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Suppose that $X_1,X_2,\dots$ is a sequence of random variables on some probability space. The tail $\sigma$-algebra $\mathcal{T}$ is defined as the intersection of $\sigma$-algebras $\mathcal{T}:=\bigcap_n\mathcal{F}_n$, where $\mathcal{F}_n=\sigma(X_n,X_{n+1},\dots)$ is the $\sigma$-algebra generated by $X_n,X_{n+1},\dots$.

I know that $\{\sum_{k=1}^n X_n\ \text{converges}\}$ is a tail event. But is $\{\frac1n\sum_{k=1}^n X_k\ \text{converges}\}$ a tail event?

Xiang Yu
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    Basically, if you can change values of any finite collection of the $X_k$ without changing whether the event occurs, it's a tail event. That's the idea behind the definition. – Ian May 29 '16 at 13:25

1 Answers1

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For every fixed $m\in\mathbb N$ we have:

$$\frac1n\sum_{k=1}^nX_k\text{ converges}\iff\frac1n\sum_{k=m}^nX_k\text{ converges}$$

or equivalently $$\{\frac1n\sum_{k=1}^nX_k\text{ converges}\}=\{\frac1n\sum_{k=m}^nX_k\text{ converges}\}$$

while evidently $\{\frac1n\sum_{k=m}^nX_k\text{ converges}\}\in\mathcal F_m$.

This allows the conclusion that this event is an element of $\mathcal T:=\bigcap_m\mathcal F_m$.

drhab
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