Discuss when $\{\limsup \sum_{k=1}^n X_k> 0 \}$ is a tail event for the sequence $(X_n)_{n\in \mathbb{N}}$

Question

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of random variables on a measure space $(\Omega,\mathcal{F})$. Denote the tail $\sigma$-algebra $\mathcal{T} = \cap_{n=1}^{\infty}\mathcal{T}_n$. Let $S_n = \sum_{k=1}^n X_k$ and $\mathcal{T}_n=\sigma(X_k \textrm{, }k\ge n)$. Please give examples when $\{\limsup S_n> 0 \} \in \mathcal{T}$, and when $\{\limsup S_n > 0 \} \notin \mathcal{T}$.

Answer 1

The event can be in the tail sigma field in some special cases: for example, if $X_i \leq 0$ for all $i$ then the event is the empty set so it belongs to the tail sigma field. What you can show is that the event does not always belong to the tail sigma filed. You can do this with a counter-example: take $X_2=X_3=... =0$ and $X_1$, say has normal distribution. [The tail sigma field has only the empty set and the whole space in this case].