TailSigmaAlgebra

A TailSigmaAlgebra is a concept in probability theory and measure theory that describes a collection of events whose occurrence or non-occurrence is determined by the "tail" of an infinite sequence of random variables. More formally, let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables. The tail sigma-algebra, denoted by $\mathcal{T}$, is the set of all events $A$ such that for any $k \in \mathbb{N}$, the event $A$ is independent of the events $X_1, \dots, X_k$. In other words, knowing the values of the first $k$ random variables does not provide any information about whether event $A$ occurs.

The tail sigma-algebra can be constructed as the intersection of sigma-algebras generated by the "tails" of the sequence. Specifically, $\mathcal{T} = \bigcap_{k=1}^{\infty} \sigma(X_k, X_{k+1}, \dots)$, where $\sigma(X_k, X_{k+1}, \dots)$ is the sigma-algebra generated by the random variables from index $k$ onwards.

A fundamental result related to the tail sigma-algebra is the Borel-Cantelli lemma, specifically its second part. The second Borel-Cantelli lemma states that if the sum of probabilities of a sequence of independent events converges, then the probability that infinitely many of these events occur is zero. This lemma is often used to show that the tail sigma-algebra is trivial, meaning it only contains events of probability 0 or 1, under certain conditions, such as when the random variables are independent and identically distributed. This triviality property is crucial in many advanced probability theorems.