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ANNALES DE L’I. H. P., SECTION B EMMANUEL RIO Covariance inequalities for stron

ANNALES DE L’I. H. P., SECTION B EMMANUEL RIO Covariance inequalities for strongly mixing processes Annales de l’I. H. P., section B, tome 29, no 4 (1993), p. 587-597 <http://www.numdam.org/item?id=AIHPB_1993__29_4_587_0> © Gauthier-Villars, 1993, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce ﬁchier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 587- Covariance inequalities for strongly mixing processes Emmanuel RIO URA n° 743 CNRS, Universite de Paris-Sud, Bat. n° 425 Mathématique, 91405 Orsay Cedex, France. Ann. Inst. Henri Poincaré, Vol. 29, n° 4, 1993, p. 597. Probabilités et Statistiques ABSTRACT. - Let X and Y be two real-valued random variables. Let a denote the strong mixing coefficient between the two a-fields generated respectively by X and Y, and Qx (u) = inf {t: P ( ] X I > t) ~ M} be the quan- tile function of We prove the following new covariance inequality: which we show to be sharp, up to a constant factor. We apply this inequality to improve on the classical bounds for the variance of partial sums of strongly mixing processes. Key words : Strongly mixing processes, covariance inequalities, quantile transformation, maximal correlation, stationary processes. RESUME. - Soient X et Y deux variables aleatoires reelles. Notons a le coefficient de melange fort entre les deux tribus respectivement engendrees par X et Y. Soit la fonction de quantile de I X I. Nous etablissons ici l’inégalité de covariance suivante : et nous montrons son optimalite, a un facteur constant pres. Cette inegalite est ensuite appliquee a la majoration de la variance d’une somme de variables aleatoires d’un processus melangeant. Classification A.M.S. : 60 F 05, 60 F 17. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques - 0246-0203 Vol. 29/93/04/$ 4,00/ (é) Gauthier-Villars 588 E. RIO 1. INTRODUCTION AND RESULTS Let (~ be a probablility space. Given two a-fields d and r1I in (Q, ~ P), the strong mixing coefficient a (d, is defined by [notice that ~) ~ 1/4]. This coefficient gives an evaluation of the dependance between d and ~. The problem of majorizing the covariance between two real-valued r.v.’s X and Y with given marginal distributions and given strong mixing coefficient was first studied by Davydov (1968). He proved that, for any positive reals p, q, and r such that l/r== 1, where 0’ (X) denotes the a-field generated by X. Davydov obtained C = 12 in (1.0). Davydov’s inequality has the following known application to the control of the variance of partial sums of strongly mixing arrays of real-valued random variables. Let (Xi)i E 7L be a weakly stationary array of zero-mean real-valued r.v.’s [i. Q. Cov (X~, Xt)= Cov (Xo, for any s and any t in Z~]. For any n E Zd, we define a strong mixing coefficient 0153" by where c (Xi) denotes the c-field generated by ~~. We shall say that the array is strongly mixing iff lim an = O. Then inequality (1 0) yields the folloBving result. THEOREM 1 . 0 (DaVydOV). - Let d ~_ 1 and let be a weakly stationary array of real-valued random variables. Suppose that Under the additional assumption ~ 2/r + ao y the series £ Cov X~) is absolutely convergent, has a nonnegative sum 0-2, and t ~ Zd lim n-d Var Sn = 0’2. n - + o0 Up to now, inequality (1 . 0) and his corollaries were the main tool for studying mixing processes. We have in view to improve on Davydov’s inequality. Let 2(1(F, G) denote the class of bivariate r.v.’s (X, Y) with Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 589 COVARIANCE INEQUALITIES . given marginal distributions functions F and G satisfying the mixing constraint Let denote the usual inverse function of F. In order to maximize Cov (X, Y) over the class G), it is instructive to look at the extremal case a== 1/4 (that is, to relax the mixing constraint). In that case, M. Frechet (1951, 1957) proved that the maximum of Cov (X, Y) is obtained when (X, y)=(F-1 (U), G -1 (U)), where U is uniformly distributed over [0, 1 ] (actually, Frechet gives a complete proof of this result only when F and G are continuous). In other words, we have: In view of ( 1.1 ), one may think that the maximum of the covariance function over G) should depend on 0153, and rather than on the moments of X and Y. Unfortunately, the exact maximum has a more complicated form in the general case than in the extremal case a.