Let $N,k \geq 1$ be two integers and consider the following Markov chain on $[0,k\times N]^k$. It starts at $X^0=(N, N, \ldots, N)$ and $X^{n+1}$ is the realisation of a multinomial distribution with $kN$ trials and probability distribution $(X^n_1, \ldots, X^n_k) / (kN)$. I am interested (for large $N \gg 1$) in the expectation of first time when one of the coordinates of the Markov chain vanishes. A good upper bound would be great.
A scaling argument should show that for any $\epsilon>0$ it takes $O(N)$ steps for one of components to be of order $\epsilon N$. But for one component to be zero, I believe that it should be of order $N \ln(N)$.
For example, for $k=2$, the scaled process $Z^{(N)}_t = N^{-1} \, X^{[Nt]}_1$ behaves (when away from $0$ or $1$) as the diffusion $dZ = \sqrt{Z(1-Z)}\,dW$ so that one can compute the expected hitting time of $0$ or $1$ by noticing that $t \mapsto F(Z_t)-t$ is a martingale with $F(z) = z \ln(z) + (1-z) \ln(1-z)$. This approach does to seem to generalize well for the original Markov chain.