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In the below passage, I was wondering what $\mathbf{1}$ signifies:

How much time does a Markov chain spend in state $i$, in the long term? That is, what is the long term fraction of time that $X_n=i$ ? We can write this long term fraction of time as $ \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{m=0}^{n-1} \mathbf{1}${$X_m=i$}

The text goes on to say that $\frac{1}{n} \sum_{m=0}^{n-1} \mathbf{1}${$X_m=i$} counts the number of steps among {$0,1,\ldots,n-1$} where $X_m= i$ - however, being unsure of what $\mathbf{1}$ means, notationally, I have been unable to see this.

I would greatly appreciate an explanation as to the meaning of $\mathbf{1}$ and how such means that expressions counts the number of steps among that set where $X_m = i$.

Princess Mia
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    Indicator / characteristic function. Alternatively written by $\chi_{{X_m=i}}$ – FShrike Nov 28 '23 at 12:28
  • It means $1$ if $X_m=i$, and $0$ if $X_m\ne i$. (In case you don't know what an indicator / characteristic function is!) – TonyK Nov 28 '23 at 12:34
  • Note: the sum counts the number of such occurrences, but your expression has a factor of $\frac 1n$ so it counts the portion of ${0,1,\cdots, n-1}$ that passes the test. – lulu Nov 28 '23 at 12:36
  • https://en.wikipedia.org/wiki/Indicator_function – Filippo Nov 28 '23 at 12:36

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$\mathbf 1\{X_m = i\}$ is the indicator function or characteristic function of the event $\{X_m = i\}$, defined by $$\mathbf 1\{X_m = i\} = \begin{cases} 1 & X_m = i, \\ 0 & X_m \ne i.\end{cases}$$ So $\sum\limits_{m=0}^{n-1} \mathbf 1\{X_m = i\}$ counts the number of visits by the chain to state $i$ up to time $n-1$. Thus $\frac 1n \sum_\limits{m=0}^{n-1} \mathbf 1\{X_m = i\}$ is the proportion of time before $n$ spent in state $i$.

In the context of Markov chains, given some irreducible chain with invariant distribution $\pi$, the ergodic theorem states that $$\frac 1n \sum_{m=0}^{n-1} \mathbf 1\{X_m = i\}\to \pi(i) \quad \text{as } n \to \infty $$ with probability $1$. That is, the long-term proportion of time spent in each state is described precisely by the invariant distribution.

kipf
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