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In Durrett's Probability (4th edition), an example of a tail event (an event in the tail sigma-field $\bigcap_n \sigma(X_n, X_{n+1}, \dots)$) is the following: given independent random variables $X_1, X_2, \dots,$ and their partial sums $S_n = \sum_{i=1}^n X_i$, the following event is a tail event (Example 2.5.2):

$$ \{ \limsup_n S_n > x c_n \}, \; c_n \to \infty. $$

I understand the high level idea of a tail event (i.e. only depends in the asymptotic behavior of the sum since $c_n$ go to infinity) but I cannot articulate a rigorous explanation. Is there a concrete way to show this?

VHarisop
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1 Answers1

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It's pretty much just like your intuitive explanation. For a any $n$, we can write $$ \frac{S_{n+k}}{c_{n+k}} = \frac{X_1+\ldots+X_n}{c_{n+k}} + \frac{X_{n+1}+\ldots +X_{n+k}}{c_{n+k}} $$ Then, since the first term converges to zero as $k\to\infty$, we have $$ \limsup_m \frac{S_m}{c_m} = \limsup_{k} \frac{S_{n+k}}{c_{n+k}} = \limsup_k \frac{X_{n+1}+\ldots+X_{n+k}}{c_{n+k}} $$ so is clearly $\sigma(X_{n+1},\ldots)$-measurable.

spaceisdarkgreen
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  • Thank you for your time. Why is it obvious that $\limsup_m \frac{S_m}{c_m} = \limsup_k \frac{S_{n+k}}{c_{n+k}}$? – VHarisop Feb 27 '19 at 04:38
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    For any $n$, $\limsup_m x_m = \limsup_{k} x_{n+k}.$ This is because (similarly to the limit) the limsup only depends on the tail of the sequence. It is the upper limit that the sequence reaches infinitely many times: any amount of initial values tells you nothing about what happens infinitely many times. – spaceisdarkgreen Feb 27 '19 at 06:18
  • How is it guaranteed that $\lim_{k -> \infty} \frac{X_1+\ldots+X_n}{c_{n+k}} =0$ when $X_i$s may take infinite values. Or are those cases excluded. – somitra Jan 08 '21 at 16:32
  • @somitra I would assume that case is excluded in this context. Otherwise the assertion (and not just the proof) is wrong. Consider for instance $X_1$ being zero with 50% probability and $\infty$ the other 50% of the time, and all other $X_i$ zero with 100% probability. – spaceisdarkgreen Jan 08 '21 at 16:45
  • @spaceisdarkgreen Thanks for the clarification. I was splitting my hairs for hairs over it for days. – somitra Jan 08 '21 at 16:46
  • Isn't lim sup is always a tail event? Why is the 1st part of your explanation even needed? See here https://math.stackexchange.com/a/1281014/342736 for example. – Maverick Meerkat Dec 13 '21 at 09:07
  • @Maverick That shows that the limsup of a sequence of events is in the events’ tail field. I don’t see how this problem (which directly involves limsup of reals, not events) follows immediately from that, but I haven’t thought about it much. – spaceisdarkgreen Dec 13 '21 at 15:21