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Uncorrelated.tex
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Uncorrelated.tex
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\documentclass[a4paper,10pt]{scrbook}
\usepackage{hyperref}
\usepackage{mathematix}
\usepackage[margin=1in,right=1in]{geometry}
\usepackage{lipsum}
\usepackage{enumerate}
\usepackage{caption}
\captionsetup{font=footnotesize,labelfont=bf}
\usepackage{natbib}
\usepackage{bibentry}
\nobibliography*
\renewcommand{\cov}{\operatorname{cov}}
\newcommand{\comingSoon}{Work in progress (this section will be updated in the upcoming version). Stay tuned on Twitter for updates (\texttt{@isToxic}).}
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bookmarks=true, % show bookmarks bar?
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pdfmenubar=true, % show Acrobat’s menu?
pdffitwindow=false, % window fit to page when opened
pdfstartview={FitH}, % fits the width of the page to the window
pdftitle={Probability Theory Cookbook}, % title
pdfauthor={Pantelis Sopasakis}, % author
pdfsubject={}, % subject of the document
pdfcreator={P. Sopasakis}, % creator of the document
pdfproducer={P. Sopasakis}, % producer of the document
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% Update \bibentry so that bibentries appear in blue
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\title{Probability Cookbook}
\author{Pantelis Sopasakis}
\begin{document}
\maketitle
\tableofcontents
\addchap{Abstract}
This document is intended to serve as a collection of important results in general probability
theory, stochastic processes, uncertainty quantification, risk measures and a lot more.
It can be used for a quick brush up or as a quick reference or
cheat sheet for graduate students and researchers in the domain of mathematics and
engineering.
The readers may find a list of bibliographic references with comments at the end
of this document. This is still work in progress, so several important results are
still missing.
This document, as well as its future versions, will be available at
\url{https://mathematix.wordpress.com/probability-cookbook}.
% \end{abstract}
\chapter{Probability Theory}
\section{General Probability Theory}
\subsection{Measurable and Probability spaces}
\begin{enumerate}
\item ($\sigma$-algebra). Let $X$ be a nonempty set. A collection $\F$ of subsets of $X$
is called a $\sigma$-algebra if (i) $X \in \F$, (i) $A^c\in\F$ whenever $A\in\F$,
(ii) if $A_1,\ldots, A_n\in\F$, then $\bigcup_{i=1,\ldots,n}A_i\in\F$. The space $X$
equipped with a $\sigma$-algebra $\F$ is called a \textit{measurable space}.
\item (d-system) A collection $\mathcal{D}$ of subsets of $X$ is called a d-system or a Dynkin class if
(i) $X\in\mathcal{D}$,
(ii) $A\setminus B\in\mathcal{D}$ whenever $A,B\in\mathcal{D}$ and $A\supseteq B$,
(iii) $A\in\mathcal{D}$ whenever $A_n\in \mathcal{D}$ and $A_n \uparrow A$ (meaning,
$A_{k}\subseteq A_{k+1}$ and $\bigcup_{k\in\N}A_k=A$).
\item (p-system). A collection of sets $\mathcal{P}$ in $X$ is called a p-system
if $A\cap B\in \mathcal{P}$ whenever $A,B\in\mathcal{P}$.
\item A collection of sets is a $\sigma$-algebra if and only if it is both a p- and a d-system.
\item (Smallest $\sigma$-algebra). Let $\HH$ be a collection of sets in $X$. The smallest collection of sets
which contains $\HH$ and is a $\sigma$-algebra exists and is denoted by $\sigma(\HH)$.
\item (Monotone class theorem). If a d-system $\mathcal{D}$ contains a p-system $\mathcal{P}$, then is also contains $\sigma(\mathcal{P})$.
\item \label{mps1311949}
(Borel $\sigma$-algebra). On $\Re$, the $\sigma$-algebra $\sigma(\{(a,b); a<b\})$ is called the Borel $\sigma$-algebra on $\Re$
which we denote by $\B_\Re$. For topological spaces $(X,\tau)$, the Borel $\sigma$-algebra is
defined as $\B_X = \sigma(\tau)$, i.e., it is the smallest $\sigma$-algebra which contains
all open sets. $\B_\Re$ is generated by:
\begin{enumerate}[i.]
