  We give bounds on the Poincare (inverse spectral gap) constant of a
non-negative, integer-valued random variable W, under negative dependence
assumptions such as ultra log-concavity and total negative dependence. We show
that the bounds obtained compare well to others in the literature. Examples
treated include some occupancy and urn models, a random graph model and small
spacings on the circumference of a circle. Applications to Poisson convergence
theorems are considered.
