How Do I Know If A Markov Chain Follows The Markov Property? In Probability and Statistics, a Markov Chain is said to have the "Markov Property" if the next state of this Markov Chain only depends on the current state of this Markov Chain.
This being said, I had the following question: Suppose I have some data about a Discrete Markov Process, such as the weather each day (e.g. Sun, Rain, Snow) over a period of days. I believe that my data has 1st Order Markov Property, i.e. the next day's weather only depends on the current weather. I go ahead and fit a 3-State Discrete Markov Chain to this data and estimate the transition probabilities corresponding to the 3 x 3 transition matrix. But how exactly do I determine if the data I have collected actually obeys the Markov Property?
The only thing that comes to mind are simulation methods - that is, if I create a Markov Model and then test this model, and the state sequence transitions reasonably match the weather in the real world, I could informally conclude that my data obeys the Markov Property. However, this seems informal and I was looking for a more "mathematically rigorous" way to determine if my data follows the Markov Property.
Could someone please comment on this?
Note: Apparently, there also exist the "2nd Order Markov Property" in which the next state of this Markov Chain only depends on the current and the previous state of this Markov Chain. I would be curious to know how we could test to see if the same data we collected earlier might obey the "2nd Order Markov Property".

An initial and simple test for this would be to see if the data show evidence of the weather being affected by the weather two days ago when you're already conditioning on the weather one day ago.  To do this you would form a $3 \times 3 \times 3$ contingency table for three consecutive days and use this to conduct a test of conditional independence to see if the Markov property is violated.  If you find evidence that the weather two days ago is not independent of the weather today, conditional on the weather one day ago, that is a violation of the Markov property (and suggests that there is auto-correlation in the data that goes back more than one period).
