Estimates for the probability of the coincidence of the frequency vectors of outcomes of independent multinomial schemes

We consider the problem to estimate the probability that vectors whose components are the frequencies of outcomes of independent polynomial schemes with $N$ outcomes and $n$ trials each coincide. We present local limit theorems for this probability under various constraints imposed on the growth of the parameters $n$, $N$ and on the distributions of outcome frequencies in each scheme. By the use of these results we derive asymptotic estimates for the mean number of coinciding components of the outcome frequency vectors in the case of two independent polynomial schemes.