Chapter Questions
For each of the following decide whether $f_{n} \rightarrow 0$ (i) in $L^{p}$, (ii) uniformly, (iii) pointwise, (iv) a.e.(a) $f_{n}=\mathbf{1}_{\left[n, n+\frac{1}{n}\right]}$,(b) $f_{n}=n \mathbf{1}_{\left[0, \frac{1}{n}\right]}-n \mathbf{1}_{\left[-\frac{1}{n}, 0\right]}$
Go back to the proof of Theorem $8.1$ (with $E=[0,1]$ ) to see which of the sequences of random variables constructed there converge in probability.
Find an example of a sequence of random variables on $[0,1]$ that does not converge to 0 in probability.
Using Chebyshev's inequality find a lower bound for the probability that the average number of heads in 100 tosses of a coin differs from $\frac{1}{2}$ by 0.1.
Find a lower bound for the probability that the average number shown on a die in 1000 tosses differs from $3.5$ by $0.01$.
Find $\lim \sup _{n \rightarrow \infty} A_{n}$ for a sequence $A_{1}=[0,1], A_{2}=\left[0, \frac{1}{2}\right], A_{3}=\left[\frac{1}{2}, 1\right]$, $A_{4}=\left[0, \frac{1}{4}\right], A_{5}=\left[\frac{1}{4}, \frac{1}{2}\right]$ etc.
Let $S_{n}=X_{1}+X_{2}+\ldots+X_{n}$ describe the position after $n$ steps of a symmetric random walk on $\mathbb{Z}^{d}$. Using the asymptotic formula: $n ! \sim$ $\left(\frac{n}{c}\right)^{n} \sqrt{2 \pi n}$ and the Borel-Cantelli lemmas show that the probability of $\left\{S_{n}=0\right.$ i.o. $\}$ is 1 when $d=1,2$ and 0 for $d>2$.We have the following simple but fundamental fact.
Use the Central Limit Theorem to estimate the probability that the number of Heads in 1000 independent tosses differs from 500 by less than $2 \%$
How many tosses of a coin are required to have the probability at least $0.99$ that the average number of Heads differs from $0.5$ by less than $1 \% ?$
Show that$$a_{n} \rightarrow\left(r-\frac{1}{2} \sigma^{2}\right) T$$For each $n$ we have a sequence of $n$ independent identically distributed random variables $\xi_{n}(i)=\ln \eta_{n}(i)$ forming the so-called triangular array. We have the following version of Central Limit Theorem. It can be proved in exactly the same way as Theorem $8.30$ (see also [2]).