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\def\eps{\varepsilon}
\def\EE {{\mathbb E}}
\def\NN {{\mathbb N}}
\def\PP {{\mathbb P}}
\def\RR {{\mathbb R}}
\def\Var {\mathrm {Var}\,}

\begin{document}

\section*{MAT 1600 : Probability I\\
Assignment 5, due October 22, 2020}

\bigskip\bigskip

\begin{enumerate} 
\bigskip\item {\em (Panchenko 2.2.4)}\quad If $\EE|X|<\infty$ and
$\lim\PP(A_n)=0$, show that $\lim \EE XI_{A_n}=0$.\\
{\em Hint:}  Use the Borel-Cantelli lemma over some subsequence.

\bigskip \item We have recently proved {\em Jensen's inequality}: If
$g$ is a convex real-valued function on $\RR$ and $X$ 
a random variable with $\EE|X|<\infty$, then
$\EE\, g(X) \ge g(\EE X)$.

\begin{enumerate}
\item When is there equality in Jensen's inequality?
Give a precise characterization in terms of $g$ and the distribution
of $X$. \\
(Think about the special cases $g(x)=x^2$ and
$g(x)=|x|$.)
A sketch will help.)

\item Justify  the statement that `Under the hypotheses of
Jensen's inequality, $\EE g(X)$ is always well-defined,
though it may take the value $+\infty$'.

\end{enumerate}

\bigskip\item {\em (Panchenko 2.2.7)}\quad Suppose that
$\{X_n\}_{n\ge 1}$ are independent random variables. Show that
$$
\PP\left(\sup_{n\ge 1}X_n<\infty\right)=1
\quad \Longleftrightarrow\quad
\sum_{n\ge 1}\PP(X_n>M)<\infty\ \text{for some} M>0\,.
$$


\bigskip \item Let $\{X_n\}_{n\ge 1}$ be
i.i.d., and $S_n=X_1+\cdots+ X_n$.
\begin{enumerate}
\item {\em (Panchenko 2.2.6)} If $S_n/n\to 0$ almost surely, show that
$\EE |X_1|<\infty$. 

({\em Hint:} Use the idea in Eq. (2.2.2) and Borel-Cantelli).

\item {\em (Panchenko 2.2.8)} 
If, on the other hand, $X_i\ge 0$ and $\EE X_1=\infty$,
show that $S_n/n\to\infty$ almost surely.
\end{enumerate}


\bigskip\item {\em (Durrett 2.2.5)} Let $X_1,X_2,\dots$ be i.i.d. with
$P(X_i>x)= \frac{e}{x\log x}$ for $x\ge e$. 
Construct a sequence of constants
$\mu_n\to\infty$ such that $S_n/n-\mu_n\to 0$ in probability.\\
{\em Hint:} To get $\mu_n$, use the truncation $XI_{X_i\le n}$
for $i=1,\dots, n$, and apply the union bound.  
(Remarkably, $\EE X_1=\infty$!).

\bigskip \item {\em (Panchenko 2.2.10)}\ Let $\{X_n\}_{n\ge 1}$ be independent
and exponentially distributed, i.e., with distribution function 
$F(x)=1-e^{-x}$ for $x\ge 0$. Show that
$$
\PP\left( \limsup \frac{X_n}{\log n}=1\right)= 1\,.
$$


\end{enumerate} 
\end{document}

\bigskip\item Use Jensen's inequality to give an alternative prove of 
H\"older's inequality: If $1<p,q<\infty$
with $\frac1p +\frac1q=1$ are H\"older dual exponents, then
$$
\EE|XY| \le \left(\EE|X|^p\right)^\frac1p \left(\EE|Y|\right)^\frac1q\,,
$$
in particular, the left hand side is finite whenever the right hand side is.

{\em Hint:}
In the special case $||Y||_q=1$, use $Y$ to define
a probability measure; then rescale to
get the general statement. 

\bigskip\item 
(a) {\em Change of variables (Durrett 1.2.5)} \\
Suppose $X$ has continuous density $f$,
that $\PP(\alpha\le X\le \beta)=1$, and that $g$ is a strictly increasing,
differentiable function on $(\alpha,\beta)$. Show that $Y=g(X)$ has density
$$
\left\{
\begin{array}{ll}
\frac{f(g^{-1}(y))}{g'(g^{-1}(y))}\,,\quad & \text{if}\ g(\alpha)<y<g(\beta)\,,
\\
0\,,& \text{otherwise}\,.
\end{array}\right.
$$
In particular, when $g(x)=ax+b$ with $a>0$, then the density of
$Y$ is $\frac1a f\left(\frac{y-b}{a}\right)$.

(b) {\em Log-normal distribution (Durrett 1.2.6)}\\
Find the density of $\exp(X)$ when $X$ has a standard normal distribution.


\bigskip\item 
(a)  {\em (Durrett 1.2.7)} \quad Suppose $X$ has density function
$f$. Compute the distribution function of $Y=X^2$
and then differentiate it to find its density function.

(b) {\em Chi-square distribution}\\
Find the density of $X^2$ when $X$ has a standard normal
distribution.

\bigskip\item {\em (Durrett 1.2.3)} \quad Show that a distribution function 
$F(x)=\PP(X\le x)$ has at most countably many discontinuities.

\bigskip\item We say that two probability
measures $\mu,\nu$ on a measurable space $(\Omega, \mathcal{B})$
are {\em mutually singular} (and write $\mu\perp \nu$)
if there exists a partition
$\Omega=A\cup B$ with $A\cap B=\emptyset$ such that
$\mu(B)=\nu(A)=0$.
You will construct a large family of probability measures
$\mu_p$ on $I=(0,1)$ that are mutually singular continuous with 
respect to Lebesgue measure.

For $0<p<1$, let $\PP_p$ be the probability measure on
$\Omega=\{0,1\}^\NN$ (the space of binary sequences)
defined by repeatedly tossing a coin with bias $p$..

\begin{enumerate}
\item Prove that $\PP_p$ and $\PP_q$ are mutually
singular when $p\ne q$. \\
(Use the Strong Law of Large Numbers.)

\item Conclude that the measures $\mu_p$ on $I:=(0,1)$,
obtained by pushing $\PP_p$ forward with
$\omega\mapsto X(\omega):=\sum_{i\ge 1} \omega_i e^{-i}$
are mutually singular ($\mu_p\perp\mu_q$)
when $p\ne q$).

\item Prove that for each $p$, the distribution function
$$
F_p(x):=\PP_p(X\le x)= \mu_p((0,x])
$$
is continuous in $x$, that is, $X$ is continuously distributed
with respect to $\PP_p$.
\end{enumerate} 
{\em Remark.}\ We have seen (Panchenko p. 14, Step 1) that 
$\mu_{\frac12}$ agrees with Lebesgue measure on $I$,
in other words, $F_{\frac12}=x$.

\end{enumerate}


