(i) Let P be any family of distributions X = (X1,...,Xn) such that P{(Xi, Xi+1,...,Xn, X1,...,Xi−1) ∈ A} = P{(X1,...,Xn) ∈ A}

for all Borel sets A and all i = 1,...,n. For any sample point (x1,...,xn)
define (y1,...,yn)=(xi, xi+1,...,xn, x1,...,xi−1), where xi = x(1) = min(x1,...,xn). Then the conditional expectation of f(X) given Y = y is f0(y1,...,yn) = 1 n[f(y1,...,yn) + f(y2,...,yn, y1)
+ ··· + f(yn, y1,...,yn−1)].
(ii) Let G = {g1,...,gr} be any group of permutations of the coordinates x1,...,xn of a point x in n-space, and denote by gx the point obtained by applying g to the coordinates of x. Let P be any family of distributions P of X = (X1,...,Xn) such that P{gX ∈ A} = P{X ∈ A} for all g ∈ G. (2.39)
For any point x let t = T(x) be any rule that selects a unique point from the r points gkx, k = 1,...,r (for example the smallest first coordinate if this defines it uniquely, otherwise also the smallest second coordinate, etc.). Then E[f(X) | t] = 1 r r k=1 f(gkt).
(iii) Suppose that in (ii) the distributions P do not satisfy the invariance condition (2.39) but are given by dP(x) = h(x) dµ(x), where µ is invariant in the sense that µ{x : gx ∈ A} = µ(A). Then E[f(X) | t] = r k=1 f(gkt)h(gkt)
r k=1 h(gkt)
.
Section 2.5