Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / N / 1 / d

Linearity of expectation and the vanishing of cross covariances for independent variables giveThe expectation formula does not require independence; independence is used for the variance sum.

Write ${\bar{X}}_n=n^{-1}{\sum }_{j=1}^n{X}_j$ and apply the stated second-moment Markov inequality to $Y={\bar{X}}_n-\mu$. The preceding calculation gives $\mathbb{E}{Y}^2={\sigma }^2/n$. Thus for every $\epsilon >0$,This is convergence in probability to the common mean, proving the finite-variance weak law of large numbers. It also covers ${\sigma }^2=0$, when the bound is identically zero.