A COARSENING OF THE STRONG MIXING CONDITION

Transcription

1 Communications on Stochastic Analysis Vol. 8, No. 3 (2014) Serials Publications A COARSNING OF TH STRONG MIXING CONDITION BRNDAN K. BAR Abstract. We consider a generalization of the α-mixing condition of Rosenblatt, which we term γ-mixing. Whereas α-mixing is defined in terms of entire σ-fields of sets generated by random variables in the distant past and future, γ-mixing is defined in terms of a more coarse collection of sets. We provide a Rosenthal inequality and central limit theorem for γ-mixing processes. 1. Introduction Let {X t : t Z} be a collection of random variables defined on some probability space (Ω,F,P). Mixing conditions provide one way to formalize the notion that these random variables are only weakly dependent on one another. There are many ways to define mixing; the monographs by Doukhan [8] and Bradley [5] list five classical definitions. The oldest and most general of these is the α-mixing condition of Rosenblatt [13, 4], also known as strong mixing. For any nonempty set of integers T, let F T F denote the σ-field generated by the random variables {X t : t T}. The α-mixing coefficients {α r : r N} associated with {X t } are given by α r = sup sup P (A B) P (A)P (B), (1.1) S,T A F S,B F T where the first supremum is taken over all nonempty finite sets of integers S,T such that mint maxs r. If α r 0 as r, then {X t } is said to be α-mixing. In this paper we investigate a generalization of α-mixing obtained by coarsening the families F S and F T appearing in (1.1). For any nonempty set of integers T, let H T F denote the class of sets of the form t T {X t x t }, where each x t ranges over R. We define a sequence of γ-mixing coefficients {γ r : r N} by γ r = sup S,T sup P (A B) P (A)P (B), (1.2) A H S,B H T where, once again, the first supremum is taken over all nonempty finite sets of integers S,T such that mint maxs r. If γ r 0 as r, we say that {X t } is γ-mixing. Several other authors [12, 11, 7, 6] have investigated a coarsening of F S and F T in (1.1). The discussion in Dedecker and Prieur [7] is especially relevant. Those authors consider, among other dependence coefficients, a generalized α-mixing coefficient α r proposed originally by [12]. This mixing coefficient is introduced in Received ; Communicated by the editors Mathematics Subject Classification. Primary 37A25; Secondary 60F05. Key words and phrases. Mixing, weak dependence, Hardy-Krause variation. 317

2 318 BRNDAN K. BAR Definition 2 of [7] using the notation α(r). After dividing by a constant factor of two, we may write α r as α r = sup S,T sup P (A B) P (A)P (B), (1.3) A F S,B H T wherethis time the firstsupremumis takenoverallnonempty finite sets ofintegers S,T such that mint maxs r, and such that T is a singleton. Compared to (1.2), the set A in (1.3) is drawn from a larger collection of sets, while the set B is drawn from a smaller collection of sets. Clearly, α r α r. We will shortly give an example of a process that is γ-mixing but not α-mixing, demonstrating that the γ-mixing property is more general than α-mixing, and distinct from α-mixing. The main results of our paper are a Rosenthal inequality and central limit theorem for γ-mixing processes. The key to establishing them is a covariance inequality given in [3], which allows us to bound the covariance between two functions of a γ-mixing process by a quantity depending on the Hardy-Krause total variation norms of those functions. Our Rosenthal inequality represents a strict improvement over existing results for α-mixing processes: there is no cost to the coarsening of F S and F T that we adopt. The same cannot be said of our central limit theorem, which requires a much faster mixing rate than comparable results under α-mixing. The paper is structured as follows. In section 2, an example of a process that is γ-mixing but not α-mixing is given. Covariance inequalities applicable to γ-mixing processes are discussed in section 3. Our Rosenthal inequality is proved in section 4, and our central limit theorem in section A Process That Is γ-mixing But Not α-mixing Let {ε t : t Z} be an iid sequence of random variables that each take the value 0 with probability 1/2 and the value 1/2 with probability 1/2. For t Z, define X t as the limit in mean square of the series k=0 2 k ε t k. One may show that the marginaldistributionofeachx t isuniformon[0,1]bywritingx t = (1/2)X t 1 +ε t and using a simple argument with characteristic functions. In [1] it is shown explicitly that {X t } is not α-mixing by the construction of a set A σ(x 0 ) and a sequence of sets {B r }, B r σ(x r ), such that P (A B r ) P (A)P (B r ) 1/4 (2.1) for all r N. Let W r = {w r,1,...,w r,mr } denote the support of the random variable X r 2 r X 0, and note that m r 2 r. Let A = {X 0 1/2}, and let { } m r B r = X r [w r,k,w r,k +2 r 1 ]. k=1 Now, since X 0 U(0,1), we have P (A) = 1/2. And since X r = 2 r X 0 +w r,k for some k = 1,...,m r, we have A B r. Consequently, P (A B r ) P (A)P (B r ) = 1 2 (1 P (B r)). But since X r U(0,1), we have P (B r ) m r 2 r 1 1/2. Thus (2.1) holds, and {X t } cannot be α-mixing.

