Limit theorems arising from sequences of multinomial random vectors (Q1118900)

Set for independent and identically distributed random vectors \(X_ 1,...,X_ n\), \(X_ i\sim Mn(1,p_ 1,...,p_ k)\), \(K_ n=\sum^{n}_{i=1}X_ i\), \(S_ n=\sum^{n}_{i=1}iX_ i\). Then after normalizing \(S_ n-((n+1)/2)K_ n\) by suitable matrices the distribution converges to a multivariate normal distribution (Theorem 1). If \(p_{\ell}=p_{\ell}^{(n)}\) and \(p_{\ell}^{(n)}\) is away from 0 and 1 for \(1\leq \ell \leq u\) whereas \(np_ j^{(n)}\to \lambda_ j>0\), \(j>u\), then after a suitable linear transformation of \(S_ n\) the distribution converges to a product distribution where the first distribution is a normal one and the second is a product of Poisson distributions.