Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 6 / c / i

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- Solution i

Solution

Write

A_{i} = Y_{i - 1} Y_{i}

, so

S_{n} = \sum_{i = 1}^{n} A_{i}

. Each product has mean one, and its variance is

Var (A_{i}) = E (Y_{i - 1}^{2}) E (Y_{i}^{2}) - 1 = 2 \cdot 2 - 1 = 3.

Adjacent products share one factor, giving covariance one by part (a)(iii). Products whose indices differ by at least two involve disjoint independent random variables, so their covariance is zero. There are

n - 1

adjacent pairs. By the variance of a sum,

E S_{n} = n, Var (S_{n}) = 3 n + 2 (n - 1) = 5 n - 2, n \geq 1.

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