Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2009 / B / 4 / a / ii / Solution

The assumptions implyEventually ${\mu }_n>0$, and $X_n=0$ implies $|X_n-{\mu }_n|\ge {\mu }_n$. Thus Chebyshev's inequality givesNo independence between the different $X_n$ is needed; this is a second moment method.