Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / A / 2 / b

Expand the fourth power and take expected values. independence factors products involving distinct indices, and centering makes every term with an index occurring just once have expected value zero. The only surviving index patterns are four equal indices and two pairs. There are six orderings of each pair pattern, so

E S_{n}^{4} = \sum_{i = 1}^{n} E X_{i}^{4} + 6 \sum_{i < j} E X_{i}^{2} E X_{j}^{2} .

(1)

Uniform boundedness gives

E X_{i}^{4} \leq c^{4}

and

E X_{i}^{2} \leq c^{2}

. Hence

E S_{n}^{4} \leq n c^{4} + 6 (2 n) c^{4} = (3 n^{2} - 2 n) c^{4} \leq 3 c^{4} n^{2} .

(2)

Identical distributions were never used: only independence, the common bound, and zero means are needed.

ii

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Solution

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The summands are nonnegative, so monotone convergence theorem, equivalently Tonelli's theorem for a sum, gives

E \sum_{n = 1}^{\infty} (\frac{S _{n}}{n})^{4} = \sum_{n = 1}^{\infty} \frac{E S _{n}^{4}}{n ^{4}} \leq 3 c^{4} \sum_{n = 1}^{\infty} n^{- 2} < \infty.

(1)

A nonnegative random variable with finite expected value is finite almost surely. Thus the displayed nonnegative series converges almost surely, forcing its individual terms to tend to zero. Taking fourth roots proves