Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 26K / a / Solution

The strong law of large numbers states that if $(X_i{)}_{i\ge 1}$ are independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, thenalmost surely. Under the stronger fourth-moment hypothesis, it has a short proof. Put $\mu =\mathbb{E}{X}_1$, $Y_i=X_i-\mu$, and $S_n={\sum }_{i=1}^n{Y}_i$. Independence and centering make every term in the expansion of $S_n^4$ vanish unless each index occurs at least twice, soFor every $\epsilon >0$, the Markov inequality givesThe probabilities are summable, so the first Borel-Cantelli lemma says that $|S_n|>\epsilon n$ occurs only finitely often almost surely. Apply this simultaneously to $\epsilon =1,1/2,1/3,\dots$ to obtain $S_n/n\to 0$ almost surely, which is the claimed law.