  We study the local eigenvalue statistics $\xi_{\omega,E}^N$ associated with
the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices
with independent, identically distributed, real random variables and band width
growing as $N^\alpha$, for $0 < \alpha < \frac{1}{2}$. We consider the limit
points associated with the random variables $\xi_{\omega,E}^N [I]$, for $I
\subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random
variables with $0 \leq \alpha < \frac{1}{7}$, we prove that this family of
random variables has nontrivial limit points for almost every $E \in (-2,2)$,
and that these limit points are Poisson distributed with positive intensities.
The proof is based on an analysis of the characteristic functions of the random
variables $\xi_{\omega,E}^N [I]$ and associated quantities related to the
intensities, as $N$ tends towards infinity, and employs known localization
bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami
estimates \cite{peled, et. al.}. Our more general result applies to random band
matrices with random variables having absolutely continuous distributions with
bounded densities. Under the hypothesis that the localization bounds hold for
$0 < \alpha < \frac{1}{2}$, we prove that any nontrivial limit points of the
random variables $\xi_{\omega,E}^N [I]$ are distributed according to Poisson
distributions.