= 1/4. However, we can provide an upper bound for I Cov(X, Y) I, which is optimal, up to a constant factor. THEOREM 1. l. - Let X and Y be two integrable real-valued r.v.’s. Let a = a (cr (X), a (Y)). Let Qx (u) = inf { t : P ( X > t) __ u ~ denote the quantile function of I X I. Assume furthermore that Qx oy is integrable on [0, 1]. Then Conversely, for any symmetric law with distribution function F, and any a E ]0, 1 /4J, there exists two random variables X and Y with common distribu- tion function F, satisfying the strong mixing condition 0153(O’(X), O’(Y))~0153 and such that Remarks. - Using the same tools as in the proof of inequality (a), one can prove the following inequality: Inequality ( 1. 2) is more intrinsic than inequality (a), for the upper bound in (1.2) depends only on the "dispersion function" (s, t ) - F -1 (t) Vol. 29, n° 4-1993. 590 E. RIO - (s) of X and on the dispersion function of Y. However, inequality (a) is more tractable for the applications. Theorem 1. 1 implies ( 1. 0) with C=21+1/p, which improves on Davy- dov’s constant (note that, when U is uniformly distributed over [0, 1 ], Qx(U) has the distribution of and apply Holder inequality). The assumptions of moment on the r.v.’s X and Y in Davydov’s covariance inequality can be weakened as follows. Assume that Then, it follows from Theorem 1.1 that Of course by Markov’s inequality. Hence, we obtain a similar inequality under weaker assumptions on the distribution functions of X and Y than Davydov’s one. We now derive from Theorem 1.1 the following result, which improves on Theorem 1. O. THEOREM 1. 2. - Let be an array of real-valued random variables. Define a -1 (t) _ ~ For any positive integer n, let Qn denote the i e 7Ld nonnegative quantile function defined by: Then, Moreover, is weakly stationary and if then, and denoting by a2 the sum of the series ~ Cov (Xo, Xt), we have: Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 591 COVARIANCE INEQUALITIES In particular, if is a strictly stationary array, then Q" = QXO = Q, and so, if then, (b) and (c) hold with M = a’’ (u) [Q (2 u)]2 du. Remark. - In a joint paper with P. Doukhan and P. Massart (1992), we prove that the functional Donsker-Prohorov invariance principle holds for a strictly stationary sequence if a condition related to ( 1. 5) is fulfilled. Applications. - Let r > 2. If the tail functions of the r.v.’s X~ are uniformly bounded as for any positive u and any Then, A-t for some constant C depending on r and Cr. Hence the conclusions of Theorem 1.0 are ensured by a weaker condition on the d.f.’s of the r.v.’s Xy than Davydov’s one [this is not surprising in view i e lld of (1. 3)]. Set-indexed partial sum processes. - Let be a strongly mixing array of identically distributed r.v.’s satisfying condition (1.5). Let A c [0, be a Borel set and let where [i -1, i] denotes the unit cube with upperright vertice i and X denotes the Lebesgue measure. Then, we can derive from (a) of Theorem 1.2 the following upper bound: [Apply (~) of Theorem 1.2 to the array defined by We now study the applications of Theorem 1.2 to arrays of r.v.’s satisfying moment constraints. So, we consider the class of functions $’ = { ~ : IR + - IR + : ~ convex, increasing and differentiable, 03C6(0)=0, lim03C6(x) x = oo}. +00 ~ J Vol. 29, n° 4-1993. 592 E. RIO and, for we define the dual function ~* by ~* (y) = sup [xy - ~ (x)]. When the Cesaro means of the ~-moments of the x>o random variables Xf are uniformly bounded, Theorem 1.2 yields the following result. COROLLARY 1. 2. - Let (Xi)i E a be a stongly mixing array of real-valued random variables. Let 03C6 be some element of g- such that E (j> (X2i)) + 00 uploads/Litterature/ rio-covariance-inequalities-for-strongly-mixing-processes.pdf