Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / A / 4 / b

The tail sigma-algebra isFix $S\in \mathcal{T}$. independence of the variables and the pi-systems criterion imply independence of $\sigma (X_1,\dots ,X_n)$ from $\sigma (X_{n+1},X_{n+2},\dots )$. Hence $S$ is independent of every finite-head sigma-algebra ${\mathcal{H}}_n=\sigma (X_1,\dots ,X_n)$.

The increasing union ${\bigcup }_n{\mathcal{H}}_n$ is an algebra, hence a pi-system, and generates ${\mathcal{H}}_{\infty }=\sigma (X_1,X_2,\dots )$. Another pi-lambda argument extends the equality $\mathbb{P}(S\cap A)=\mathbb{P}(S)\mathbb{P}(A)$ from that union to all of ${\mathcal{H}}_{\infty }$. Since $S$ itself belongs to ${\mathcal{H}}_{\infty }$, take $A=S$:ThereforeThis is Kolmogorov zero-one law; no moment assumptions on the variables are needed.

Let $C$ be the event that the sample averages have a limit in the extended real number line. It is measurable: the bounded sequence $\arctan S_n$ converges exactly when it is Cauchy, and the Cauchy condition is a countable combination of measurable inequalities. Limits at $\pm \pi /2$ correspond to extended limits $\pm \infty$.

For each fixed $m$, remove the first $m-1$ terms and setThe difference $S_n-S_n^{(m)}=n^{-1}{\sum }_{k<m}{X}_k$ tends to zero for every outcome, since the finitely many variables are real-valued. Thus these two sequences have an extended limit simultaneously. Equivalently their arctangents differ by a quantity tending to zero, because arctangent is Lipschitz. The event $C$ is therefore expressible using $X_m,X_{m+1},\dots$ alone. Since this holds for every $m$, $C\in \mathcal{T}$.

Kolmogorov's law gives $\mathbb{P}(C)=0$ or one. In the latter case the measurable extended variable $\mathrm{limsup}{S}_n$, redefined on a null set if needed, is the almost sure convergence. In the former there is almost surely no extended-real limit. Hence extended convergence occurs almost surely, or divergence occurs almost surely. The law need not make the value of a possible limit finite.