When is an i.i.d Bernoulli Process indepent increment?

Question

I was reading anarticle that i.i.d bernoulli process are markov and independent increment process are markov too. So, I was wondering if an i.i.d. bernoulli process can be independent increment.

Answer 1

No, it does not have independent increments (except in the trivial case where $p=0$ or $1$). $$\operatorname{Cov}(X_3-X_2, X_2-X_1) = E[(X_3-X_2)(X_2-X_1)] -0\\=E(X_1)E(X_2)+E(X_2)E(X_3) - E(X_1)E(X_3)-E(X_2^2)\\=E(X)^2-E(X^2)\\=-\operatorname{Var}(X)$$

Put more intuitively, for example, if you know you jumped up, you know you're at $1,$ so you are either staying put or jumping down, certainly not up.