An obvious remark:
If a sequence of independent random variables converge almost surely to some limit , this limit must be a constant (almost surely).
I’ve been thinking about the Central Limit Theorem about related Large Deviations results this afternoon, and wasted almost an hour worrying about situations which were effectively well-disguised special cases of the above.
Why is it true? Well, suppose each is -measurable. But by independence, we might as well take . Then the limit variable is independent of for all , and thus independent of . If is independent of itself, it must be almost surely constant.
- Weak Law of Large Numbers and Central Limit Theorem via the Levy’s continuity theorem. (maikolsolis.wordpress.com)
- republicans reject the central limit theorem? (orgtheory.wordpress.com)
- Poisson processes appropriate for today (gottwurfelt.wordpress.com)