Maxima of the cells of an equiprobable multinomial (Q2461016)

A sequence of random vectors \(Y_n\) is considered. The \(n\)-dimensional random vector \(Y_n\) follows multinomial \((m_n,1/n,\dots,1/n)\) distribution for each \(n\). Limit behaviour of maxima \(M_n\) of coordinates of the vector \(Y_n\) is studied. It is shown that if \(m_n\) goes to infinity fast enough then \(M_n\) under a suitable affine transformation tends to Gumbel distribution. The same limit distribution is proved for Binomial distribution with increasing number of trials and possibly nonconstant probability of success. As an auxiliary result a new large deviation result for equiprobable multinomial distribution is proved.