[kernel_sanders]

Thank you for the clear explanation.

In a previous comment, I sought to further understand how the Lottery possesses the Markov property. Based on your definition above, I can see that it does simply because the distribution X_t of winning numbers has the same dependence on X_{t-1} as it does on X_1, ..., X_{t-1}, that is, zero. Do I have that correct?

[NarcolepticFrog]

Yes, exactly. More generally (and for the same reason), any sequence of iid random variables will form a Markov chain.

For an example without the Markov property, consider the sequence of random variables X_1, X_2, ... with X_1 being either -1 or 1 with equal probability, and X_t being normally distributed with mean X_1 and standard deviation 1.

Knowing the history X_1, X_2, ..., X_{t-1} gives you the exact distribution of X_t (since you know X_1), while only knowing X_{t-1} gives you much less information. This fails to be a Markov chain because the state X_{t-1} doesn't "remember" which of the two possible distributions is being used.

[kernel_sanders]

Awesome. Thanks! Again, very clear. This is interesting stuff. It makes me curious about applications of stochastic processes in general - time to read the course notes you linked. They look like fun problems to program and model.

Edit: Wikipedia, of course, has a good overview of applications of Markov chains. http://en.wikipedia.org/wiki/Markov_chain#Applications