(i) LetP 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 gk x, 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 (gk t).
(iii) Suppose that in (ii) the distributions P do not satisfy the invariance condition (2.39) but are given by d P(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 (gk t)h(gk t)
r k=1 h(gk t)
.