\item The open intervals $(a,b)$
\item The closed intervals $[a,b]$
\item All sets of the form $[a,b)$ or $(a,b]$
\item Open rays $(a,\infty)$ or $(-\infty,a)$
\item Closed rays $[a,\infty)$ or $(-\infty,a]$
\end{enumerate}
\item (Measure). A function $\mu: \F\to [0,+\infty]$ is called a measure if
for every sequence of disjoint sets $A_n$ from $\F$, $\mu(\bigcup_n A_n)= \sum_n \mu(A_n)$.
\item \label{mps1311960}
(Properties of measures). The following hold:
\begin{enumerate}[i.]
\item (Empty set is negligible). $\mu(\varnothing)=0$ [Indeed, $\mu(A) = \mu(A\cup \varnothing) = \mu(A) + \mu(\varnothing)$ for all $A\in\F$]
\item (Monotonicity). $A\subseteq B$ implies $\mu(A) \leq \mu(B)$ [Indeed, $\mu(B) = \mu(A\cup (B\setminus A))$]
\item (Boole's inequality). For all $A_n\in\F$, $\mu(\bigcup_n A_n) \leq \sum_n \mu(A_n)$
\item (Sequential continuity). If $A_n\uparrow A$, then $\mu(A_n)\uparrow \mu(A)$.
\end{enumerate}
\item (Equality of measures). Let $\mu,\nu$ be two measures on a measurable space $(X,\F)$ and let $\G$
be a p-system generating $\F$. If $\mu(A) = \nu(A)$ for all $A\in \G$, then $\mu(B) = \nu(B)$
for all $B\in\F$. As presented in \#\ref{mps1311949} above, p-systems are often available and
have simple forms.
\item (Completeness). A measure space $(X,\F,\mu)$ is called \textit{complete} if the following holds:
\[
A \in \F, \mu(A)=0, B\subseteq A \Rightarrow B \in \F.
\]
Of course, by the monotonicity property in \#\ref{mps1311960}--iii, if $(X,\F,\mu)$ is a complete
measure space then $\mu(B) = 0$.
\item (Completion). Let $(X,\F,\mu)$ be a measure space and define the set of \textit{negligible sets} of $\mu$ as
$Z_\mu = \{N \subseteq X: \exists N'\supseteq N, N'\in\F \text{ s.t. } \mu(N')=0\}$.
Let $\F'$ be the $\sigma$-algebra generated by $\F\cup Z_\mu$. Then
\begin{enumerate}[i.]
\item Every $B\in\F'$ can be written as $B=A\cup N$ with $A\in\F$ and $N\in Z_\mu$
\item Define $\mu'(A\cup N) = \mu(A)$; this is a measure on $(X,\F')$ which renders
the space $(X,\F',\mu')$ complete.
\end{enumerate}
\item (Lebesgue measure on $\Re$ and $\Re^n$). It suffices to define the \textit{Lebesgue measure} on $(\Re,\B_\Re)$
on the p-system $\{(a,b), a<b\}$; it is $\lambda((a,b))=b - a$. This extends to a measure on $(\Re,\B_\Re)$.
Likewise, the collection of $n$-dimensional rectangles $\{(a_1, b_1)\times\ldots \times (a_n, b_n)\}$ is a p-system
which generates $\B_{\Re^n}$; the Lebesgue measure on $(\Re^n, \B_{\Re^n})$ is
$\lambda(\prod_{i=1}^n (a_i, b_i))=\prod_{i=1}^n (b_i-a_i)$.
\item (Lebesgue measurable sets). The completion of the Lebesgue measure defines the class of Lebesgue-measurable
sets.
\item (Negligible boundary). If a set $C\subseteq \Re^n$ has a boundary whose Lebesgue measure is $0$, then
$C$ is Lebesgue measurable.
\item (Independent events). Let $E_1,E_2$ be two events from $\ofp$; we say that $E_1$ and $E_2$ are \textit{independent}
if $\prob[E_1\cap E_2] = \prob[E_1]\prob[E_2]$.
\item (Independent $\sigma$-algebras). We say that two $\sigma$-algebras $\F_1$ and $\F_2$ on $\Omega$
are independent if for any $E_1\in \F_1$ and $E_2\in\F_2$, $E_1$ and $E_2$ are independent.
Note that $E_1\cap E_2$ is a member of the $\sigma$ algebra $E_1 \wedge E_2$.
\item (Atom). Let $(\Omega, \F, \mu)$ be a measure space. A set $A\in\F$ is called an atom if
$\mu(A)>0$ and for every $B\subset A$ with $\mu(B)<\mu(A)$ it is $\mu(B)=0$.
