STOCHASTIC

EXPANDING BUSINESSES OPTIMAL 651
N-l < M, XN = XN, if RN-1> N-l > M, is such that (rj, ..., rM) is not in EM, then xN = XN,
otherwise Ijj = Ijj. Now, on EM,
OB
Z [E(log VJX-1> - E(log V^R^i)] < -
1
t
hence, lim S N/Sj$ > 0 which implies that lim ^/SJJ = 0 on EM U E. And we have that
P(EM U E) > P(E) - e = a - e , on the complement of EJJ, SN = SJJ, so that lim N/S^ « 1
except with at most probability e . Therefore, (Ij, ^2 • • •) Is an inadmissible sequence.
Conversely, suppose (B) holds, but (x,, ...)is inadmissible with respect to (Xj • • •),
and that lim S N/SN = 0 on E with P(E) > 0. Since (B) holds, lim S N/s£, > 0 almost surely,
which implies that lim ^N/S„ = 0 almost surely on E. This implies, in turn, that
lim SN/SN = » with positive probability, which violates Theorem 1.
The above Thoerem is the result referred to in the introduction, for the essential content
is, that unless ~xN is usually close to ~XN in the sense that E(log VJ.|HJJ_J)
- E(log VJJIRJ, 1) is small, then (X., ...) is not admissible.
REFERENCE
[1] J. L. Doob, Stochastic Processes, John Wiley and Sons, New York, 1953.
* * *
ADDENDUM*
Dr. Breiman was kind enough to submit the following amended proof of
Theorem 2. The proof on page 596 is incomplete because it does not consider
A c (see below).
Proof of Theorem 2 By definition
Define
then,
£(logKt-log^*|«t_,)gO.
XN = Y.{\ogVk-\ogVk* -E(\ogVk-\ogVk*\Rk^)}k=
1
XN+, -XN = logtV, -logK*+1 -E(\ogVN+l-\ogV*+i\RN)
g V < 00 by assumption,
and {XN} is a martingale.
Applying Theorem 4.1 (iv) (on pp. 319-320 of reference [1]) to the sequences
{X,,} and {-XN}, we have that limN^Q. XN exists and is finite almost surely
* Adapted by S. Larsson and W. T. Ziemba from personal correspondence with Dr. Leo
Breiman.
4. THE CAPITAL GROWTH CRITERION AND CONTINUOUS-TIME MODELS 597