  This paper introduces a new concept of stochastic dependence among many
random variables which we call conditional neighborhood dependence (CND).
Suppose that there are a set of random variables and a set of sigma algebras
where both sets are indexed by the same set endowed with a neighborhood system.
When the set of random variables satisfies CND, any two non-adjacent sets of
random variables are conditionally independent given sigma algebras having
indices in one of the two sets' neighborhood. Random variables with CND include
those with conditional dependency graphs and a class of Markov random fields
with a global Markov property. The CND property is useful for modeling
cross-sectional dependence governed by a complex, large network. This paper
provides two main results. The first result is a stable central limit theorem
for a sum of random variables with CND. The second result is a Donsker-type
result of stable convergence of empirical processes indexed by a class of
functions satisfying a certain bracketing entropy condition when the random
variables satisfy CND.
