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

Table of contents
- Solution 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.

 Ancestors (13)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version