Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S9 / 2 / b / Solution

Expand $X_t{X}_{t^{\prime }}$ into products of the independent mean-zero signs. Each increment is the sum of two monomials, $a_t$ and $a_{t-2}{a}_{t-1}$. At distinct times, no monomial from one increment has exactly the same set of sign indices as a monomial from the other. Every product therefore contains at least one sign to an odd power, whose average is zero. HenceThis argument covers overlapping driver sets at separations one and two, not just the disjoint case at larger separation. At equal times the value is $2$, so the complete autocovariance is $2{\delta }_{t{t}^{\prime }}$. It establishes absence of second-order serial correlation, not independence of returns.