3 A COARSNING OF TH STRONG MIXING CONDITION 319 Though {X t } is not α-mixing, it is α-mixing [7], with a geometric decay rate of α r. We can show that {X t } is also γ-mixing, with a geometric decay rate of γ r. Theorem 2.1. {X t } is γ-mixing, with γ r 2 1 r. Proof. Fix twofinite sets ofintegerss and T with mint maxs r. Forx R S and y R T, let A x = s S {X s x s } and B y = t T {X t y t }. Observe that P (A x B y ) P (A x )P (B y ) = (P (B y F S ) P (B y ))dp A x 1 2 P (B y F S ) P (B y ). Let s denote the maximum element of S. Using the triangle inequality and the independence of t T {X t 2 s t X s y t } and F S, we have P (B y F S ) P (B y ) P ( t T {X t 2 s t X s y t } F S ) P (By F S ) + P ( t T {X t 2 s t X s y t } ) P (B y ). Since X s is nonnegative, we know that B y t T {X t 2 s t X s y t }, and so P (B y F S ) P (B y ) P (( t T {X t 2 s t X s y t } ) ( t T {X t > y t }) F S ) from which it follows that +P (( t T {X t 2 s t X s y t } ) ( t T {X t > y t }) ), P (B y F S ) P (B y ) 2P (( t T {X t 2 s t X s y t } ) ( t T {X t > y t }) ). The fact that X s 1 now gives P (B y F S ) P (B y ) 2P (( t T {X t 2 s t y t } ) ( t T {X t > y t }) ) 2P ( t T {y t < X t y t +2 s t } ) 2 t T P(y t < X t y t +2 s t ). The marginal distribution of each X t is uniform on [0,1], and so P (B y F S ) P (B y ) 2 2 s t 2 2 s t = 2 2 r. t T t= s+r It follows that γ r 2 1 r for all r. Theorem 2.1 demonstrates that {X t } is γ-mixing. But {X t } is also α-mixing, so we have yet to provide an example of a process that is γ-mixing but not αmixing. In fact, this is now quite easy to achieve: we need merely consider the time reversed process {Xt}, where Xt = X t for each t Z. The time reversed process satisfies the dynamic equation Xt = 2Xt 1 mod(1) a.s., and has been studied as an example of deterministic chaotic dynamics [2, 9, 14]. Theorem 2.2. {X t} is γ-mixing but not α-mixing, with γ r 2 1 r and α r 1/4.

