04 | November | 2012 | Eventually Almost Everywhere

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.

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)

Eventually Almost Everywhere

A blog about probability and olympiads by Dominic Yeo

Daily Archives: November 4, 2012

An obvious remark

Aside

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