# Showing a process is a markov process

Let $$P$$ be a stochastic $$N \times N$$ matrix. We consider two independent copies of a Markov process on $$\{1,\ldots,N\}$$ with transition probabilities given by $$P$$. Both are assumed to start deterministically at state $$1$$.

The two processes can be thought of as a single process on $$\{1,\ldots,N\}^2$$. Is this a markov process? What does the transition matrix look like?

It is indeed a Markov process, on a new state space like you mentioned: $$N^2$$. What should be the transmission matrix? assuming every step in each chain is independent in the other, the probability of moving from $$(n_1, n_2)$$ to $$(m_1, m_2)$$ is the probability of moving in the first chain from $$n_1$$ to $$m_1$$, and the same idea in the other chain. By assumption, the requested probability will be $${[P]}^{n_1}_{m_1} \cdot {[P]}^{n_2}_{m_2}$$.