4 320 BRNDAN K. BAR Proof. γ r 2 1 r follows from Theorem 2.1 and the invariance of γ r under time reversal. α r 1/4 follows by precisely the same argument used in [1] to show that {X t }isnotα-mixing, repeatedinthesecondparagraphofthissection. Specifically, B r σ(x r ) and A = {X 0 1/2}, so from (1.3) we obtain α r P(B r A) P(B r )P(A) 1/4. 3. Covariance Inequalities The following covariance inequality for a random process {X t } is well known [8, 5]: for any r N, any nonempty finite sets of integers S and T such that mint maxs r 1, and any Borel measurable functions f : R S R and g : R T R, we have Cov(f (X s : s S),g(X t : t T)) 4 f g α r. (3.1) An inequality similar to (3.1) that involves γ-mixing coefficients rather than α- mixing coefficients has been proved in [3]. Before stating this inequality, we review the definitions of Vitali and Hardy-Krause variation for multivariate functions. For a more extensive discussion of these concepts, refer to [3, 10]. Let f be a real valued function defined on an n-dimensional rectangle [a,b] = {x R n : a x b}, and let R = [c,d] [a,b] be a smaller n-dimensional rectangle. Let R f = ( 1) I f (x I ), I {1,...,n} where x I is the vector in R n whose ith element is given by c i if i I, or by d i if i / I. For instance, if n = 2 then we have R f = f (d 1,d 2 ) f (c 1,d 2 ) f (d 1,c 2 )+f (c 1,c 2 ). The Vitali variation of f is given by f V = sup R A R f, with the supremum taken over all finite collections of n-dimensional rectangles A = {R 1,...,R m } such that m i=1 R i = [a,b], and the interiors of any two rectangles in A are disjoint. GivenanonemptysetI {1,...,n}, andafunctionf : [a,b] R, letf I denote the real valued function on i I [a i,b i ] obtained by setting the ith argument of f equal to b i whenever i / I. The Hardy-Krause variation of f is given by f HK = f I V. I {1,...,n} Vitali variationandhardy-krausevariationareequalwhen n = 1, butwhen n 2 Hardy-Krause variation may be greater than Vitali variation. Our covariance inequality for γ-mixing processes is as follows. Theorem 3.1. Suppose each X t takes values in a bounded interval [a t,b t ]. Let r N, and let S and T denote nonempty finite sets of integers with mint maxs r.

5 A COARSNING OF TH STRONG MIXING CONDITION 321 Then for any functions f : s S [a s,b s ] R and g : t T [a t,b t ] R that are left-continuous and of bounded Hardy-Krause variation, we have Cov(f (X s : s S),g(X t : t T)) f HK g HK γ r. Proof. Immediate from Theorem 4.2 in [3], and the definition of γ r. Theorem 3.1 is applicable to bounded random variables. Given a particular choice of f and g, it may be possible to extend Theorem 3.1 so that it is applicable to unbounded random variables. As an example, let us choose f and g to be product functions. Theorem 3.2. Fix r N, and let S and T be nonempty finite sets of integers with mint maxs r. Let A 1 = (3 S 1)(3 T 1) and A 2 = 2 S +2 T. Then for p,q [1, ] satisfying sup t S T X t p < and ( S + T )p 1 +q 1 = 1, we have ( ) ( ) Cov s s SX, X t A X t p γr 1/q, t T t S T where A = A 1 if q = 1, or A = A 1/q 1 A (q 1)/q 2 q(q 1) (1 q)/q if q > 1. Proof. If γ r = 0 then F S and F T must be independent, in which case the theorem is trivial. Assume γ r > 0. Let Xt = min{max{x t, a t },a t }, where a t = X t p c q/p γr 1/p for some constant c > 0. Begin by writing ( ) ( Cov s s SX, X t Cov X s, ) X t t T s S t T ( + Cov s s SX X s, ) X t s S t T ( + Cov s s SX, X t ) X t. (3.2) t T t T Using standard arguments with the inequalities of Hölder and Markov, we can bound the last two terms on the right-hand side of (3.2) as follows: ( Cov s s SX X s, ) ( ) X t 2 S X t p cγr 1/q, (3.3) s S t T t S T ( Cov s s SX, X t ) ( ) X t 2 T X t p cγr 1/q. (3.4) t T t T t S T We will use Theorem 3.1 to bound the first term on the right-hand side of (3.2). Clearly { } Xt is γ-mixing, with mixing coefficients bounded by those of {Xt }. Let the functions f : s S [ a s,a s ] R and g : t T [ a t,a t ] R be given by f (x s : s S) = s S x s and g(x t : t T) = t T x t. For nonempty I S we have f I (x s : s I) = ( s I x s ) ( s S\I a s ). The Vitali variation of f I is given

