Asymptotics of conditional probabilities of succesful allocation of random number of particles into cells

The article is devoted to the memory of Valentin Fedorovich Kolchin.\qquad\qquad\qquad\qquad\qquad\linebreak Let $\zeta$, $\zeta_i$ ($i\inN$) be independent identically distributed nonnegative integer-valued random variables, $(\eta_{i1},\dots, \eta_{iN})$ be the fillings of cells in the generalized scheme of allocation of $\zeta_i$ particles into $N$ cells, $1\le i\le n$, for fixed $Z_n=(\zeta_1,\ldots,\zeta_n)$ these allocation schemes are independent. We consider the conditional probabilities $P(A_{n, N}\,|\,Z_n)$ of the event\linebreak $A_{n, N}=\{\text{each cell in each of } n \text{ allocation schemes contains no more than } r \text{ particles}\}$, where $r$ is some fixed number. The sufficient conditions for the convergence of the sequence $P(A_{n, N}\,|\,Z_n)$ to a nonrandom limit with probability 1 are given. It is shown that the random variable $\ln P(A_{n, N}\,|\,Z_n)$ is asymptotically normal. Applications of the obtained results to the noise-proof encoding are discussed.