On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions (Q1819845)

Let \(n_ i\), \(i=1,2,...,r\) be positive integers and \(\underset \tilde{} X=(X_ 1,...,X_{n_ 1+...+n_ r})\) be a random vector with nonnegative integer-valued components. Define \[ Y_ j=\sum^{n_ j}_{i=1}iX_{n_ 1+...+n_{j-1}+i},\quad j=1,2,...,r. \] This paper studies essentially certain elementary properties of the random vector \((Y_ 1,...,Y_ r)\) in the case when the distribution of \(\underset \tilde{} X\) is multinomial.