Literature request on Markov Chains which state transition probability matrix evolves over time I want to know is there any literature on markov chains who's state transition probability matrix evolves over time? 
For instance, I have 2 states, 1 and 2. With 
$$P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} = \begin{bmatrix} 0.9 & 0.1 \\ 0.8 & 0.2 \end{bmatrix}$$
Where $p_{ij}$ indicates the change that you jump from state $i$ to $j$.
Now at time $k$ I jump from 1 to 2, which has a 0.1 probability. The next instance $k+1$ I have a probability of 0.2 of staying in state 2. Now if I stay at $k+1$ in state 2 then the probability of staying in state 2 would increase and the probability of jumping from state 2 to 1 would decrease. Meaning that $P_{k+2}$ would, for instance have evolved to
$$P_{k+2} = \begin{bmatrix} 0.9 & 0.1 \\ 0.5 & 0.5 \end{bmatrix}$$
If I again stay in state 2 it increases further...
$$P_{k+3} = \begin{bmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{bmatrix}$$
If I then jump from state 2 to 1 it decreases...
$$P_{k+4} = \begin{bmatrix} 0.9 & 0.1 \\ 0.6 & 0.4 \end{bmatrix}$$
Now I want to know if there is any literature regarding this?
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I believe my question is very much related to https://mathoverflow.net/questions/168398/time-inhomogeneous-markov-chains
 A: This corresponds to any bivariate renewal process where one is given some positive integer valued independent $(U_i)$ i.i.d. and $(V_i)$ i.i.d. and one is at state $1$ at time $n$ if there exists some $k$ such that$$\sum_{i=1}^k(U_i+V_i)\leqslant n\lt U_{k+1}+\sum_{i=1}^k(U_i+V_i),$$ 
and one is at state $2$ otherwise.
The distributions of every $U_i$ and every $V_i$ are as follows: for every $n$, $P(U_i\geqslant n+1\mid U_i\geqslant n)$ is the probability to stay at state $1$ one more step when one is at state $1$ for $n$ steps, likewise, $P(V_i\geqslant n+1\mid V_i\geqslant n)$ is the probability to stay at state $2$ one more step when one is at state $2$ for $n$ steps.
In the example given in the post, this means that $$P(V=1)=0.8,\quad P(V=2)=0.2\times0.5,\quad P(V=3)=0.2\times0.5\times0.2.$$
A: Yes, this is a so-called "inhomogenous Markov chain". Some authors use the term "non-homogenous".
Please note that you are interested in Markov chains with discrete time which are different from models with continuous time.
