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# Divergence of the harmonic series, in a probability context

Posted 01-27-2009 at 10:42 PM by Marid Audran
Updated 02-06-2009 at 11:54 PM by Marid Audran (Adding a remark that it's not obvious)

I thought of this while reading in Rosenthal's book about the Borel-Cantelli lemma and trying to think up an example.

I learned a long time ago that the harmonic series
$\sum_{n=1}^\infty \frac 1n$
diverges to infinity. Nonetheless, when I thought of the following example (which boils down to this fact), it felt less than completely intuitive to me.

Consider an infinite sequence of real numbers $x_1, x_2, x_3, ...$, where each number is chosen randomly from the interval $(0,1)$ with the uniform distribution. (Use the Axiom of Choice where appropriate to do this.) What is the probability that infinitely many of the $x_n$ satisfy $x_n < \frac 1n$? It must be 1!

Edited to add: Rosenthal has an exercise that's similar to this example. Imagine an infinite sequence of integers, where the nth integer is chosen randomly from the set {1,2,...,n} with the uniform distribution. What is the probability that the sequence contains infinitely many 5's?

Edited to add: I realized that my example is not as "obvious" as I might have thought it was. For one thing (in fact, this may be the only thing), my example depends on the fact that the x_i are independent. (The independence is why Borel-Cantelli can be used here.) Taken by itself, the fact that the individual probabilities sum to infinity does NOT imply that "infinitely many are true" has probability 1. Indeed, consider an alternate experiment: Suppose $x_1$ is chosen randomly from (0,1) and all $x_i, i>1$, are equal to $x_1$. What is the probability that infinitely many of the $x_n$ satisfy $x_n < \frac 1n$? It must be zero!
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