The Bienaymé-Galton-Watson (BGW) process was introduced by Irénée-Jules Bienaymé (1845) to explain mathematically the observed phenomenon (Malthus, 1817, de Châteauneuf, 1845) that family names, both among aristocracy and among bourgeoisie, tend to become extinct. The process has since found uses in many other areas, such as genetics (Ewens, 1969), epidemiology (Becker, 1977), queueing theory (Kendall, 1951), and demography (Keyfitz, 1985). Statistical methods for these branching processes were first developed by Harris (1948), and have been the target of extensive research for the last two decades.
To define the process, let 
, 
be independent, identically distributed random variables, taking
values
in the nonnegative integers, with probability generating function
.
We assume throughout that we are in the supercritical case, i.e.,
that
.
The BGW branching process 
with offspring distribution 
, starting from 
ancestors, is defined recursively by 
and
where 
is taken to mean 0. We will assume that 
.
It is clear from (1) that 
is a Markov chain with
transition probabilities given by
where 
are the probabilities of a 
-fold convolution
of
the offspring distribution.
Since, unless 
, the process 
has positive
probability of hitting the
absorbing state 0, it is not possible to estimate any parameters
consistently.
Usually, therefore, inferences are made conditional upon non-
extinction
(Sweeting, 1986, argues the approximate ancillarity of this
conditioning).
Lockhart (1982) showed that two branching processes with the same
mean,
variance, and lattice, and finite 
moments cannot be
distinguished, even on the set of non-extinction, on the basis of a
path
of observed generation sizes. In essence, large generation sizes
behave
like normal random variables, since they are the sum of a large
number
of iid random variables.
Dion (1974,1975) exhibited conditionally consistent estimators of
the
offspring mean and the offspring variance. The estimator of the
mean,
namely
which Harris derived as the maximum likelihood estimator based on observing the entire family tree, was shown to be a nonparametric maximum likelihood estimator even when only observing generation sizes by Feigin (1977) and, independently, by Keiding and Lauritzen (1978). In general, there are no (conditionally) consistent estimates of other interesting parameters such as the extinction probability or the offspring distribution.
Dion et al. (1982) discussed maximum likelihood estimation of the offspring distribution of the BGW-process. They concentrated on the case where the distribution is supported on three points. The results of Lockhart show that in this case it may be possible to consistently estimate the offspring distribution on the explosion set.
Guttorp (1991) gave an
algorithm for the computation of the offspring distribution mle
which is considerably simpler than the method proposed by
Dion et al. The algorithm is given and the
induced estimate of the offspring variance is described in section 2.
We know that the variance is consistently estimable, and that the
offspring
distribution is not, but we cannot immediately deduce that the mle
of
the variance (based on the inconsistent mle of the offspring
distribution)
is consistent. Guttorp (1991) proved consistency for the case of
distributions
with finite support and having all positive probabilities bounded
below by
some 
. The latter condition was needed to allow the
assumption of
fixed lattice size. In this paper we remove this condition.
Section 3 is devoted to a technical tool needed
to establish consistency; namely a local limit theorem for discrete
random variables which does not (as do most such results in the
literature)
assume that the lattice size of the random variables are known and
equal.
In section 4 we apply this result to show consistency, still under
fairly restrictive
regularity conditions, and section 5 consists of some discussion of
possible
extensions.