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Two processes $(X_t)_{t \in T}$, $(Y_t)_{t \in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions, i.e., for all $t_1$, $t_2$, $\ldots$, $t_n$, $n \in \mathbb{B}$, $$ (X_{t_1}, \ldots X_{t_n}) \overset{d}{=} (Y_{t_1}, Y_{t_2}, \ldots Y_{t_n}) $$

How to give sense of this by using the $\pi-\lambda$ theorem, when the process takes value on a countable state space?

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1 Answer 1

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$(\Leftarrow)$ Assuming that $X_t$ and $Y_t$ take values in $(S,\mathcal{S})$, consider the collection of all finite cylinder sets $\mathcal{C}$ of the form $\{x\in S^T:(x_{t_1},\dots,x_{t_n})\in C\}$, where $n\in \mathbb{N}$, $t_i\in T$, and $C\in \mathcal{S}^n$, and the collection $\mathcal{L}=\{A\in \mathcal{S}^T : P\{X\in A\}=P\{Y\in A\}\}$. Now $\mathcal{C}$ is a $\pi$-system, $\mathcal{L}$ is a $\lambda$-system and $\mathcal{C}\subset\mathcal{L}$ (by assumption).

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