Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 2

A real random variable is a measurable map from $(\Omega ,\mathcal{F})$ to $\mathbb{R}$ with its Borel sigma-algebra. Equivalently, $\{X\le x\}\in \mathcal{F}$ for every real $x$. Its cumulative distribution function isIt is nondecreasing, right-continuous, and tends to zero and one at the two infinities.

Let ${\mu }_X$ be the distribution measure of $X$. The measurable change-of-variables rule for expectations givesFor the second equality, let $U$ be uniform on $(0,1)$ and $Q=F^{\leftarrow }$. The quantile function is measurable and satisfiesThis equivalence uses monotonicity and right-continuity of $F$. Consequently $\mathbb{P}(Q(U)\le x)=F(x)$: $Q(U)$ has the same distribution as $X$. In particular ${\int }_0^1|Q(u)|du=\mathbb{E}|X|<\infty$, so the signed integral is legitimate. Hence the quantile expectation formula isNo continuity or strict increase of the distribution function is required.

The variables are independent and identically distributed if they all have one common probability distribution and every finite selection satisfiesfor distinct indices and arbitrary Borel sets. The condition concerns joint distributions, not merely pairwise correlations.

The strong law of large numbers states that if $X_n$ are independent and identically distributed real random variables with $\mathbb{E}|X_1|<\infty$, thenThe first absolute moment condition is part of the standard statement.

The zero means and orthogonality give the exact second momentThus $A_N\to 0$ in $L^2$. Chebyshev's inequality gives the requested weak law of large numbers:Independence is unnecessary; the stated second-moment orthogonality is sufficient.

For an explicit almost surely convergent subsequence take $N_j=2^j$. For each positive integer $m$,The First Borel-Cantelli lemma makes only finitely many such exceedances occur. Intersecting the resulting probability-one events over $m$ gives $A_{2^j}\to 0$ almost surely.