STOCHASTIC

S. L. BRUMEIXE AND R. G. VICKSON
Since X„ and Z„ have uniformly bounded range, the sequence of bivariate
c.d.f.'s Hn(x,z) of (X„,Z„) on F x S contains a subsequence H. which converges
to the c.d.f. H of a bivariate random variable (X,Z) on I r xS [3,
Chapter VIII, p. 267]. Similarly, the sequence {«'} contains a subsequence
{«"} such that Y„„ -* Y. Since EZ = $rxSz d 2 H(x,z) exists, the conditional
expectation E(Z\X) exists ^-almost surely [3, Chapter V, pp. 162-165]. Let
A cz l r be any /i-measurable set. Recalling that E(Z„„ \ Xn„) ^ 0, it follows that
0 ^ lim \ E(Z„„ | X„„ = x) dFXn„(x)
n"-*oo J A
= lim j zd 2 Hn..(x,z) = \ zd 2 H{x,z)
n"-»oo JAxS JA*S
= f E(Z\X=x)dFx(x).
Thus E(Z | X) ^ 0 /i-almost surely (and therefore also P-almost surely, where
P is the "underlying" probability measure). A simple argument shows that
P[Y£Z']= PIX+Z ^Q,ZeI r , and this completes the proof.
III.4 EXTENSION
In Theorem 3.1 it is assumed that Zand Fare bounded random variables.
In applications one is often interested in unbounded random variables, and
a generalization of Theorem 3.1 can be proved for these situations under
additional assumptions. The following results due to Strassen [13] are stated
without proof.
Theorem 3.2 If X and Y are random variables in R 1 having finite means
EX and EY, the following conditions are equivalent:
(a) Eu(X) ^ Eu(Y) for all concave nondecreasing u.
(b) There exists Z such that P\Y f] = PIX+Z £ f] for all £ e R 1 , and
E(Z\ X) ^ 0 almost surely.
Theorem 3.3 If X and Y are random variables in E r having finite means
EX and EY (with EX = EY), the following conditions are equivalent:
(a) Eu{X) ^ Eu(Y) for all concave u (whether nondecreasing or not).
(b) There exists Z such that P\Y^ £] = P\_X+Z ^ £] for all £, e R 1 , and
E{Z\X) = 0 almost surely.
The generalization of Theorem 3.3 to the case of unequal means and concave
nondecreasing u in (a), appears to be an unsolved problem.
112 PART II QUALITATIVE ECONOMIC RESULTS