 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2007/a/2/b/ii/solution

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.

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

\frac{1}{n} k = 1 \sum n X_{k} ⟶ 0 almost surely .

This is a fourth-moment proof of a strong law of large numbers for these bounded independent variables, without an identical-distribution assumption.

 New to topics? Read the docs here!