wiki uses this example to illustrate Markov chains.

The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the preceding day, can be represented by a transition matrix:

${\displaystyle P={\begin{bmatrix}0.9&0.1\\0.5&0.5\end{bmatrix}}}$

The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. The columns can be labelled "sunny" and "rainy", and the rows can be labelled in the same order.

The weather on day 1 is known to be sunny. This is represented by a vector in which the "sunny" entry is 100%, and the "rainy" entry is 0%:

${\displaystyle \mathbf {x} ^{(0)}={\begin{bmatrix}1&0\end{bmatrix}}}$

for day n + 1(Note: the original value on wiki is n, which seems to be incorrect)

${\mathbf {x}}^{{(n)}}={\mathbf {x}}^{{(0)}}P^{n}$

The superscript (n) is an index, and not an exponent.

is $P^n$ here the n-Step Transition Probabilities?


1 Answer 1


Yes, $P^n$ represents the $n$-step transition probability matrix. Here is one way to show that. According to Chapman-Kolmogorov equations, the $(n+m)$-step transition probabilities can be expressed as the sum of products of $n$-step and $m$-step probabilities,as follows:

$$ p_{ij}^{(n+m)}=\sum_{l}p_{il}^{(n)}p_{lj}^{(m)} $$

or, using matrix notation as $P^{(n+m)}$=$P^{(n)}P^{(m)}$. Substituting in the last expression $n-1$ for $n$ and $1$ for $m$ we get $P^{(n-1+1)}$=$P^{(n-1)}P^1$, that is $P^{(n)}$=$P^{(n-1)}P^1$. Applying recursion to the right side, we arrive at


in other words, the $n$-step transition matrix is the n-th power of the 1-step transition matrix.

  • $\begingroup$ Thanks, your answer is very helpful. Given day1 is non-probabilistic, consider day2, is the n of "𝑛-step" equal to 1? $\endgroup$
    – JJJohn
    Commented Aug 6, 2019 at 12:45

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