• 1. Infinitely often
•    1.1. Problem set
• 2. Borel-Cantelli lemmas
•    2.1. Problem set

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$\newcommand{\io}{~~\mathrm{i.o.}}$ $\newcommand{\ev}{~~\mathrm{ev.}}$ $\newcommand{\calF}{\mathcal{F}}$ $\newcommand{\rightarrowL}[1]{\stackrel{L_{#1}}{\longrightarrow}}$ $\newcommand{\rightarrowP}{\stackrel{P}{\longrightarrow}}$ $\newcommand{\rightarrowA}{\stackrel{a.s.}{\longrightarrow}}$

1. Infinitely often

EXAMPLE 1:  Suppose that $A_n = \left[ -\frac 1n, 1-\frac 1n \right].$ Find $\{A_n\io\}.$

• Consider all the elements which are in all the $A_n$ s. The set of all such elements is $\cup_{k\geq 1} A_k. $
• Next consider all the elements which are in all the $A_n$ s except possibly $A_1.$ All the earlier elements are already there, plus possibly some extra. The set of all these elements is $\cup_{k\geq 2} A_k. $
• Similarly, $\cup_{k\geq 3} A_k $ is the set of all elements that are in all the $A_n$ s except possible the first two.

1.1. Problem set

EXERCISE 1: If $A_n = [-n,n]$ find $\{A_n\io\}.$

EXERCISE 2: If $A_1 \subseteq A_2 \subseteq A_2\subseteq \cdots,$ then show that $\{A_n\io\} = \cup_n A_n.$

EXERCISE 3: If $A_1 \supseteq A_2 \supseteq A_2\supseteq \cdots,$ then is it true that $\{A_n\io\} = \cap_n A_n?$

EXERCISE 4: Let $B_n = \cup_{k\geq n} A_k.$ Show that $P(B_n)\rightarrow P\{A_n\io\}.$

EXERCISE 5: We can define a set called $\{A_n\ev\}$ (where $\ev$ is the abbreviation of eventually) as $$\{A_n\ev\} = \cup_n \cap_{k\geq n} A_k.$$ Do you see why we call it "eventually" (compare with the statement "We shall all be eventually dead.").

EXERCISE 6: Show that $\{A_n\ev\}\subseteq\{A_n\io\}.$

EXERCISE 7: Let $P\{A_n\ev\}=1.$ Show that $P(A_n)\rightarrow 1.$

EXERCISE 8:  We shall say $A_n\rightarrow A $ if $\{ A_n\ev\} = \{ A_n\io\} = A.$ Show that in this case $P(A_n)\rightarrow P(A).$ This is a generalisation of the theorem on continuity of probability that you have learned in Probability I.

EXERCISE 9: Let $(A_n)$ be a sequence of events with $P(\liminf A_n) = P(\limsup A_n) = p.$ Show that $P(A_n)\rightarrow p.$

2. Borel-Cantelli lemmas

Proof: $$\begin{eqnarray*} P(A_n\io) & = & P(\cap_n\cup_{k\geq n} A_k)\\ & = & P( \lim_n\cup_{k\geq n} A_k)\\ & = & \lim_n P(\cup_{k\geq n} A_k). \end{eqnarray*}$$ Now $P(\cup_{k\geq n} A_k)\leq \sum_{k\geq n} P(A_k).$

Proof:Skipped.[QED]

EXAMPLE 2:  Let $U\sim Unif(0,1)$ and for $n\in{\mathbb N}$ let $$A_n = \left\{U\in\left(0,\frac 1n\right)\right\}.$$ Notice how we are defining all the $A_n$'s based on the same $U.$ Clearly, $A_n$'s are not independent. For instance, $P(A_n)=\frac 1n$, but $P(A_n|A_{n+1})=1.$

EXAMPLE 3: Show that if a monkey types a keyboard randomly, then it will type the exact text of Hamlet infinitely often.

2.1. Problem set

EXERCISE 10: If a coin with $P(head)=0.01$ is tossed repeatedly (independently of course), then find the probability that there will be infinitely many heads.

EXERCISE 11: A fair coin is tossed independently infinitely many times. What is the chance that there would be infinitely many consecutive heads?

Hint:

Be careful: If $X_n$ denotes the outcome ofthe $n$-th toss, then $(X_1,X_2)$ is not independent of $(X_2,X_3)$.

EXERCISE 12: Let $(X_n)$ be any jointly distributed sequence of $Unif(0,1)$ random variables. Show that $P\left(X_n < \frac{1}{n^2} \io\right) = 0.$

EXERCISE 13: Let $(X_n)$ be an iid sequence of $Unif(0,1)$ random variables. Show that $P\left(X_n < \frac 1n \io\right) = 1.$

EXERCISE 14: Toss a fair coin repeatedly (independently). What is the probability that heads occur on all $n$-th tosses where $n$ is a square number?

Hint:

You really do not need the Borel Cantelli lemmas here.

EXERCISE 15:  Construct a sequence of events $(A_n)$ such that $\sum_n P(A_n) = \infty$ but $P(A_n\io) = 0.$

EXERCISE 16: Consider $[-1,1]$ as a 1-dimensional dart board, 0 being the bull's eye. A drunkard is throwing darts at this board hitting the board at any random points (all points equally likely). After the each attempt he gets a prize if he hits within $\frac 1n$ distance of the bull's eye (if it is the $n$-th attempt). What is the chance that he gets infinitely many prizes? [Assume infinite life for the guy, but don't drink to it!]

EXERCISE 17: Same problem as before except that $[-1,1]$ is replaced by the unit disc in ${\mathbb R}^2,$ the bull's eye being at the origin. What is the chance that he gets at leaast one prize? Wht is the chance he gets infinitely many prizes?

EXERCISE 18: However, if you want to just show that the monkey will surely write Hamlet at least once, then you really do not need to use the second Borel Cantelli lemma. You can prove it much more easily using continuity of probability. How?

EXERCISE 19: Can you construct an expansion for which the limit does not exist?

Hint:

You can get one using only 0's and 3's.

Proof: We shall prove it for $n=3.$ (the other cases are exactly similar).