 Solution

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

For fixed

ε > 0

and

n > ε

P (∣ X_{n} - 1∣ > ε) = \frac{1}{n l o g n} \to 0,

n

on the exceptional event, hence

E ∣ X_{n} - 1 ∣^{p} = \frac{n ^{p - 1}}{lo g n} .

For every

p > 1

this tends to infinity, so convergence in Lp fails. In contrast its first moment is

1/ lo g n \to 0

, so convergence actually holds in

L^{1}

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