Limiting conditional distribution for tests of independence in the two way table

For testing the hypothesis of independence in the r x c two way contingency table, we consider the likelihood ratio statistic -2 ln(λ) and the χ² statistic conditionally given the sufficient statistic of row and column margin totals. We use Stirling's formula in the conditional characteristic function of normalized cell counts to give a direct proof that the limiting null conditional distribution is multivariate normal Z : script N_rc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to Z^T Λ^-Z, which has a central χ_v², distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).