  The local eigenvalue statistics of large random matrices near a hard edge
transitioning into a soft edge are described by the Bessel process associated
with a large parameter $\alpha$. For this point process, we obtain 1)
exponential moment asymptotics, up to and including the constant term, 2)
asymptotics for the expectation and variance of the counting function, 3)
several central limit theorems and 4) a global rigidity upper bound.
