 Solution

ID: 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^{'}}

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. Hence

E [X_{t^{'}} X_{t}] = 0 (t^{'} \neq = t) .

This 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 δ_{t t^{'}}

. It establishes absence of second-order serial correlation, not independence of returns.

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