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

Table of contents
- Solution i

Solution

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} .

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} .

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

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