Let $X=(X_1,...,X_k)$ is multinomially distributed with index $n$ $$X\text{~} \text {Mul}(n,p_1,...,p_k)$$ Let $1\leq i_1 <,...<i_l\leq k$ .Further we define the sequence $Y_1 =\sum_{j=1}^{i_1}X_j$,$Y_2 =\sum_{j=I_1+1}^{i_2}X_j$,...,$Y_l =\sum_{j=i_{l-1}+1}^{i_l}X_j$,$Y_{l+1} =\sum_{j=i_l+1}^{k}X_j$.Compute the probability $$P(Y_1=n_1,...,Y_{l+1}=n_{l+1})$$
I thought that $Y_i$ form a Markov chain but i realize that it is not clear.I am unable to solve this.Any ideas to solve this problem and how can we think of this problem ?