Something about Markov chains

Question

We check $P(X_{n+1}\in B|\mathcal{F}_n)=P(X_{n+1}\in B|X_n)$ when we want to prove $X_n,n=1,2,\dots$ is a Markov chain.

Through this equation it seems that $X_n$ is a Markov chain if $X_{n+1}$ is independent with $X_k,k=1,2,\dots,n-1$ but I can not prove it. Can anyone tell me this is right or wrong?

In fact, I don't believe this is true but I think with extra condition this can be right. Can anyone help to find an extra condition to make it right? Thanks!

Answer 1

$ X_{n+1} $ itself is not independent of the past. It is only independent of the past when you specify $ X_n $ (or the distribution thereof). This is easier to see with something like a random walk, where the increments are independent of the past but of course the present value is not. (If you know you were at 10 a step ago, then you know you can't be at -10 next step.)

Answer 2

This is an example of conditional independence. The future is independent of the past given the present. This is not the same as asserting that future and past are independent.