Conditional Probability and Borel-Cantelli theorems

Question

I am given a sequence of events $A_1, A_2, \dots$ that satisfy

$$ \sum_{n \geq k} \mathbb{P}\left(A_n \ \middle|\ \bigcap_{i=k}^{n-1} A_i^c \right) = \infty, \forall k \in \mathbb{N} $$ and asked to conclude that $\mathbb{P}(\limsup_n A_n) = 1$. So far, I've tried:

$$ \mathbb{P}(\limsup_n A_n) = \lim_{k\to\infty} \mathbb{P}(\bigcup_{n \geq k}A_n) = \lim_{k\to\infty} \mathbb{P}\left( \bigcup_{n \geq k} A_n \setminus \bigcup_{i=k}^{n-1}A_i \right) \\ = \lim_{k\to\infty} \sum_{n \geq k} \mathbb{P}\left( A_n \cap \left( \bigcap_{i=k}^{n-1} A_i^c \right) \right) \qquad \text{(union over disjoint events)} $$

However, I'm not sure how to link the fact I'm given about the conditional probabilities with the above limit. Any ideas?

Answer 1

$\newcommand{\P}{\mathbb{P}}$It is equivalent to show the complement occurs almost never: $$\P\left(\bigcup_{k\in\Bbb N}\bigcap_{n\ge k}A_n^c\right)\overset{?}{=}0$$To do so, it is equivalent to show each tail $T_k:=\bigcap_{n\ge k}A_n^c$ has zero probability. For $N>k$ put $T_{k,N}:=\bigcap_{n=k}^NA_n^c$. $$\P(T_{k,N})=\P\left(A_N^c\cap\bigcap_{n=k}^{N-1}A_n^c\right)=\left(1-\P\left(A_N\big|\bigcap_{n=k}^{N-1}A_n^c\right)\right)\P(T_{k,N-1})$$Then: $$\P(T_k)=\lim_{N\to\infty}\P(T_{k,N})=\P(A_k^c)\lim_{N\to\infty}\prod_{m=k+1}^N\left(1-\P\left(A_m\big|\bigcap_{n=k}^{m-1}A_n^c\right)\right)=0$$Because a product of reals $\alpha_\bullet$ in the interval $[0,1)$ converges to zero iff. the series in $(1-\alpha_\bullet)$ tends to $+\infty$ as is well-known. Consider $\ln(1-x)=-x-x^2/2-\cdots\le-x$.

Intuitively, the hypothesis says that the probability of the events is not significantly reduced if you condition on the previous events failing to occur; this suggests that either the 'current' event occurs or one of the previous events should occur, with 'high' probability; that is, something will happen. The result is then not so surprising.

Answer 2

Suppose that there exists $k_0$ such that $\mathbb P\left(\bigcap_{i=k_0}^\infty A_i^c\right)>0$. Then we know, by assumption, that $$\tag{*} \sum_{n=k_0+1}^\infty \mathbb P\left(A_n\cap \bigcap_{i=k_0}^{n-1}A_i^c\right)\frac 1{\mathbb P\left(\bigcap_{i=k_0}^{n-1}A_i^c\right)}=\infty. $$ Since $\left(\mathbb P\left(\bigcap_{i=k_0}^{n-1}A_i^c\right)\right)_{n\geqslant 1}$ converges to a positive value, (*) is equivalent to $$ \tag{**} \sum_{n=k_0+1}^\infty \mathbb P\left(A_n\cap \bigcap_{i=k_0}^{n-1}A_i^c\right)=\infty. $$ This leads to a contradiction, since we sum probabilities of a sequence of pairwise disjoint sequence.

We thus derive that for each $k$, $\mathbb P\left(\bigcap_{i=k}^\infty A_i^c\right)=0$. Since a countable union of events of probability $0$ has zero probability, we get that $\mathbb P(\liminf A_n^c)=0$ and taking the complements gives the wanted conclusion.