The probabilities of large deviations for a certain class of statistics associated with multinomial distribution

Let eta= (eta(1), horizontal ellipsis ,eta(N)) be a multinomial random vector with parametersn=eta(1)+ MIDLINE HORIZONTAL ELLIPSIS +eta(N)andp(m)> 0,m= 1, horizontal ellipsis ,N,p(1)+ MIDLINE HORIZONTAL ELLIPSIS +p(N)= 1. We assume thatN ->infinity and maxp(m)-> 0 asn ->infinity. The probabilities of large deviations for statistics of the formh(1)(eta(1)) + MIDLINE HORIZONTAL ELLIPSIS +h(N)(eta(N)) are studied, whereh(m)(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.