Borel-Cantelli Lemmas

These lemmas relate summability of the series of probability masses of a given sequence $\left< A_n \right>$ of events to the probability of the tail event

$\limsup_n A_n = \cap_n \cup_{k\ge n} A_k = [A_n i.o.]$

First lemma:

$\sum_n P[A_n] < \infty$ implies $P[\limsup_n A_n] = 0$ .

Since $P[\limsup_n A_n] = \lim_{n \rightarrow \infty} P[\cup_{k \ge n} A_k] \le \lim_{n \rightarrow \infty}\sum_{k\ge n} P[A_k] = 0$ the conclusion follows.

Second lemma:

$\sum_n P[A_n] = \infty$ and $\left< A_n \right>$ is an independent sequence of events, then $P[\limsup_n A_n] = 1$

$P[\limsup_n A_n] = 1$ iff $P[\liminf_n A_n^C] = 0$ . Since

$P[\liminf_n A_n^C] = \lim_n P[\cap_{k\ge n} A_k^C] \le \lim_n \lim_m P[\cap_{k = n..(n+m)} A_k^C] \\ = \lim_n \lim_m \prod_{k = n..n+m} P[A_k^C] = \lim_n \lim_m \prod_{k = n..n+m} (1 - PA_k) \\ \le \lim_n \lim_m \prod_{k = n..n+m} e^{-PA_k} = \lim_n \lim_m e^{- \sum_{k=n..n+m} PA_k} = 0$

where the equality $1 - x \le e^{-x}$ when $x \ge 0$ has been used.

Bounding lemma:

Let $\liminf_n A_n = \cup_n \cap_{k\ge n} A_k = [A_n a.b.f.m]$

$P[\liminf_n A_n] \le P[\limsup_n A_n]$

Follows since $P[\liminf_n A_n] = \lim_n P[\cap_{k\ge n} A_k] \le \lim_n P[\cup_{k \ge n} A_k] = P[\limsup_n A_n]$

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