Definition of limiting distribution in a Markov chain -- why do we condition on the initial state? Given a Markov chain $\{X_n \mid n \in \{0, 1, \ldots\}\}$ with states $\{0, \ldots, N\}$, define the limiting distribution as
$$
\pi = (\pi_0, \ldots, \pi_N)
$$
where
$$
\pi_j = \lim_{n \to +\infty} \mathbb{P}\{X_n = j \mid X_0 = i\}
$$
I am confused as to why we condition on $X_0 = i$. What kind of a role does the initial state play? My textbook offers no explanation.

Consider the following system, with $N=3$ :
$$M= \begin{bmatrix}
0.5 & 0.5 & 0 & 0 \\
0.4 & 0.6 & 0 & 0 \\
0 & 0 & 0.3 & 0.7 \\
0 & 0 & 0.2 & 0.8 \\
\end{bmatrix}$$
where
$$M_{i, j} = \mathbb{P}\left\{X_n = j \mid X_{n-1} = i\right\}$$
Notice that
$$M^\infty = \begin{bmatrix} 
\frac 49 & \frac 59 & 0 & 0 \\
\frac 49 & \frac 59 & 0 & 0 \\
0 & 0 & \frac 29 & \frac 79 \\
0 & 0 & \frac 29 & \frac 79 \\ \end{bmatrix}$$
Can you say that $M^\infty\,_{i,j}$ doesn't depend on $i$?
