By M. Frank Norman (Eds.)

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**Example text**

1. 2, and recalling our assumption that R, < 00 and r, c 00, we see that Rj < 00 and 5 < 00 for all j 2 1 in a distance diminishing model. The assumption that r, < 1 permits us to say more. 1 r* = lim rjlj < 1 j-r Q) and Proof. 2, for any n 2 j 2 1, r,, < ri'r,,,, where q = [ n / j ] and m = n-jq

In addition, and 1. IL, (b) H=supnBolUnI,< CQ; and (c) there is a k 2 1, an r < 1, and an R < 00 such that II UYII G r I l f II + R I f I for a l l f e L. Only (b) and (c) are needed for the next lemma. 1. 2) where R' = (1 -r)-'RH. Furthermore, J = supnBo11 U"II < 00. Proof. l), implies that sup, 11 U'"'f11 < 00 for each f E L. 21). 1 is applicable in the present context. The following useful condition for aperiodicity generalizes Lemma 1 of Norman (1970a). 1. 3) for all n 2 I , then U is aperiodic.

Here we describe this procedure in abstract terms and note its properties. The starting point is a learning model ((X, W), (E, Y),p, u), with which are associated state and event sequences Xn and En. In addition, there are measurable spaces ( X * , W*) and ( E * , Y*), and measurable transformations @ and Y of X and E onto X * and E*, respectively. Intuitively, x* = @(x) and e* = Y ( e ) represent simplified state and event variables. In the full ZHL model, they are projections: @ ( w , z ) = (V,Y) and Y(S, a, r ) = (a,r ) 9 where s = (B, W) or (W, B), a = br or PO, and r = B or W.