STOCHASTIC

THE ASYMPTOTIC VALIDITY OF QUADRATIC UTILITY
there exists a t0 such that
0 ^ (1/0/UO ^ (l/'o)Aom(-)
for all y2, ...,Yme R'(W). The integration operation over R'{W) changes
nothing; it follows immediately that (l/t)pt(W) ^ (l/t0)pt0{W); WeSu
0 < t < t0. Subsequently, it will be shown that £|W— 1| = o(t) and, using
Fatou's lemma, this implies p,(W) = o(t) except when W = 1. (This is also
simple to verify directly.) Consequently, the condition (ii) in Theorem II is
satisfied since s < 1 and
where 0 < t < t0 and W e St = [0, e).
0 = lim(l/t)pt(W) ^ (llt)pto(W),
r->0
Next, the behavior of the moments around unity are analyzed as t-> 0;
as previously, the case when A4 = 1 is considered first, and generalizations are
developed immediately thereafter.
Theorem IV Let ^C(X) ~ MLX (t/x, to 1 ) with /, ii, a 2 > 0. Then
and
E(X-\) _ ix+l/2a 2
~,Z~oE(X-l) 2 ~ V 1
Hm
r/v