/3 α n 1 and make about nα n blocks of size αn 1, I 1, I 2,..., I nαn. De ne: Q i,i+1 = Q(α n ) for 1 i αn 1 = Q(1/2) for i = jαn 1, 1 j nα n = Q(1 α n ) in rest T n = j 1 f1g (X i ) = i2i j j T n,j vart n,1 αn 2 V 1 var( T n,j ) nα n V 2 j2 If n 1/3 α n! then CLT. If n 1/3 α n = 1 the limiting distribution is a convolution X N, where X is non-normal nondegenerate. (Institute) April / 30

10 Maximal coe cient of correlation Maximal coe cient of correlation A, B be two sub σ-algebras ρ(a, B) = sup jcorr (f, g)j, f 2L 2 (A),g 2L 2 (B) For a vector of random variables (Y k ) 1kn we de ne ρ k = max ρ(σ(y i, i s), σ(y j, j s + k)). 1s,s+kn For a nonhomogeneous Markov chain of length n, (ξ i ) 1in, ρ k = max ρ(σ(ξ s ), σ(ξ s+k )). 1s,s+kn if the joint distribution of (ξ s, ξ s+k ) is bivariate normal ρ k = max jcorr(ξ s, ξ s+k )j. 1s,s+kn (Institute) April / 30

11 Relation to Dobrushin s coe cient Relation to Dobrushin s coe cient (Bradley 2011) For any 0 < a < b < 1 there is a Markov chain with ρ 1 = a and δ 1 = b. Therefore, for a triangular array, λ n = 1 ρ n,1 and α n = 1 δ n,1 can converge to 0 at di erent rates. Sethuraman and Varadhan (2005) ρ 1 δ 1/2 1. For a Markov chain ρ k ρ k 1. (Institute) April / 30

12 Relation to operator norm Relation with operator norm in L 2 ρ 1 (X, Y ) = supf jje(g(y )jx )jj 2 ; jjg(y )jj 2 < and Eg(Y ) = 0g, g jjg(y )jj 2 For stationary Markov chains with values in (X, B(X )) and transition probability Q(x, A) = P(ξ 1 2 Ajξ 0 = x), de ne the operator Q Z (Qu)(x) = X u(y)q(x, dy), u 2 L 2 (X, B(X ), π) Denote L 0 2 (π) = fg 2 L 2(X, B(X ), π) with R gdπ = 0g. The coe cient ρ 1 is simply the norm operator of Q : L 0 2 (π)! L0 2 (π), ρ 1 = jjqjj L 0 2 (π) = sup jjq(g)jj 2. g 2L 0 2 (π) jjgjj 2 (Institute) April / 30

13 Some history Some history on the maximal coe cient of correlation Conditions imposed to the maximal coe cient of correlation make possible to study the asymptotic behavior of many dependent structures including classes of Markov chains and Gaussian sequences. This coe cient was used by Kolmogorov and Rozanov and further studied by Rosenblatt, Ibragimov, Shao among many others. An introduction to this topic, mostly in the stationary setting, can be found in the Chapters 9 and 11 in Bradley. Application to the central limit theorem (CLT) for various stationary Markov chains with ρ 1 < 1 are surveyed in Jones. In the nonstationary setting and general triangular arrays a central limit theorem was obtained by Utev, assuming ρ mixing coe cients converging to 0 uniformly at a logarithmic rate. (Institute) April / 30

14 CLT under conditions on the maximal coe cient of correlation Theorem (P-2010) Suppose that (X n,i ) 1in are Markov chains and max jx n,i j C n a.s. and C n(1 + j log(λ n )j)! 1in λ 3/2 0 as n!. n b n Then n i=1 X n,i σ n D! N(0, 1) as n!. Corollary Assume that max 1in jx n,i j C a.s. and also var X n,i c > 0 for all n 1 and 1 i n. Then CLT holds provided λ 3 nn(1 + j log(λ n )j) 2!. (Institute) April / 30