A space without atoms is called non-atomic%
\footnote{A special class of spaces with (only) atoms are the discrete probability spaces where
$\F$ is generated by a discrete --- often finite --- set of events. Several
results in measure theory require that the space be non-atomic. However, we may
often prove these results for discrete or finite spaces.}.
\end{enumerate}
\subsection{Random variables}\label{sec:random_variables}
\begin{enumerate}
\item (Measurable function). A function $f:(X,\F)\to (Y,\G)$ (between two measurable spaces) is
called \textit{measurable} if $f^{-1}(G) \in \F$ for all $G\in\G$ (i.e., if it inverts all
measurable sets to measurable ones).
\item (Measurability test). Let $\F,\G$ be $\sigma$-algebras on the nonempty sets $X$ and $Y$. Let $\G'$ be
a p-system which generates $\G$. A function $f: (X,\F)\to (Y,\G)$ is measurable
if and only if $f^{-1}(G')\in \F$ for all $G'\in\G'$ (it suffices to check the
measurability condition on a p-system).
\item ($\sigma$-algebra generated by $f$). Let $f:(X,\F)\to (Y,\G)$ (between two measurable spaces) be a measurable
function. The set
\[
\sigma(f) \dfn \{f^{-1}(B){}\mid{} B\in \G\},
\]
is a sub-$\sigma$-algebra of $\F$ and is called the $\sigma$-algebra generated by $f$.
\item (Preservation of measurability). Let \(f,g:\Omega\to\Re\) be two measurable functions on \((\Omega, \F)\).
Then, the functions \(h_1(x) = f(x) + g(x)\), \(h_2(x) = f(x) - g(x)\),
\(h_3(x) = \max\{f(x), g(x)\}\), \(h_4(x) = \min(f(x), g(x))\),
\(h_5(x) = f(x)g(x)\) are measurable. For all \(\alpha\in\Re\), \(h_6(x) = \alpha f(x)\) is measurable.
\item (Measurability of supermum/infimum). Let \((f_n)_n\) be a sequence of real-valued measurable
functions. Then \(\sup_n f_n\) and \(\inf_n f_n\) are measurable.
\item (Sub/sup-level sets) Let $f:(X,\F)\to \Re$. The following are equivalent:
\begin{enumerate}[i.]
\item $f$ is measurable,
\item Its \textit{sub-level sets}, that is
sets of the form $\lev_{\leq \alpha} f \dfn \{x\in X: f(x) \leq \alpha\}$ are measurable,
\item Its \textit{sup-level sets},
that is sets of the form $\lev_{ \geq \alpha} f \dfn \{x\in X: f(x) \geq \alpha \}$ are
measurable.
\end{enumerate}
\item \label{rv220000}
(Random variable).
A real-valued random variable $X:\ofp \to (\Re, \B_\Re)$ is a measurable function $X$
from a probability space $\ofp$ to $\Re$, equipped with the Borel $\sigma$-algebra, that is,
for every Borel set $B$, $X^{-1}(B)\in\F$.
\item \label{rv221030}
Every nonnegative (real-valued) random variable $X$ on $(\Re_+, \B_{{\Re}_+})$
is written as
\[
X(\omega) = \int_0^{+\infty} 1_{X(\omega)\geq t}\,\d t.
\]
\item (Increasing functions). Every increasing function $f:\Re\to\barre$ is Borel-measurable.
\item (Semi-continuous functions). Every lower semi-continuous function $X:\Omega\to\Re$ (where $\Omega$ is assumed
to be equipped with a topology) is Borel-measurable.
\item (Push-forward measure)~[\ref{cite:cinlar2011}]. Given measurable spaces $(\mathcal{X},\F)$ and $(\mathcal{Y}, \mathcal{G})$,
a measurable mapping $f: X \to Y$ and a (probability) measure $\mu$ on $(\mathcal{X},\F)$, the \textit{push-forward} of $\mu$
is defined to be a measure $f∗(\mu)$ on $(\mathcal{Y}, \mathcal{G})$ given by
\[
(f_*\mu)(B) = \mu(f^{-1}(B)) = \mu(\{\omega\mid f(\omega)\in B\}),
\]
for $B\in\mathcal{G}$.
\item (Change of variables). Let $F$ be a random variable on the probability space $\ofp$ and $F_*\prob$
is the push-forward measure. random variable $X$ is integrable with respect to the push-forward measure $F_*\prob$
if and only if $X\circ F$ is $\prob$-integrable. Then, the integrals coincide
\[
\int X \d(F_*\prob) = \int (X\circ F) \d \prob.
