  In this paper, we consider Anderson type operators on a separable Hilbert
space where the random perturbations are finite rank and the random variables
have full support on $\mathbb{R}$. We show that spectral multiplicity has a
uniform lower bound whenever the lower bound is given on a set of positive
Lebesgue measure on the point spectrum away from the continuous one. We also
show a deep connection between the multiplicity of pure point spectrum and
local spectral statistics, in particular, we show that spectral multiplicity
higher than one always gives non-Poisson local statistics in the framework of
Minami theory.
  In particular, in higher rank Anderson models with pure-point spectrum, with
the randomness having support equal to $\mathbb{R}$, there is a uniform lower
bound on spectral multiplicity and in case this is larger than one the local
statistics is not Poisson.