6 322 BRNDAN K. BAR by the L 1 norm of the mixed partial derivative obtained by differentiating f I once with respect to each argument: f I V = a s ( ) dx s = 2 I a s. s I [ as,as] s I s S s S\I Thus, using the binomial theorem, the Hardy-Krause variation of f is given by ( ) ( ) S f HK = a s S! ( ) ( S s)!s! 2s = a s 3 S 1, s S s=1 and similarly g HK = ( t T a t)( 3 T 1 ). It now follows from Theorem 3.1 that ( Cov X s, ) ( ) ( ) X t A 1 a t γ r = A 1 X t p c 1 q γr 1/q. s S t T t S T t S T (3.5) Combining (3.2) through (3.5), we obtain ( ) Cov X t ( A 1 c 1 q +A 2 c )( ) X t p γr 1/q. s SX s, t T Minimizing A 1 c 1 q +A 2 c over c yields the constant A. s S t S T Note that if we choose S and T to be singletons containing t and t+r respectively, and set q = 1, then Theorem 3.2 states that Cov(X t,x t+r ) 4 X t X t+r γ r. (3.6) If instead q > 1, then the constant term A = 4q(q 1) (1 q)/q achieves a maximum value of 8 at q = 2, and so we have Cov(X t,x t+r ) 8 X t p X t+r p γ 1/q r. (3.7) Inequalities(3.6) and(3.7) resemble the classic covariance inequalities for α-mixing processes [5, Theorems 1.11 and 3.7], achieving the familiar constant terms of 4 and 8 in the bounded and unbounded cases respectively. Since our inequalities involve γ-mixing coefficients rather than α-mixing, they constitute a refinement of the classic inequalities. 4. Rosenthal Inequality Givenconstantsp 0andε > 0, andasequenceofrandomvariablesx = {X t }, define W n (p,ε,x) and D n (p,ε,x) as follows: n W n (p,ε,x) = X t p p+ε D n (p,ε,x) = W n (p,0,x) for p 1 = W n (p,ε,x) for 1 < p 2 { = max W n (p,ε,x),(w n (2,ε,X)) p/2} for p 2.

7 A COARSNING OF TH STRONG MIXING CONDITION 323 The random variables{x t } are said to satisfy a Rosenthal inequality if there exists a constant b < such that n 1 X t p bd n (p,ε,x) for all n. A Rosenthal inequality for α-mixing processes is given in [8]. When p 1, the Rosenthal inequality is a trivial consequence of the inequality (a+b) p a p +b p, which holds for any positive a,b. When p > 1, the Rosenthal inequality for α-mixing processes is proved in two steps. First, using a covariance inequality for α-mixing processes, the Rosenthal inequality is proved for any even integer p. Second, the so-called interpolation lemma [15, 8] is used to extend the inequality to all real p > 1. To prove a Rosenthal inequality for γ-mixing processes, we modify the arguments used in the α-mixing case in the following way. First, in place of the covariance inequality for α-mixing processes, we employ Corollary 3.1 from above, which applies to γ-mixing processes. Second, we modify the interpolation lemma so that it is applicable under γ-mixing. The following lemma provides this modification. We will say that one sequence of numbers {γ r } dominates another sequence {γ r} if γ r γ r for all r. Lemma 4.1. Fix k 0, ε > 0, and a sequence of nonnegative real numbers {γ r }. Suppose there exists a constant b < such that any centered sequence of random variables X = {X t } whose γ-mixing coefficients are dominated by {γ r } satisfies n k X t bv n (k,ε,x) for all n, where V n (k,ε,x) = W n (k,ε,x) for k 2 { = max W n (k,ε,x),(w n (2,ε,X)) k/2} for k 2. Then for any p [0,k] there exists a constant b < such that any centered sequence of random variables X = {X t } whose γ-mixing coefficients are dominated by {γ r } satisfies n p X t b V n (p,ε,x) for all n. Proof. The lemma is trivial for p 1, so we assume k,p 1. Suppose X = {X t } is a centered sequence of r.v.s whose γ-mixing coefficients are dominated by {γ r }. Set a = V n (p,ε,x) 1/p, X t = min{max{x t, a},a}, X t = X t X t, Y t = X t X t, Z t = X t X t.