\]
\item (Measures from random variables). Let $X$ be a random variable on $\ofp$.
We may use $X$ to define the following measure
\[
\nu(A) = \int_A X\d \prob,
\]
defined for $A\in\F$. This is a positive measure which for short we denote as $\nu=X\prob$
and it satisfies:
\[
\int_A Y\d \nu = \int_A XY\d \prob,
\]
for all random variables $Y$.
\item (Compositions). Let $f:(X,\F_X)\to (Y, \F_Y)$ and $g:(Y,\F_Y)\to (Z,\F_Z)$ be two measurable functions.
Then, the function $h:(X,\F_X)\ni x\mapsto h(x) \dfn f(g(x)) \in (Z,\F_Z)$ is measurable.
\item (Simple function; definition). A simple function is one of the form
\[
f(x) = \sum_{k=1}^{n}\alpha_k 1_{A_k}(x),
\]
where $1_{A_k}$ is the characteristic function of a measurable set $A_k$, that is
\[
1_{A_k} = \begin{cases}
1,&\text{if }x\in A_{k}\\
0,&\text{otherwise}
\end{cases}
\]
\item (Characterization of measurability). A function $f:(X,\F)\to\Re$ is $\F$-measurable if and only if
it is the point-wise limit of a sequence of simple functions. A function $f:(X,\F)\to\Re_+$ is
$\F$-measurable if and only if it is the point-wise limit of an increasing sequence of simple functions.
\item (Continuity and measurability). Every continuous function $f:(X,\F)\to\barre$ is Borel-measurable.
\item (Monotone class of functions). Let $M$ be a collection of functions $f:(X,\F)\to\barre$; let $M_+$
be all positive functions in $M$ and $M_b$ all bounded functions in $M$. We say that $M$ is a \textit{monotone class}
of functions if (i) $1\in M$, (ii) if $f,g\in M_b$ and $a,b\in \Re$, then $af+bg\in M$ and (iii)
if $(f_n)_n\subseteq M_+$ and $f_n \uparrow f$, then $f\in M$.
\item (Monotone class theorem for functions). Let $M$ be a monotone class of functions on $(X,\F)$. Suppose
that $\F$ is generated by some p-system $\mathcal{C}$, $1_A \in M$ for all $A\in\mathcal{C}$.
Then, $M$ includes all positive $\F$-measurable functions and all bounded $\F$-measurable functions.
\item (Simple function approximation theorem).
Let $X$ be an extended-real-valued Lebesgue-measurable function defined on a Lebesgue measurable set $E$.
Then there exists a sequence $\{\phi_k\}_{k\in\N}$ of simple functions%
\footnote{A simple function is a finite linear combination of indicator functions of measurable sets, that is,
simple functions are written as $\phi(x)=\sum_{i=1}^{n}\alpha_i 1_{A_i}(x)$. }
on $E$ such that
\begin{enumerate}[i.]
\item $\phi_k\to X$, point-wise on $E$
\item $|\phi_k| \leq |X|$ on $E$ for all $k\in\N$
\end{enumerate}
If $X\geq 0$ then there exists a sequence of point-wise increasing simple functions with these properties.
\item (Simple function approximation trick).
Let $f$ be a real-valued measurable function, $f:\ofp\to(\Re,\mathcal{B}_\Re)$. Define
\[
\phi_k(x) = \begin{cases}
\frac{j-1}{2^k}, &\frac{j-1}{2^k}\leq f(x) < \frac{j}{2^k}\\
k, &f(x) \geq k
\end{cases}
\]
Then,
\begin{enumerate}[i.]
\item The sets $\{x: f(x) \geq k\}$ and $\{x: \frac{j-1}{2^k}\leq f(x) < \frac{j}{2^k}\}$ are measurable because $f$ is measurable
\item $\phi_k$ are measurable for all $k\in\N$
\item $\phi_k(x) \leq \phi_{k+1}(x)$ for all $k\in\N$ and for all $x\in\Omega$
\item Let $E\subseteq \Omega$ so that $\sup_{x\in E}f(x) \leq M$. Then $\sup_{x\in\Omega}|f(x) - \phi_k(x)|\leq \nicefrac{1}{2^k}$ for all $k\geq M$
\end{enumerate}
\end{enumerate}
\subsection{Limits}
\subsubsection{Limits of sequences of events}
\begin{enumerate}
\item (Nested sequences and probabilities). Let $(E_n)_n$ be a non-increasing sequence of events ($E_n\supseteq E_{n+1}$ for all $n\in\N$).
