 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2011/a/2/c/iii/solution

Let

E_{n} = {∣ X_{n} - 1∣ = n}

. These events are independent and

\sum_{n \geq 2} P (E_{n}) = \sum_{n \geq 2} 1/ (n lo g n) = \infty

S_{n} / n

had any finite sample-path limit, then

\frac{X _{n + 1}}{n} = \frac{S _{n}}{n} - \frac{n - 1}{n} \frac{S _{n - 1}}{n - 1} ⟶ 0.

But on

E_{n + 1}

∣ X_{n + 1} ∣/ n

is either

1

(n + 2) / n

, contradicting this convergence infinitely often. Therefore the averages have no finite almost sure limit. This is a rare-jump counterexample to a strong law with convergent first-moment averages.

 New to topics? Read the docs here!