Independence of past and future states in Markov Chains

Question

I have seen this statement in a quiz:

Let $X_i$ denote state $i$ in a Markov chain. It is necessarily true that $X_{i+1}$ and $X_{i-1}$ are uncorrelated.

Apparently, this statement is false but I can't figure out why. I thought that for Markov Chains the past and future states are independent given the present. Did I misunderstand this?

Answer 1

The key phrase is "given the present". If past and future are independent given the present, it doesn't follow that past and future are unconditionally independent.

For example, consider the simple random walk that takes a step either left or right with equal probability. If you know where I am today, then the knowledge of where I was yesterday won't affect where you think I'll be tomorrow. OTOH if you don't know where I am today, then knowing where I was yesterday will affect where you think I will be tomorrow, since tomorrow I can be at most two steps away from where I was yesterday.

Answer 2

Conditioned on $X_i$, $X_{i+1}$ and $X_{i-1}$ are indeed uncorrelated (and actually are much stronger: they are independent; you can check this).

However, think of the following chain: $X_n = B$ where $B$ is some Bernoulli random variable. You can check this is Markov (If I tell you the state $X_{n-1}$, I told you what $X_n$ is so no need to further condition) Note that $X_i = X_j$ for any $i,j$, and these are clearly positively correlated!.