Then $\lim_n \prob[E_n]$ exists and
\[
\prob\left[\bigcap_n E_n\right] = \lim_n \prob[E_n].
\]
If $(E_n)_n$ is a nondecreasing sequence ($E_n \subseteq E_{n+1}$ for all $n\in\N$), then
\[
\prob\left[\bigcup_n E_n\right] = \lim_n \prob[E_n].
\]
\item (Limits inferior). For a sequence of events $E_n$, the \textit{limit inferior} of $(E_n)_n$
is denoted by $\liminf_n E_n$ and is defined as
\[
\liminf_n E_n = \bigcup_{n\in \N}\bigcap_{m\geq n}E_n = \{x:\ x\in E_n\ \text{for all but finitely many } n\in \N\}.
\]
\item (Limit superior). The \textit{limit superior} of $(E_n)_n$, $\limsup_n E_n$, is
\[
\limsup_n E_n = \bigcap_{n\in \N}\bigcup_{m\geq n}E_n = \{x:\ x\in E_n\ \text{infinitely often} \}.
\]
\item (Limits of complements). The limit (super/inferior) of a sequence of complements is the complement of the limit
\begin{align*}
\liminf_n E_n^c &= (\limsup_n E_n)^c,\\
\limsup_n E_n^c &= (\liminf_n E_n)^c.
\end{align*}
\item (Relationship between limits). It is
\[
\liminf_n E_n \subseteq \limsup_n E_n.
\]
\item (Probabilities of $\liminf E_n$ and $\limsup E_n$). The sets $\liminf_n E_n$ and $\limsup_n E_n$ are measurable and
\[
\prob[\liminf_n E_n] \leq \liminf_n \prob[E_n] \leq \limsup_n \prob[E_n] \leq \prob[\limsup_n E_n].
\]
\item (A result reminiscent of Baire's category theorem). Let $(E_n)_n$ be a sequence of almost sure events. Then
$\prob[\cap_n E_n] = 1$.
\item (Borel-Cantelli lemma). Let $(E_n)_n$ be a sequence of events over $\ofp$. The following hold
\begin{enumerate}[i.]
\item If $\sum_{n=1}^{\infty}\prob[E_n]<\infty$, then $\prob[\limsup_n E_n] = 0$
\item If $(E_n)_n$ are independent events such that $\sum_{n=1}^{\infty}\prob[E_n]=\infty$, then
$\prob[\limsup_n E_n] = 1$.
\end{enumerate}
\item (Corollary: Borel 0-1 law). If $(E_n)_n$ is a sequence of independent events,
then $\prob[\limsup_n E_n]\in\{0,1\}$ (according to the summability of $(\prob[E_n])_n$).
\item (Kochen-Stoone lemma). Let $(E_n)_n$ be a sequence of events. Then,
\[
\prob[\limsup_n E_n] \geq \limsup_n \frac{\left(\sum_{k=1}^n \prob[A_k]\right)^2}{\sum_{k=1}^{n}\sum_{j=1}^{n}\prob[A_k\cap A_j]}
\]
\item (Corollary of Kochen-Stoone's lemma). If for $i\neq j$, $E_i$ and $E_j$ are either independent
or $\prob[E_i\cap E_j] \leq \prob[E_i]\prob[E_j]$ and $\sum_{n=1}^{\infty}\prob[E_n]=\infty$,
then $\prob[\limsup_n E_n] = 1$.
\end{enumerate}
\subsubsection{Limits of sequences of random variables}
\begin{enumerate}
\item (Lebesgue's monotone convergence theorem). Let $(f_n)_n$ be an increasing sequence of
nonnegative Borel functions and let $f \dfn \lim_n f_n$ (in the sense $f_n\to f$ point-wise a.e.).
Then $\E[f_n] \uparrow \E[f]$.
\item (Lebesgue's Dominated Convergence Theorem). Let $X_n$ be real-valued RVs over $\ofp$.
Suppose that $X_n$ converges point-wise to $X$ and is \textit{dominated} by a
$Y\in\mathcal{L}_1\ofp$, that is $|X_n|\leq Y$ $\prob$-a.s for all $n\in\N$.
Then, $X\in\mathcal{L}_1\ofp$
and
\[
\lim_n \E[|X_n-X|] = 0,
\]
which implies
\[
\lim_n \E[X_n] = \E[X].