8 324 BRNDAN K. BAR Jensen s inequality allows us to bound n 1 X t p by 2 ( p 1 n p n p) Y t + Z t 2 ( p 1 n p Y t +2 ( p 1 n p n p)) Z t 1 {Zt 0} + Z t 1 {Zt<0} 2 p 1 n k p/k ( n ) k Y t +2 2p 2 Z t p/k 1 {Zt 0} ( n k +2 2p 2 Z t p/k 1 {Zt<0}). Define the random variables ( ) ξ t = Z t p/k 1 {Zt 0} Z t p/k 1 {Zt 0} ( ) ζ t = Z t p/k 1 {Zt<0} + Z t p/k 1 {Zt<0}, and observe that ( n k ( n Z t p/k 1 {Zt 0}) = ξ t + and ( n k ( n Z t p/k 1 {Zt<0}) = ζ t + We thus have n p X t n ( ) ) k Z t p/k 1 {Zt 0} 2 k 1 n k ( n ) k ξ t + Z t p/k n ( ) ) k Z t p/k 1 {Zt<0} 2 k 1 n k ( n ) k ζ t + Z t p/k. 2 p 1 n k Y t p/k +2 2p+k 3 n k ξ t +2 2p+k 3 n k ( n ) k ζ t +2 2p+k 2 Z t p/k. Y t, ξ t and ζ t are all nondecreasing transformations of X t, and therefore all have γ-mixing coefficients that are dominated by {γ r }. Thus, under the hypothesis of

9 A COARSNING OF TH STRONG MIXING CONDITION 325 the lemma, there exists b 1 < such that n p X t 2 p 1 (b 1 V n (k,ε,y)) p/k +2 2p+k 3 b 1 V n (k,ε,ξ) ( n ) k +2 2p+k 3 b 1 V n (k,ε,ζ)+2 2p+k 2 Z t p/k. In [15, 8] it is shown that V n (k,ε,y) 2 k V n (p,ε,x) 4 k/p, that V n (k,ε,ξ) 2 k+p V n (p,ε,x), that V n (k,ε,ζ) 2 k+p V n (p,ε,x), and, for p k ε, that ( n Z t p/k ) k 2 p V n (p,ε,x). We thus obtain n X t p b 2 V n (p,ε,x) for some b 2 0 not depending on n or X. This completes the proof for the case where p k ε. But if the theorem is true for p k ε, then it must also be true for p k 2ε, and so on for all p [0,k]. With Lemma 4.1 in hand, we may state our Rosenthal inequality for γ-mixing processes. Theorem 4.2. Fix p 0 and ε > 0, and let k denote the smallest even integer equal to or greater than p. Let {X t } satisfy X t = 0 and X t p+ε < for each t, and have γ-mixing coefficients satisfying r=1 (r+1)k 2 γr ε/(k+ε) <. Then there exists a constant b < not depending on ε such that, for all n, n p X t bd n (p,ε,x). Proof. The proof of this theorem differs from the proof under α-mixing see e.g. [8, Section 1.4.1] in only two respects. First, Theorem 3.2 is used in place of the covariance inequality for α-mixing processes. Second, Lemma 4.1 is used in place of the interpolation lemma [15, 8] for α-mixing processes. Note that the only difference between Theorem 4.1 and the Rosenthal inequality for α-mixing processes stated in [8] is that we have replaced α-mixing coefficients with γ-mixing coefficients. Theorem 4.1 thus represents a strict refinement of that result. 5. Central Limit Theorem In this section we prove a central limit theorem for stationary γ-mixing processes. Theorem 5.1. Suppose {X t } is stationary, and satisfies X 0 = 0, X 0 2+ε < for some ε > 0, and γ r = O ( exp ( r δ)) for some δ > (4+ε)/(4+2ε) and all r N. Then r=1 X 0X r <, and if σ 2 := X r=1 X 0X r > 0, then n 1/2 n X t d N(0,σ 2 ) as n. Proof. Absolute convergence of r=1 X 0X r follows from Theorem 3.2. Suppose σ > 0. We will show that n 1/2 n X t d N(0,σ 2 ) using a lemma of Withers

