On the -mixing condition for stationary random sequences

For strictly stationary sequences of random variables two mixing conditions are studied which together form the i//-mixing condition. For the dependence coefficients associated with these two mixing conditions this article gives results on the possible limiting values and possible rates of convergence to these limits. For strictly stationary random sequences, the "«/--mixing" (or "*-mixing") condition was introduced by Blum, Hanson, and Koopmans [2]. They showed that for Markov chains satisfying this condition the mixing rate had to be exponential; later Kesten and O'Brien [5] showed that in the general case the mixing rate could be arbitrarily slow and that a large class of mixing rates could occur for stationary ip-mixing random sequences. This article will probe further into the nature of this mixing condition. Let (Í2, *3, P) be a probability space, and for any collection Y of random variables let <$>(Y) denote the Borel field generated by Y. For any two a-fields & and ® define ^*i&,^) = sapPj\n.B\, A E&,BE9,,P(A)P(B)>0; P{A)P{B)' P(A PB) PiA)PiB)> Vi&,®>) = mîPlA.n.B\, A E&,BE§,PiA)PiB)>0. Obviously i//(éE, $) = 1 = \¡/*(&, %) if (t and % are independent a-fields; otherwise }'i&,<&)< 1 <<//*($, ®). Given a strictly stationary sequence (Xk, k = .. .,-1,0,1,...) of random variables, define for -oo</<L^t» the a-fields %L = <$>(Xk, J < K =c L), and for n = 1,2,3,... define «//* = ^*(Si0cc, ̂ ) and «/*„' = «f-'CÍ0«,, %?)■ The «//-mixing condition is lim «//* = lim «// = 1. n-*cc n-*cc A strictly stationary sequence (Xk) is called "mixing" if VA, B E <$xx, lim P{A n T~"B) = F(^)F(t5) Received by the editors May 23, 1980. 1980 Mathematics Subject Classification. Primary 60G10, 60F20.