  Staircase tableaux are combinatorial objects that were first introduced due
to a connection with the asymmetric simple exclusion process (ASEP) and
Askey-Wilson polynomials. Since their introduction, staircase tableaux have
been the object of study in many recent papers. Relevant to this paper, the
distri- bution of parameters on the first diagonal was proven to be
asymptotically normal. In that same paper, a conjecture was made that the other
diagonals would be asymptotically Poisson. Since then, only the second and the
third diagonal were proven to follow the conjecture. This paper builds upon
those results to prove the conjecture for fixed k. In particular, we prove that
the distribution of the number of alphas (betas) on the kth diagonal, k > 1, is
asymptotically Poisson with parameter 1\2. In addition, we prove that symbols
on the kth diagonal are asymptotically independent and thus, collectively
follow the Poisson distribution with parameter 1.
