Expectation of transition in a Markov process I have the following transition matrix for a Discrete Markov process with 3 states, say states A, B and C:
\begin{bmatrix}0.985992 &  0.0134092 & 0.000599272\\
               0.0265225 &  0.949748 & 0.0237297 \\
               0.00196398 & 0.0895138 & 0.908522
\end{bmatrix}
As I was doing some calculations I noticed that if I simulated data the number of transitions from any two states is roughly the same going both ways (e.g. the number of states going from A to B, A->B, is roughly the same as from B to A, B->A), for example for 100000 iterations:

*

*A->B: 8164 and B->A: 8340

*B->C: 7602 and C->B: 7779

*A->C: 365 and C->A: 188

In an intuitive way, Why is this? I understand it if I calculate the expectations, for example:
$\mathbb{E}(A \rightarrow B)=\pi_A P(B | A)$
and this is actually very close to $\mathbb{E}(B \rightarrow A)=\pi_B P(A | B)$, where $\pi_X$ is the stationary distribution for state $X$.
I just don't quite grasp why this happens. Any insights? Any reference or leads appreciated.
 A: This is a peculiar property of your transition matrix, but it doesn't have to be that way.
If your system satisfies $\pi_A P(B|A) = \pi_B P(A|B)$ for all states $A$ and $B$, your system is said to satisfy the property of detailed balance. Transition matrices that satisfy this property, are called reversible.
But there are Markov chains with stationary distributions that don't satisfy this property. E.g., you could try to find a chain that has as stationary distribution the uniform distribution but a transition matrix that is not symmetric. That means, find a transition matrix where all rows and all columns sum to one, but that is not symmetric. A simple example would be:
$$
T = \left[\begin{matrix}
              0.1 & 0.1 & 0.8 \\
              0.2 & 0.8 &   0 \\
              0.7 & 0.1 & 0.2 \\
          \end{matrix}\right].
$$
Clearly, the stationary distribution is the uniform distribution $\pi_U = (1/3, 1/3, 1/3)$, and this Markov chain is not reversible.
A good, short introduction to Markov chains can be found e.g. in

Bishop, Christopher M., and Nasser M. Nasrabadi. Pattern recognition and machine learning. Vol. 4. No. 4. New York: springer, 2006.

