# Check memoryless property of a Markov chain

I suspect that a series of observed sequences are a Markov chain…

However how could I check that they indeed respect the memoryless property of

Or at the very least prove that they are Markov in nature? Note these are empirically observed sequences. Any thoughts?

EDIT

Just to add, the aim is to compare a predicted set of sequence from the observed ones. So we’d appreciate comments on as to how best to compare these.

First Order Transition matrix where m=A..E states

Eigenvalues of M

Eigenvectors of M

I wonder if the following would give a valid Pearson $\chi^2$ test for proportions as follows.

1. Estimate the one-step transition probabilities — you’ve done that.
2. Obtain the two-step model probabilities:
3. Obtain the two-step empirical probabilities
4. Form Pearson test statistic

It is tempting for me to think that each $T_U \sim \chi^2_3$, so that the total $T\sim \chi^2_{12}$. However, I am not entirely sure of that, and would appreciate your thoughts on this. I am not likewise not co sertain about whether one needs to be paranoid about independence, and would want to split the sample in halves to estimate $\hat p$ and $\bar p$.