Suppose i have a random vector $N_1,...,N_{2n}$ following a multinomial distribution with $k$ trials and an even number of (mutualy exclusive) outcomes $2n$ such that the probabilities of each outcome are paired up as follows :
$$\mathbb{P}(\text{outcome}_i) = ap_i \text{ and } \mathbb{P}(\text{outcome}_{n+i}) = (1-a)p_i$$
with $a, p_1,...,p_n$ all in [0,1] and $\sum\limits_{i=1}^{n} p_i = 1$.
Question: Will the vector $(\frac{N_1}{a}+\frac{N_{n+1}}{1-a},...,\frac{N_n}{a}+\frac{N_{2n}}{1-a})$ be multinomial itself (up to a multiplicative factor if needed) ? Otherwise, what is this distribution ?
No, the vector doesn’t follow a multinomial distribution; the most direct way to see this is to note that the components of a multinomially distributed vector are integers and the components of this vector aren’t. I doubt that the distribution falls under any standard distribution; it’s a complicated distribution where each component can take values $\frac ja+\frac k{1-a}$ with $j$ and $k$ non-negative integers.