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Could anyone just explain to me what does it mean by mathematically, $P_n(x, dy)$ is the law of $X_n$ here in the page $46$.

https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf

Thanks for helping.

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They full quote is "The kernel $P_n(x, dy)$ is the law of $X_n$ given that $X_0 = x$." This means that the conditional distribution of $X_n$ given $X_0 = x$ is given by $$ \Pr(X_n \in B \mid X_0 = x) = P_n(x, B) $$ for all measurable subsets $B$ of the state space. The word "law" is a synonym for "distribution" in this context. The "$dy$" notation comes from the fact that for a sufficiently nice measurable function $g$ we have $$ E[g(X_n) \mid X_0 = x] = \int_{\mathcal{X}} g(y) P_n(x, dy), $$ where $\mathcal{X}$ is the state space (I didn't check what notation the link uses for the state space).

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