Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 25J (Probability and Measure)

Events $(A_n)$ are independent events if for every finite list of distinct indices $i_1,\dots ,i_k$, $\mathbb{P}({\bigcap }_j{A}_{i_j})={\prod }_j\mathbb{P}(A_{i_j})$. Random variables are independent random variables if for every such list and Borel sets $B_j\subseteq \mathbb{R}$, $\mathbb{P}(X_{i_j}\in B_j\text{ for all }j)={\prod }_j\mathbb{P}(X_{i_j}\in B_j)$. This is equivalent to independence of their generated sigma-algebras; pairwise independence alone is insufficient.

The tail sigma-algebra isKolmogorov zero-one law states that if the random variables are mutually independent, every event in $\mathcal{T}$ has probability zero or one. No identical-distribution assumption is needed.

The first event is a tail event: for any $m$ it equals ${\bigcup }_{N\ge m}{\bigcap }_{n\ge N}\{X_n\le 0\}$, measurable in $\sigma (X_m,X_{m+1},\dots )$.

Convergence of the partial sums is also a tail event. For any $m$, subtracting the finite real sum $X_1+\cdots +X_{m-1}$ shows that convergence is equivalent to convergence of ${\sum }_{j=m}^n{X}_j$. The Cauchy criterion expresses the latter using countable unions and intersections of inequalities involving these tail sums, so it belongs to every tail sigma-algebra.

The event that the partial sums are nonpositive infinitely often is not generally a tail event. For a counterexample let $X_1$ take values $-1$ and one on a two-point probability space, each with positive probability, and let $X_n=0$ for $n\ge 2$. The event is then $\{X_1=-1\}$, while $\sigma (X_2,X_3,\dots )$ is trivial. The variables in this example are even independent. Its status can be different for particular sequences, but it is not tail-measurable in general.

Split the overlapping products into two nonoverlapping families:Within each sum the summands depend on disjoint independent pairs and are independent identically distributed Bernoulli variables of mean $1/4$. The two families need not be independent of each other. The strong law of large numbers says that averages of independent identically distributed integrable random variables converge almost surely to their expectation. Applying it separately to each family and intersecting the two probability-one events gives