Exercise 10. Borel-Cantelli Lemmas.
Let (A_n) be a sequence of events of some probability space ((Omega), (F), (P)).
Assume that (sum_n P(A_n) < +infty).
(a) Show that (P(cup_{k geq n} A_k) longrightarrow_{n o +infty} 0).
(b) Conclude that (P(lim sup A_n) = P(A_n ext{ i.o}) = 0).
Exercise 10. Borel-Cantelli Lemmas Let (A_n) be a sequence of events of some probability space ((A), (P)).
1. Assume that (sum_n P(A_n) < +infty).
(a) Show that (P(cup_{k > n} A_k)
ightarrow 0).
(b) Conclude that (P(lim sup A_n) = P(A_n ext{ i.o}) = 0).
2. Assume the events (A_n) are mutually independent and (sum_n P(A_n) = +infty).
(a) Show that (P(N_k > n A_n) = 0) for all (k). You can use without proof (1 - x < e^{-x}) for (x > 0).
(b) Conclude that (P(lim sup A_n) = P(A_n ext{ i.o}) = 1).
(c) Give an example to illustrate the failure of the result for dependent events.
3. We toss a fair coin infinitely many times independently. Let (A) be the event ({H ext{ appears infinitely many times}}). Show that (A) is a certain event.