\]
\item (Dominated convergence in $\mathcal{L}^p$).
For $p\in[1,\infty)$ and a sequence of random variables $X_k:\ofp\to\barre$,
assume that $X_k\to X$ almost everywhere ($X(\omega)=\lim_k X_k(\omega)$ $\prob$-a.e.)
and there is $Y\in\mathcal{L}^p\ofp$ so that $X_k\leq Y$. Then,
\begin{enumerate}[i.]
\item $X_k\in\mathcal{L}^p\ofp$ for all $k\in\N$,
\item $X\in\mathcal{L}^p\ofp$
\item $X_k\to X$ in $\mathcal{L}^p\ofp$, that is $\lim_k \|X_k-X\|_p = 0$.
\end{enumerate}
\item (Consequence of the dominated convergence theorem)~[\ref{cite:DWalnut2011}].
Let $\{E_k\}_{k=1}^{\infty}$ be a collection of disjoint events and let $E=\bigcup_{k}E_k$.
Then,
\[
\int_E f = \sum_{k=1}^{\infty} \int_{E_k} f.
\]
\item (Bounded convergence). If $X_k \to X$ almost surely and $\sup_k |X_k| \leq b$
for some constant $b>0$, then $\E[X_k]\to \E[X]$ and $\E[|X|] \leq b$.
\item (Fatou's lemma). Let $X_n\geq 0$ be a sequence of random variables.
Then,
\[
\E[\liminf_n X_n] \leq \liminf_n \E[X_n].
\]
\item (Fatou's lemma with varying measures). For a sequence of nonnegative random variables $X_n\geq 0$ over $\ofp$,
and a sequence of (probability) measures $\mu_n$ which converge strongly to a (probability)
measure $\mu$ (that is, $\mu_n(A)\to\mu(A)$ for all $A\in\F$), we have
\[
\E_\mu[\liminf_n X_n]\leq \liminf_n \E_{\mu_n}[ X_n]
\]
\item (Reverse Fatou's lemma). Let $X_n\geq 0$ be a sequence of nonnegative random variables over $\ofp$ and
assume there is a $Y\in\mathcal{L}_1\ofp$ so that $X_n\leq Y$. Then
\[
\limsup_n \E[X_n] \leq \E[\limsup_n X_n]
\]
\item (Integrable lower bound).
Let $X_n$ be a sequence of random variables over $\ofp$. Suppose, there exists a
$Y\geq 0$ such that $X_n\geq -Y$ for all $n\in\N$. Then,
\[
\E[\liminf_n X_n] \leq \liminf_n \E[X_n].
\]
\item (Beppo Levi's Theorem).
Let $X_k$ be a sequence of nonnegative random variables on $\ofp$ with $0 \leq X_1 \leq X_{2} \leq \ldots$.
Let $X(\omega) = \lim_{k\to\infty}X_k(\omega)$. Then $X$ is a random variable and
\[
\lim_{k\to\infty} \E[X_k] = \E[\lim_{k\to\infty} X_k].
\]
\item (Beppo Levi's Theorem for series).
Let $X_k$ be a sequence of nonnegative integrable random variables on $\ofp$
and let $Y_k = \sum_{j=0}^k X_k$. Assume that $\sum_{k=1}^{\infty} \E[Y_k]$ converges.
Then $Y_k$ satisfies the conditions of the BL theorem and
\[
\sum_{k=1}^{\infty} \E[Y_k] = \E \left[\sum_{k=1}^{\infty}Y_k\right].
\]
\item (\hypertarget{link:uniformly_integrable}{Uniform integrability} -- definition)~[\ref{cite:KSigman2009}]. A collection $\{X_k\}_{k\in T}$ is said to be \textit{uniformly
integrable} if $\sup_{t\in T}\E[|X_t| 1_{|X_t|>x}] \to 0$ as $x\to\infty$.
\item (Constant absolutely integrable sequences as uniformly integrable)~[\ref{cite:KSigman2009}]. The sequence $\{Y\}_{t\in T}$
with $\E[|Y|]<\infty$ is uniformly integrable.
\item (Uniform boundedness in $\mathcal{L}^p$, $p>1$, implies uniform integrability).
If $\{X_t\}_{t\in T}$ is uniformly bounded in $\mathcal{L}^p$, $p>1$ (that is,
$\E[|X_k|^p] < c$ for some $c>0$), then it is uniformly integrable.
\item (Convergence under uniform integrability)~[\ref{cite:KSigman2009}].