10 326 BRNDAN K. BAR [16, Lemma 3.1]. Split {X t } into k Bernstein blocks of length n 1, separated by gaps of length n 2, as follows: n k k+1 [ n X t = η in + ν in, k = η in = ν in = ν k+1,n = i=1 i=1 in 1+(i 1)n 2 t=(i 1)(n 1+n 2)+1 i(n 1+n 2) t=in 1+(i 1)n 2+1 n t=k(n 1+n 2)+1 X t. X t, X t, n 1+n 2 ] i = 1,...,k i = 1,...,k The sequences n 1 (n) and n 2 (n) are chosen to satisfy n 1 n β and n 2 n α, where 0 < α < β < 1. Withers lemma states that n 1/2 n X t d N ( 0,σ 2) if the following four conditions are satisfied for φ, ψ being either sine or cosine functions: ( ( k Cov φ j=2 ωn 1/2 j 1 i=1 1 n 1 n ( k+1 ) 2 ν in 0 (5.1) i=1 k ηin 2 1( ηin 2 > nǫ) 0 for all ǫ > 0 (5.2) i=1 k 1 1 k Cov(η in,η jn ) 0 (5.3) n i=1 j=i+1 ) ( ) ) η in,ψ ωn 1/2 η jn 0 for all ω > 0.(5.4) (We have simplified Withers conditions by noting that ( n X t) 2 nσ 2 ; see e.g. [5, Prop. 8.3(IV)]). To verify (5.1), we note that Theorem 4.1 implies that ( k+1 i=1 ν in) 2 = O(kn 2 ) = o(n). To verify (5.2), we use the inequalities of Hölder and Markov to obtain η 2 in 1( η 2 in > nǫ) η in 2 2+ε P ( η 2 in > nǫ) ε/(2+ε) (nǫ) ε/2 η in 2+ε 2+ε. Theorem 4.1 implies that η in 2+ε = O(n 1/2 1 ), and so the left-hand side of (5.2) is O(k(n 1 /n) 1+ε/2 ) = O((n 1 /n) ε/2 ) = o(1). To verify (5.3), we use (3.7) to obtain Cov(η in,η jn ) 8n 2 1 X ε γε/(2+ε) (j i)n 2 for 1 i < j k. It follows that the left-hand side of (5.3) is O(nγn ε/(2+ε) 2 ) = o(1). It remains to verify (5.4). Let n 3 = n 3 (n) be an increasing sequence satisfying n 3 n κ for some κ > 0, and let X tn = min{n 3,max{X t, n 3 }}. For j = 2,...,k, define S j = j 1 i=1 {(i 1)(n 1 +n 2 )+1,...,in 1 +(i 1)n 2 } T j = {(j 1)(n 1 +n 2 )+1,...,jn 1 +(j 1)n 2 }.

11 A COARSNING OF TH STRONG MIXING CONDITION 327 Using Markov s inequality and the boundedness of φ and ψ, we may show that ( ( ) Cov j 1 ( ) ) φ ωn 1/2 η in,ψ ωn 1/2 η jn i=1 Cov φ ωn 1/2 sn,ψ ωn 1/2 tn s SjX t TjX +4jn 1 n 2 ε 3 X 0 2+ε 2+ε. (5.5) We will use Theorem 3.1 to bound the first term on the right-hand side of (5.5). Let f : [ n 3,n 3 ] Sj R and g : [ n 3,n 3 ] Tj R be given by f (x s : s S j ) = φ ωn 1/2 s s Sjx g(x t : t T j ) = ψ ωn 1/2 t. t Tjx Clearly, for nonempty I S j, we have ( ( )) f I (x s : s I) = φ ωn 1/2 x s +n 3 S j \I. s I The function obtained by differentiating f I once with respect to each argument is bounded in absolute value by ( ωn 1/2) I. Thus, fi V (2ωn 3 n 1/2 ) I. Using the binomial theorem, we now have f HK (2ωn 3 n 1/2) I I S j = = S j s=1 S j! (2ωn 3 n 1/2) s ( S j s)!s! ( 1+2ωn 3 n 1/2) S j 1. We can show similarly that g HK (1+2ωn 3 n 1/2 ) Tj 1. Thus, since the γ- mixing coefficients of {X tn } are dominated by those of {X t }, Theorem 3.1 implies that the first term on the right-hand side of (5.5) is bounded by f HK g HK γ n2 ( 1+2ωn 3 n 1/2) S j + T j γn2 = ( 1+2ωn 3 n 1/2) jn 1 γn2. It follows that the quantity on the left-hand side of (5.4) is bounded by ( k 1+2ωn 3 n 1/2) kn 1 γn2 +4k 2 n 1 n 2 ε 3 X 0 2+ε 2+ε. Recall that n 1 n β, n 2 n α, n 3 n κ and k n 1 β for parameters α,β,κ satisfying 0 < α < β < 1 and κ > 0, and recall that X 0 2+ε < and γ r =

