 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/6/c/ii/solution

0 \leq P (\frac{S _{n}}{n} > 1 + ϵ) \leq P (\frac{S _{n}}{n} - 1 \geq ϵ) \leq \frac{Var ( S _{n} )}{n ^{2} ϵ ^{2}} = \frac{5 n - 2}{n ^{2} ϵ ^{2}} ⟶ 0.

This actually proves the stronger statement

S_{n} / n P 1

, namely convergence in probability, despite the statistical dependence between adjacent products.

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