Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 2 / c / i / Solution

For fixed $\epsilon >0$ and $n>\epsilon$,so there is convergence in probability to one. The absolute difference is exactly $n$ on the exceptional event, henceFor every $p>1$ this tends to infinity, so convergence in Lp fails. In contrast its first moment is $1/\log n\to 0$, so convergence actually holds in $L^1$.