12 328 BRNDAN K. BAR O(exp( r δ )) as r. We therefore have ( k 1+2ωn 3 n 1/2) kn 1 ( ( γn2 = O n 1 β 1+2ωn κ 1/2) n ) exp( n αδ ) (5.6) 4k 2 n 1 n 2 ε 3 X 0 2+ε 2+ε = O ( n 2 β 2κ εκ). (5.7) If we choose κ < 1/2, then ( 1+2ωn κ 1/2) n 1/2 κ exp(2ω), and the expression in (5.6) is O(n 1 β exp(n κ+1/2 n αδ )). We may ensure that it vanishes by choosing α,κ to satisfy κ < αδ 1/2. If, in addition, κ > (2 β)/(2 + ε), then the expression in (5.7) also vanishes, and (5.4) is satisfied. We can find κ to satisfy these conditions whenever α,β are such that (2 β)/(2+ε)< 1/2 and (2 β)/(2+ ε) < αδ 1/2. These two inequalities may be satisfied by choosing α,β sufficiently closetoone,sincetheassumptionsofourtheoremimplythat1/(2+ε) < δ 1/2 Note that the rate of γ-mixing required in Theorem 5.1 is substantially stronger than would be required under α-mixing. Using the central limit theorem given in [5, Theorem 10.7], we see that our γ-mixing condition may be replaced with the α-mixing condition r=1 αε/(2+ε) r <. Thus, in the case of bounded random variables, the memory condition α r = O(r δ ), δ > 1, is sufficient for stationary α-mixing processes to satisfy a central limit theorem, whereas the analogous condition under Theorem 5.1 is γ r = O(exp( r δ )), δ > 1/2. Acknowledgment. Parts of this paper derive from my doctoral research at Yale University. An earlier version was circulated under the title A new mixing condition. I thank Donald Andrews, Yuichi Kitamura, Peter Phillips and Alessio Sancetta for helpful comments. References 1. Andrews, D. W. K.: Non-strong mixing autoregressive processes, J. Appl. Probab. 21 (1984), no. 4, Bartlett, M. S.: Chance or chaos? J. Roy. Statist. Soc. Ser. A 153 (1990), no. 3, Beare, B. K.: A generalization of Hoeffding s lemma and a new class of covariance inequalities, Statist. Probab. Lett. 79 (2009), no. 5, Blum, R. and Rosenblatt, M.: A class of stationary processes and a central limit theorem, Proc. Nat. Ac. Sc. U.S.A. 42 (1956), no. 7, Bradley, R. C.: Introduction to Strong Mixing Conditions, vol. 1, Kendrick Press, Heber City, Dedecker, J., Doukhan, P., Lang, G., León, J. R., Louhichi, S. and Prieur, C.: Weak Dependence: With xamples and Applications. Lecture Notes in Statistics vol. 190, Springer, New York, Dedecker, J. and Prieur, C.: New dependence coefficients. xamples and applications to statistics. Probab. Theory Related Fields 132 (2005), no. 2, Doukhan, P.: Mixing: Properties and xamples. Lecture Notes in Statistics vol. 85. Springer- Verlag, New York, Lawrance, A. J.: Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators, J. Appl. Probab. 29 (1992), no. 4, Owen, A. B.: Multidimensional variation for quasi-monte Carlo. Technical report no , Department of Statistics, Stanford University, Peligrad, M.: Some remarks on coupling of dependent random variables, Statist. Probab. Lett. 60 (2002), no. 2,

13 A COARSNING OF TH STRONG MIXING CONDITION Rio,.: Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Springer- Verlag, Berlin, Rosenblatt, M.: A central limit theorem and a strong mixing condition, Proc. Nat. Ac. Sc. U.S.A. 42 (1956), no. 1, Tong, H. and Chen, B.: A note on one-dimensional chaotic maps under time reversal, Adv. Appl. Probab. 24 (1992), no. 1, Utev, S.: Inequalities for sums of weakly dependent random variables and estimates of the convergence rate in the invariance principle, in: Borovkov, A. A. (ed.), Advances in Probability Theory: Limit Theorems for Sums of Random Variables, Optimization Software Inc., New York, Withers, C. S.: Central limit theorems for dependent variables, I, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 4, Brendan K. Beare: Department of conomics, University of California - San Diego, La Jolla, CA 92101, USA -mail address: bbeare@ucsd.edu