Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 1 / b / i / Solution

Independent random variables satisfy, for every finite selection of distinct indices $n_1,\dots ,n_m$ and Borel sets $C_1,\dots ,C_m$,The Second Borel-Cantelli lemma says that if events $E_n$ are independent and ${\sum }_n\mathbb{P}(E_n)=\infty$, thenHere ${\mathrm{limsup}}_n{E}_n={\bigcap }_N{\bigcup }_{n\ge N}{E}_n$ is the event that infinitely many of the $E_n$ occur.