Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / A / 2

A real-valued random variable is a map $X:\Omega \to \mathbb{R}$ measurable from $(\Omega ,\mathcal{F})$ to the Borel real line. Equivalently,Its values are finite real numbers here; extended-real-valued random variables require a correspondingly enlarged target.

For two real-valued random variables and each real $a$,If $X+Y<a$, density of the rationals supplies $q$ with $X<q<a-Y$; conversely either pair of inequalities in the union implies $X+Y<a$. Every set in the countable union is measurable, so the half-line criterion proves measurability of $X+Y$. Its values are real, and induction gives every finite sum $X_1+\cdots +X_n$ is a real-valued random variable.

Expand the fourth power and take expected values. independence factors products involving distinct indices, and centering makes every term with an index occurring just once have expected value zero. The only surviving index patterns are four equal indices and two pairs. There are six orderings of each pair pattern, soUniform boundedness gives $\mathbb{E}{X}_i^4\le c^4$ and $\mathbb{E}{X}_i^2\le c^2$. HenceIdentical distributions were never used: only independence, the common bound, and zero means are needed.

The summands are nonnegative, so monotone convergence theorem, equivalently Tonelli's theorem for a sum, givesA nonnegative random variable with finite expected value is finite almost surely. Thus the displayed nonnegative series converges almost surely, forcing its individual terms to tend to zero. Taking fourth roots provesThis is a fourth-moment proof of a strong law of large numbers for these bounded independent variables, without an identical-distribution assumption.