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I am reading the Durret's Probability Theory and Examples. In the book, tail $\sigma$ field is defined as $\mathcal T=\cap_n^\infty\sigma(X_n,X_{n+1},...)$. Then the book shows some easy example as follows:

Let $S_n=X_1+X_2+...+X_n$ $$\{\lim_{n\to\infty}S_n\ exists\}\in\mathcal T$$ $$\{\limsup_{n\to\infty}S_n>0\} \notin\mathcal T$$ $$\{\limsup_{n\to\infty}\frac{S_n}{c_n}>x\}\in\mathcal T,c_n\to \infty$$ I pondered these examples for 2 hours, but still could not understand. I am so damn, can anyone give some intuitive interpretations or a written out proof? thanks in advance.

Falrach
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Fellow InstituteOfMathophile
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  • For the second example look here: https://math.stackexchange.com/questions/2713080/events-in-tail-sigma-algebra?rq=1 and https://math.stackexchange.com/questions/707647/about-example-1-8-2-in-durrett-probability-theory-and-examples?rq=1 For the third look here: https://math.stackexchange.com/questions/3128310/tail-event-example?rq=1 – Falrach Nov 13 '19 at 11:18
  • thanks, I understand now. – Fellow InstituteOfMathophile Nov 13 '19 at 14:21

1 Answers1

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For the first example note that for a $n\in\Bbb N$ $$\sum_{i=1}^\infty X_i \text{ exists} \Leftrightarrow \sum_{i=n}^\infty X_i \text{ exists}$$ where the right side is clearly $\sigma (X_n , X_{n+1} , \ldots )$ measurable. Since this holds for every $n$, the left side is $\mathcal T$ measurable.

Falrach
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