STOCHASTIC

(a.s.) on the set
lim XN < oo > u < ljm XN > — oo (El)
By Theorem 1, O^SK/Ss* converges a.s. to a finite value, and hence
log Sf,/Sn* converges a.s. to a finite value or to — oo.
Let A be the set on which limN(5iv/5N*) > 0. We are then required to prove
that
f £(logK„|J»K_1)-£(log^*|/fA(_I)# -oo.
Now, \im„(SNISN*) > 0 implies that limNlog(SN/SN*) > - oo, or, equivalently,
limNY.^Jog(VJVk*)>-co, and since XN ^ Z~=1 \og(VJVk*), it follows
that limjv XN > — oo, and by (El) lim XN exists and is finite. Hence
t £(logKw|J?«_1)-£(logK„*|/l*_1) * -oo.
N=I
On A c either (1) limA'iv / )i|/!A_l) = «,.
4=1
Hence in all cases lim/v(.SJV/.S/v*) = 0 iff
| JJdogK/I^.J-Edog^l/f*.,) = 00.
* = i
We would also like to record the following typographical and other minor
errors:
Page" Line In paper Change to
647 (593)
648 (594)
649 (595)
650 (596)
651 (597)
6 in abstract
5,12
27
last
10
last
8 from bottom
1
are independent
s,
VII > 0
h
vNtv„*
Theorem 3
*« =
" Numbers in parentheses indicate page numbers in this volume.
may be dependent
•S,
£1
>0
A,
V^-dVS-x
Theorem 2
v=
PART V. DYNAMIC MODELS