If $X_k \to X$ a.s. and $\{X_k\}_k$ is \hyperlink{link:uniformly_integrable}{uniformly integrable} then
\begin{enumerate}[i.]
\item $\E[X]< \infty$
\item $\E[X_k] \to \E[X]$
\item $\E|X_k-X|\to 0$
\end{enumerate}
\end{enumerate}
\subsection{The Radon-Nikodym Theorem}
\begin{enumerate}
\item (Absolute continuity).
Let $(\mathcal{X}, \mathscr{G})$ be a measurable space and $\mu$ and $\nu$ two measures on it.
We say that $\nu$ is \textit{absolutely continuous} with respect to $\mu$ if
for all $A\in\mathscr{G}$, $\nu(A)=0$ whenever $\mu(A)=0$. We denote this by $\nu\ll\mu$.
\item (Radon-Nikodym). Let $(\mathcal{X}, \mathscr{G})$ be a measurable space, let $\nu$ be a \textit{$\sigma$-finite}
measure on $(\mathcal{X}, \mathscr{G})$ which is {absolutely continuous} with respect
to a measure $\mu$ on $(\mathcal{X}, \mathscr{G})$. Then, there is a measurable function $f:\mathcal{X}\to[0,\infty)$
such that for all $A\in \mathcal{G}$
\[
\nu(A) = \int_A f \d \mu.
\]
This function is denoted by $f=\frac{\d\nu}{\d \mu}$.
\item (Linearity). Let $\nu$, $\mu$ and $\lambda$ be $\sigma$-finite measures on $(\mathcal{X}, \mathscr{G})$ and $\nu\ll\lambda$, $\mu\ll\lambda$.
Then
\[
\frac{\d(\nu+\mu)}{\d \lambda} = \frac{\d \nu}{\d \lambda} + \frac{\d \mu}{\d \lambda},\ \lambda\text{-a.e.}
\]
\item (Chain rule). If $\nu\ll\mu\ll\lambda$,
\[
\frac{\d\nu}{\d\lambda} = \frac{\d\nu}{\d\mu} \frac{\d\mu}{\d\lambda},\ \lambda\text{-a.e.}
\]
\item (Inverse). If $\nu\ll\mu$ and $\mu\ll\nu$, then
\[
\frac{\d \mu}{\d \nu} = \left( \frac{\d \nu}{\d \mu}\right)^{-1},\ \nu\text{-a.e.}
\]
\item (Change of measure).
If $\mu\ll\lambda$ and $g$ is a $\mu$-integrable function, then
\[
\int_{\mathcal{X}} g \d \mu = \int_{\mathcal{X}} g \frac{\d \mu}{\d \lambda}\d \lambda.
\]
\item (\hypertarget{link:lotus}{Change of variables in integration}). This was addressed using the push-forward.
\[
\E[g(X)] = \int g\circ X\d\prob = \int_\Re g \,\d(X_*\prob).
\]
If the measure $\d(X_*\prob)$ is absolutely continuous with respect to the Lebesgue
measure $\mu$ (on $(\Re, \B_\Re)$, then, the Radon-Nikodym derivative $f_X\dfn \frac{\d(X_*\prob)}{\d \mu}$,
where $f_X:\Re\to\Re$ exists. Then
\[
\E[g(X)] = \int_\Re g\, \d(X_*\prob) = \int_\Re g f_X\, \d\mu = \int_\Re g(\tau)f_X(\tau)\,\d \tau.
\]
This is known as the \textit{law of the unconscious statistician} (LotUS).
\end{enumerate}
\subsection{Probability distribution}
\begin{enumerate}
\item (Probability distribution). Let $X:\ofp \to (Y, \G)$ be a random variable. The measure
\[
F_X(A) = \prob[X\in A] = \prob[\{\omega\in\Omega\mid X(\omega) \in A\}] = \prob[X^{-1}A] = (X_*\prob)(A),
\]
is called the \textit{probability distribution} of $X$ and it is a measure . Note that for all $A\in\G$, $X^{-1}A\in\F$
since $X$ is measurable.
\item \label{rv221088}
(Probability distribution of real-valued random variables).
The \textit{probability distribution} or \textit{cumulative distribution function} of a random variable $X$ on a space
$\mathcal{L}^p\ofp$ is $F_X(x) = \prob[X\leq x]$ for $x\in\Re$. The inverse cumulative
distribution of $X$ is $F_X^{-1}(p)$ for $p\in[0,1]$ is defined as
$F_X^{-1}=\inf\{x\in\Re: F_X(x) \geq p\}$.
