Let {Zn, n > 1}
be a single type Galton-Watson process initiated by a single ancestor
with mean
EZ1 º m.
This paper develops the local limit theory
of Zn, viz., the behavior of
P(Zn = vn)
for 0 < vn ¥ as n ®
¥. These
results are used to study the large deviations of
{Zn+1/Zn:
n > 1} conditioned on Zn >
vn. Some general questions on averages indexed
by random sequences are discussed.