Borel-Cantelli Lemma and Its Connection to Ergodic Theory

The Borel-Cantelli Lemmas

First Borel-Cantelli Lemma

Let ${E_n}$ be a sequence of events in a probability space. If the sum of their probabilities is finite, i.e.

$$ \sum_{n=1}^{\infty} P(E_n) < \infty$$

then the probability that infinitely many of these events occur is zero:

$$ P(\limsup E_n) = P\left(\bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k\right) = 0. $$

Second Borel-Cantelli Lemma

For a sequence of independent events ${E_n}$, if the sum of their probabilities is infinite, i.e.

$$ \sum_{n=1}^{\infty} P(E_n) = \infty$$

then the probability that infinitely many of these events occur is one:

$$ P(\limsup E_n) = P\left(\bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k\right) = 1. $$

Connection to Ergodic Theory

Ergodic theory studies the long-term average behavior of systems over time. A key result in ergodic theory is the Ergodic Theorem, which essentially states that for ergodic systems, the time average of a function along the orbits of almost every point converges to the space average of that function.

Link with Borel-Cantelli Lemma

• The Borel-Cantelli Lemma and ergodic theory intersect in the concept of recurrence and the behavior of certain events or sequences over time.
• In ergodic theory, it can be shown for certain systems that the measure of a set being revisited infinitely often is either zero or one, similar to the binary outcomes in the Borel-Cantelli Lemma.
• The second part of the Borel-Cantelli Lemma, which deals with the occurrence of events under conditions of independence and infinite sum of probabilities, mirrors the almost sure recurrence in ergodic systems.