The usual local central limit theorem provides an approximation for the
probability 
for an iid sequence 
.
The approximation is proportional to the
lattice size of the underlying distribution of the and is not a
continuous function of the underlying distribution. The probability
in question is a continuous function of this underlying
distribution and so the approximation cannot be good uniformly in the class of
possible distributions for the . Approximate likelihood methods demand
uniform approximations; in this section we provide such an approximation.
We are unable to handle the case of distributions nearly concentrated on a
single point gracefully. Moreover, it is convenient to impose the stringent
hypothesis that all distributions studied be supported on a fixed finite
set of indices 
for some fixed 
. Fix 
and let 
be the set of all probability distributions
supported on the integers such that 
and such that 
for 
and 
. The conditions
guarantee that the variance 
(where it seems convenient we
indicate explicitly the dependence of the mean and variance on the underlying
distribution in question) of the distribution is bounded
away from 
over and that each in has a lattice size dividing 
. (Note that could be
replaced throughout by the least common multiple of 
.)
If 
are
independent integer valued random variables write 
where 
are integer valued random variables and 
.
That is, 
is the residue class modulo of the sum. Our
theorem is
that 
is approximately normal and that and are
approximately
independent, uniformly for probabilities in . The
distribution of is available via an expression which may
be computed analytically for small values of and approximately
in some other cases.
Let 
be the characteristic function of
.
Let 
for 
. Then
In later sections the computability of the approximation is of less importance than the formal nature of the approximation.
We begin with the weaker form of the theorem
For a single fixed the asymptotic distribution of is
uniform on the set of possible residue classes and so the result
can be converted to an approximation of the conditional distribution
of 
given 
- a trivial consequence of the usual local central
limit theorem. However, under the condition that the
variates are bounded the conditional form of the result is also valid
uniformly in .
The local central limit theorem is established by analyzing the inversion formula for the characteristic function. We prove Theorem 1 following Petrov(??) taking care to make error estimates uniform.
The Fourier inversion formula is
where 
is the characteristic function of and
. We will split the range of integration into a number of
pieces and bound some and expand others.
Define 
for and let 
where
It follows from the lemma that
for every
such that
.
The integral over
will be broken into two pieces, integrating separately over
and the complement
where
is the sequence
Over
we will expand the function
about
. Over
we will use the following extension of Cramér's lemma.
It follows from the lemma that
In the usual proof of the local central limit theorem either the quantity
less than 1 and therefore the integral over
is exponentially small or
and the integral over
is identical to that over
where a standard Taylor expansion is used. To get a uniform result, however, we must contemplate the
where
is slightly less than 1 so that neither of the two cases just mentioned is applicable.
Thus for any
such that
we have
Consider now a
for which
. Define
(For
the ratios in the logarithm are uniformly close to 1 permitting use of the principal branch of the logarithm throughout the following.) We can write, for
, using a Taylor expansion
where
where
and, for all
so large that
,
Define
and
Then since
we have
for all
larger than some integer
depending only on
and
.
On
the quantity
is bounded by
so that On the set
we have the Taylor expansion
where
for every
.
