Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 2

Kolmogorov zero-one law states that every event in the tail sigma-algebra of an independent sequence has probability zero or one. Removing any fixed number of summands changes $S_n/n$ by a finite random constant divided by $n$, so does not change its extended-real limsup. Therefore $\overline{X}$ is measurable with respect to the tail sigma-algebra. Each event $\{\overline{X}\le q\}$ for rational $q$ has probability zero or one. Monotonicity of these events forces the distribution to be concentrated at one extended-real value, allowing either infinity.

The strong law of large numbers says that for independent identically distributed variables with $\mathbb{E}|X_1|<\infty$,

Pointwise, the positive integers $n$ with $Z\ge n$ are exactly $1,\dots ,\lfloor Z\rfloor$. Thus $Y=\lfloor Z\rfloor ={\sum }_{n\ge 1}{1}_{\{Z\ge n\}}$. The monotone convergence theorem gives $\mathbb{E}Y={\sum }_{n\ge 1}\mathbb{P}(Z\ge n)$. Since $Y\le Z<Y+1$,These inequalities also hold for infinite expectations. This is the tail-sum formula comparison for a nonnegative real variable.

For any $\lambda >0$, the variable $Z=|X_1|/\lambda$ has infinite expectation. Part (a)(ii) therefore forces ${\sum }_n\mathbb{P}(|X_1|\ge \lambda n)=\infty$. Identical distributions give

For each positive integer $m$, the events $\{|X_n|\ge mn\}$ are independent and have divergent total probability by part (i). The Borel-Cantelli second lemma makes them occur infinitely often. Taking the countable intersection over $m$ givesIf ${\mathrm{limsup}}_n|S_n|/n$ were finite along any such sample path, then for some finite path-dependent $C$, $|S_n|\le Cn$ eventually. But $X_n=S_n-S_{n-1}$ would give $|X_n|/n\le C+C(n-1)/n\le 2C$ eventually, a contradiction. ThereforeThis is the divergence of normalized iid partial sums without a first absolute moment.