TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 17

Chapter 17 CONDITIONS OF WEAK DEPENDENCE FOR STATIONARY PROCESSES The past history of the process Xt is described by the a-algebras SJX- S, the future by the o-algebras 9J1 +S. It may be that these o-algebras are independent, in the sense that, for all A E 9J1`7' | Chapter 17 CONDITIONS OF WEAK DEPENDENCE FOR STATIONARY PROCESSES The past history of the process Xt is described by the r-algebras the future by the a-algebrasW s. It may be that these cr-algebras are independent in the sense that for all Be9Jlt s P AB -P A P B 0. In the general case the magnitude of the left-hand side measures the dependence between past and future and it may be useful to assume this to be small in some sense. In this chapter we examine some of the possible ways of limiting the dependence. 1. Regularity Definition . A stationary process Xt is said to be regular if the oalgebra t is trivial in the sense that it contains only events of probability zero or one. The famous zero-one law for independent random variables see for example 59 31 implies that for instance a sequence of independent identically distributed random variables is regular. In the Hilbert space terminology of the last chapter regularity simply means that the subspace 0 h - t which consists of the random variables measurable with respect to yjl- Tc contains only the constant functions. 302 WEAK DEPENDENCE FOR STATIONARY PROCESSES Chap. 17 Theorem . In order that a stationary process Xt be regular it is necessary and sufficient that for all lim sup P AB -P A P B I O. i- OO A e JJ1 a Proof. To prove the necessity of the condition write xA for the indicator function of an event A t I J zxH j j toe A and set xa-P A rj xB-PB so that P AB -P A P B E . Since is measurable with respect to equation gives EtfEtffiM_ J E 2 E E E W .J2 0 as t go in virtue of the theorem of Appendix 3. To prove the sufficiency suppose to the contrary that is satisfied but that Xt is not regular. Then there is an event A e SCR _ w with 0 P A 1 and then sup P AB -P A P B P X -P A 2 0 which contradicts . Corollary . A regular process Xt is metrically transitive. Proof. Let A be an invariant event. For any s 0 we can find a finite t and an event such that p a-4 u

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