Typo in Ross' Introduction to Probability Models 11th ed p. 205?

Question

possible typo

I have problems understanding the text marked in red. Is this supposed to say something else? The syntax just seems off to me.

Answer 1

I expect the word "chain" has been inadvertentently omitted from immediately after "Markov". Apart from that, and perhaps a slight awkwardness of expression, I can't see much else wrong with the statement. It's simply explaining what $\ P^n_{i,j}\ $ means in terms of the long-term frequency interpretation of probability.

If $\ S_t\ $ is the state of the chain at time $\ t\ $, then I take "the long-run proportion of time when the Markov chain is in state $\ i\ $ that it will be in state $\ j\ $ after $\ n\ $ transitions" to be referring loosely to the quantity $$ \lim_{T\rightarrow\infty} \frac{\left\vert\left\{t\in\left\{0,1,\dots,T\,\right\}\vert \,S_t = i\, \&\, S_{t+n}=j\,\right\} \right\vert}{\left\vert\left\{t\in\left\{0,1,\dots,T\,\right\}\vert \,S_t = i\, \right\} \right\vert}\ .$$ While this limit—in theory—might not exist, and be a random quantity when it does, the strong law of large numbers tells us that with probability 1 it's equal to $\ P^n_{i,j}\ $.