  Let I_1,...,I_n be independent but not necessarily identically distributed
Bernoulli random variables, and let X_n=\sum_{j=1}^nI_j. For \nu in a bounded
region, a local central limit theorem expansion of P(X_n=EX_n+\nu) is developed
to any given degree. By conditioning, this expansion provides information on
the high-order correlation structure of dependent, weighted sampling schemes of
a population E (a special case of which is simple random sampling), where a set
d\subset E is sampled with probability proportional to \prod_{A\in d}x_A, where
x_A are positive weights associated with individuals A\in E. These results are
used to determine the asymptotic information, and demonstrate the consistency
and asymptotic normality of the conditional and unconditional logistic
likelihood estimator for unmatched case-control study designs in which sets of
controls of the same size are sampled with equal probability.
