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I am reading LS book "Recursive Macroeconomics Theory".

In this book, as well as in some online lecture notes on the Internet. They often denote, for example, $\pi_t(s^t)$ for probability that history $s^t$ happens at time $t$.

But I wonder, if we are mentioning $s^t$, it implies time is $t$. So, why don't we just make it simple by writing $\pi(s^t)$ instead?

Thank in advance for any help!

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You are right, it is the same thing. Actually, later on the book the density over the history $s^t = [s_t, s_{t-1}, \cdots, s_0]$ is written as

$$ \pi(s^t) = \pi(s_t|s_{t-1}) \cdots \pi (s_1|s_0)\pi(s_0) \tag{2.3.1} $$

where $\pi(s_0)$ denotes the probability of the initial state (or $\pi_0(s_0)$ if you wish), and $\pi(s|s')$ is a transition probability.

Note that Eq. (2.3.1) encapsulates the the probability of obtaining a given history in a Markov process, since ${\rm Prob}(s_t|s_{t-1}, s_{t-2}\cdots,s_0) = {\rm Prob}(s_t|s_{t-1}) = \pi(s_t|s_{t-1})$. This just to show that $\pi_t(s^t)\color{blue}{\equiv}\pi(s^t)$

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