An obvious remark


An obvious remark:

If a sequence of independent random variables X_n converge almost surely to some limit X, 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 X_n is \mathcal{F}_n-measurable. But by independence, we might as well take \mathcal{F}_n=\sigma(X_n). Then the limit variable X is independent of \mathcal{F}_n for all n, and thus independent of \cup F_n=\mathcal{F}\supset \sigma(X). If X is independent of itself, it must be almost surely constant.