15 Integral form Integral form 1 λ 2 nb 2 n n EXn,i 2 I (jx n,i j > εh(λ n )b n )! 0 as n! i=1 where h(λ n ) = λ 3/2 n (1 + j log(λ n )j) 1. Uniformly bounded ρ n,1 mixing coe cients. There is a positive number ρ such that sup n ρ n,1 < ρ < 1. Then, the CLT holds provided that for every ε > 0 1 b 2 n n EXn,i 2 I (jx n,i j > εb n )! 0. i=1 (Institute) April / 30

16 Tools: Martingale related central limit theorem Tools: Martingale related central limit theorem Assume (X n,j ) 1jn is an array of centered random variables that are square integrable and adapted to an increasing array of sigma elds (F n,j ) 1jn. Theorem Assume ES 2 n = 1. De ne the projector operator P n,j Y = E(Y jf n,j ) E(Y jf n,j 1 ). Assume and Then the CLT holds. max jp n,js n j! P 0 as n!. 1jn n j=1 (P n,j S n ) 2! P 1 as n!. (Institute) April / 30

17 Tools: CLT for general triangular arrays Tools: CLT for general triangular arrays For 0 j n denote where S n,j = j i=1 X n,i Theorem Assume ES 2 n = 1. Also assume that and A n,j = E(S n S n,j jf n,j ), max (jx n,j j + ja n,j j)! P 0 as n! 1jn n (Xn,j 2 + 2X n,j A n,j )! P 1 as n!. j=1 Then S n converges in distribution to N(0, 1). (Institute) April / 30

18 Bounds for the variance of partial sums of Markov chains Bounds for the variance of partial sums of Markov chains Sharp upper and lower bounds for the variance of partial sums of a Markov chain in function of the maximal coe cient of correlation 1 ρ ρ 1 n EXi 2 ESn ρ 1 i=1 1 ρ 1 and in function of Dobrushin s coe cient 1 δ 1 (1 + p δ 1 ) 2 n EXi 2. i=1 n EXi 2 ESn 2 (1 + p δ 1 ) 2 i=1 1 δ 1 n EXi 2. i=1 (Institute) April / 30

19 Quenched Invariance principle for Stationary processes and Markov chains Markov processes with homogeneous transitions that do not start from equilibrium. -Quenched Central Limit Theorem for stochastic processes -Quenched Central Limit theorem for Reversible Markov processes -Stationary Martingale Approximation (Institute) April / 30

20 Stationary Sequences Stationary sequence as a functional of a Markov chain with a general state space. - (ξ n ) n2z is a stationary ergodic Markov chain de ned on a probability space (Ω, F, P) with values in a measurable space. - The marginal distribution is denoted by π(a) = P(ξ 0 2 A). -S n = S n (g) = n k=1 g(ξ k ) - Any stationary sequence (Y k ) k2z can be viewed as a function of a Markov process ξ k = (Y j ; j k) with the function g(ξ k ) = Y k. (Institute) April / 30

21 Central limit theorem started at a point Central limit theorem started at a point For any continuous and bounded f E x (f (S n / p n))! E (f (cn)), a.s. where N is standard normal. Equivalently, E(f (S n / p n)jf 0 )! E (f (cn)), a.s. For t 2 [0, 1] de ne W n (t) = S [nt] / p n, where [x] denotes the integer part of x. By quenched functional CLT we mean that for any function f continuous and bounded on D(0, 1) E(f (W n )jf 0 )! E (cf (W )) a.s. where W is the standard Brownian motion on [0, 1]. (Institute) April / 30

22 Almost sure martingale approximations Almost sure martingale approximations Martingales with stationary and ergodic di erences have both almost sure CLT and invariance principle started at a point. Type of approximations needed to transport the Quenched CLT from martingale to the stationary sequence. E((S n M n ) 2 jf 0 )/n! 0 a.s. E( max 1in (S i M i ) 2 jf 0 )/n! 0 a.s. (S n M n ) 2 /n! 0 a.s. (Institute) April / 30

