Chapman-Kolmogorov Equation

Memoryless Variable stochastic
For n-step transition
To define propabilities $P_{ij}^{n}$ to be the probability that a process in state $i$ will be in state $j$ after $n$ transitions.
That is $P_{ij}^{n} = P {X_{n + k = j} ∣ X_{k} = i}, n \geq 0, i, j > 0$
- after n steps
The equation gives us $P_{ij}^{n + m} = k \sum P_{ik}^{n} . P_{kj}^{m}$
- Think of going from $i$ to $j$ in $n + m$ steps with an intermediate strop in state $k$ after $n$ steps, like the end of the trees
- then sum all possible $k$ states
If $P^{(n)}$ denotes the matrix of $n$ -step transition probabilities $P_{ij}^{n}$ then $P^{(n + m)} = P^{(n)} . P^{(m)}$
- implies that n-step transition matrix maybe optained by multiplying the matrix $P$ by itself n times
example:

StrixTheKiet Notes