Let $N(t)$ be a Poisson process of rate $\lambda$. Suppose that we wanted to construct a continuous-time Markov chain with a finite state space $S$ by setting $X_t = N(t) \text{ (mod } n\text{)}, n \in \mathbb{N}, \left| S \right| = n + 1$. What do we know about the periodicity of the chain and the probabilities of its transition matrix?
2026-03-31 12:22:30.1774959750
What properties does a Markov chain formed from a Poisson process modulo m have?
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