23 Quenched functional CLT under projective criteria Quenched functional CLT under projective criteria Here is a short history of the quenched CLT under projective criteria. - A result in Borodin and Ibragimov (1994) states that if jje(s n jf 0 )jj 2 is bounded, then the CLT in its functional form started at a point holds. - Later work by Derriennic and Lin (2001) improved on this result imposing the condition jje(s n jf 0 )jj 2 = O(n 1/2 ɛ ) with ɛ > 0. - Improvements by Wu and Woodroofe (2004); Zhao and Woodroofe (2008). - A step forward was made by Cuny (2009) who improved the condition to jje(s n jf 0 )jj 2 = O(n 1/2 (log n) 2 (log log n) 1+δ ) by using sharp results on ergodic transforms in Gaposhkin (1996). -Peligrad and Cuny (2011) under the condition jje(x k jf 0 )jj p 2 k1 k jjq = k f jj p 2 <. k1 k (Institute) April / 30

24 Additive functionals of reversible Markov chains Additive functionals of reversible Markov chains. Theorem Let (ξ i ) i2z be a stationary and ergodic reversible Markov chain, f 2 L 2 0 (π) with the property var(s n ) lim! σ 2 jje(s n jf 0 )jj n! f < (equivalently n 2 2 n n 2 < ). Then E max 1jn (S j M j ) 2 /n! 0 and the functional CLT holds. Kipnis and Varadhan (1986), Gordin Peligrad (2011). -Q = Q where Q is the adjoint operator de ned by < Qf, g >=< f, Q g >, for every f and g in L 2 (π). (Institute) April / 30

25 Quenched Invariance principle for Reversible Markov chains Quenched Invariance principle for Reversible Markov chains Conjecture: Kipnis and Varadhan. For any square integrable function of stationary, ergodic, reversible Markov chains under centering and normalization p n, the functional central limit theorem started from a point holds under the condition lim n! var(s n )/n! σ 2 f < jje(s (equivalently n jf 0 )jj 2 2 n < ). The same question for continuous time n 2 reversible Markov chains. (Institute) April / 30

26 Quenched CLT for Reversible Markov chains Quenched CLT for Reversible Markov chains Cuny Peligrad (2011). Theorem For a stationary reversible Markov chain satisfying n (log log n) 2 jje(s n jf 0 )jj 2 2 n 2 <. the central limit theorem started at a point holds. (Institute) April / 30

27 Method of proof Method of proof: Construction of approximating martingale - E(S n jf 1 ) E(S n jf 0 )! D 0 in L 2 and a.s. -Then one estimates S n n k=1 D k in L 2 - almost sure-type martingale approximation based on maximal inequalities -spectral calculus (Institute) April / 30

28 Fourier series Application of martingale approximation to Fourier Series S n (θ) = n X j exp(jiθ) where i 2 = 1, j=1 periodogram I n (θ) = 2πn 1 js n (θ)j 2 Wiener and Wintner (1941), Lacey and Terwilleger (2008): For all stationary sequences (X j ) j2z in L 1 there is a set Ω 0 of probability 1 such that S n (θ) converges for all θ and ϖ 2 Ω 0 n (Institute) April / 30

29 Almost sure central limit theorem for Fourier series Almost sure central limit theorem for Fourier series Theorem (X k ) k2z is a stationary ergodic sequence and E(X 0 jf ) = 0 almost surely. Then for λ almost all θ 2 (0, 2π) lim EjS n(θ)j 2 /n = σ 2 (θ) n! and 1 p n (Re(S n (θ)), Im(S n (θ)))! d (N 1 (θ), N 2 (θ)) where N 1 (θ) and N 2 (θ) are independent identically distributed normal random variables mean 0 and variance 1 2 σ2 (θ). MP-Wei Biao Wu (2009) (*) Regularity can be replaced by jje 0 (S n (θ))jj 2 = o( p n). (Institute) April / 30

30 Method of proof of central limit theorem for Fourier series Method of proof of central limit theorem for Fourier series Carleson and Hunt theorems imply for almost all θ in [0, 2π] E 1 (S n (θ)) E 0 (S n (θ))! D 0 in L 2 Martingale approximation. Spectral density theory. (Institute) April / 30