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I am reading LS book, on Chapter 8 - Complete Markets (3rd version). There is an example on page 264 that I am quite confused.

I attach the picture of the example here under.

I don't understand why we have a history (0,0,0,...,0,1,1)? since if $s_t=0$, then $s_{t+1} = 0$ for sure, right?

Or the authors means the history here is in order $(s_t, s_{t-1},....,s_1,s_0)$, so it makes sense since we know that $s_0=1, s_1=1$, but isn't it quite weird, since we usually write a history $h_t=(s_0,s_1,...s_t)$ right?

Anyone has an idea? Really appreciate your help!

enter image description here

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Since this is a Markov chain

\begin{eqnarray} \pi_t(0,0,\cdots,1,1) &=& \color{blue}{\pi(s_t = 0 | s_{t - 1} = 0)} \cdots \color{magenta}{\pi(s_2 = 0 | s_{1} = 1)} \color{red}{\pi(s_1 = 1 | s_{0} = 1)}\color{orange}{\pi(s_0 = 1)} \\ &=& \color{blue}{1} \times \cdots\times \color{magenta}{0.5}\times\color{red}{1}\times \color{orange}{1} = 0.5 \end{eqnarray}

But you are right if $t > 2$ then $\pi(s_{t + 1} = 0 | s_{t} = 0) = 1$, so if $s_t = 0$ then $s_{t + 1} = 0$, but at $t = 2$ there's a 50/50 chance that the state changes from $1$ to $0$, so the state may remain $1$ as in the first history, or switch to $0$ as the second one

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