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.

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Yes. This is a simple consequence of the Kolmogorov 0-1 law as the event {X > x} is in a tail event. The proof of the 0-1 law is nearly identical to the one given here.