\item (Push-forward).
\label{rv221089}
The probability distribution of a random variable $X$ with values in $(\mathcal{X},\mathscr{G})$,
is the push-forward measure $X_*\prob$ on $(\mathcal{X},\mathscr{G})$ which is
a probability measure on $(\mathcal{X},\mathscr{G})$ with $X_*\prob = \prob X^{-1}$.
\item
\label{rv221137}
(Associated $p$-system). We associate with $F_X:\Re\to[0,1]$ the measure $\mu$ which is defined on
the $p$-system $\{(-\infty,x]\}_{x\in\Re}$ as $\mu((-\infty, x]) = F_X(x)$.
\item
\label{rv231132}
(Properties of the cumulative and the inverse cumulative distributions). The notation
$X\sim Y$ means that $X$ and $Y$ have the same cumulative distribution, that is
$F_X = F_Y$.
\begin{enumerate}[i.]
\item If $Y\sim U[0,1]$, then $F_X^{-1}(Y) \sim X$.
\item $F_X$ is c\`adl\`ag
\item $x_1<x_2 \Rightarrow F_X(x_1) \leq F_X(x_2)$
\item $\prob[X>x] = 1 - F_X(x)$
\item $\prob[\{x_1 < X \leq x_2\}] = F_X(x_2) - F_X(x_1)$
\item $\lim_{x\to-\infty}F_X(x) = 0$, $\lim_{x\to\infty}F_X(x) = 1$
\item $F_X^{-1}(F_X(x)) \leq x$
\item $F_X(F_X^{-1}(p)) \geq p$
\item $F_X^{-1}(p) \leq x \Leftrightarrow p \leq F_X(x)$
\end{enumerate}
\end{enumerate}
\subsection{Probability density function}
\begin{enumerate}
\item (Definition).
The probability density function $f_X$ of a random variable $X:\ofp\to (\mathcal{X},\mathscr{G})$
with respect to a measure $\mu$ on $(\mathcal{X},\mathscr{G})$ is the Radon-Nikodym derivative
\[
f_X = \frac{\d (X_*\prob)}{\d \mu},
\]
which exists provided that $X_*\prob \ll \mu$, and $f_X$ is measurable and $\mu$-integrable. Then,
\begin{align*}
\prob[X\in A] = \int_{X^{-1}A}\d \prob
= \int_{\Omega} 1_{X^{-1}A}\d\prob
= \int_{\Omega} (1_{A}\circ X)\d\prob
= \int_{A}\d(X_*\prob)
= \int_A f_X \d \mu.
\end{align*}
\item (Probability distribution).
If $X$ is a real-valued random variable and its range ($\Re$) is taken with the
Borel $\sigma$-algebra, then
\begin{align*}
\prob[X\leq x] = \int_{(-\infty, x]}X\d \prob
= \int_{\{\omega\in\Omega: X(\omega) \leq x\}}\d \prob
= \int_{-\infty}^x f_X\d \mu
\end{align*}
Note that the first integral is written with a slight abuse of notation as the
integration with respect to $\prob$ is carried out over the set $\{\omega\in\Omega: X(\omega) \leq x\}$;
The first integral can be understood as shorthand notation for the second integral.
\item (Expectation).
Let a real-valued random variable $X$ have probability density $f_X$. Let $\iota$
be the identity function $\iota:x\mapsto x$ on $\Omega$. Then
\[
\E[X] = \int_\Omega X\d\prob
= \int_\Omega (\iota\circ X)\d\prob
= \int_\Re \iota\d(X_*\prob)
= \int_\Re \iota(x) f_X(x) \d\mu
= \int_\Re x f_X(x) \d x.
\]
\item (Distribution of transformation). Let $g:\Re\to\Re$ be a strictly increasing function.
Let $X$ be a real-valued random variable with probability density function $f_X$
and let $Y(\omega) = g(X(\omega))$ be another random variable. Then
\begin{align*}
F_Y(y) &= F_X(g^{-1}(y)),\\
f_Y(y) &= f_X(g^{-1}(y))\frac{\partial g^{-1}(y)}{\partial y}.
\end{align*}
\item (Expectation of transformation). Let $X$ be a real-valued random variable on \(\ofp\) with
probability density function \(f_X\) and let $Y(\omega) = g(X(\omega))$
be another random variable. Then
\[
\E[Y] = \int_{-\infty}^{\infty} f_X(\tau) g(\tau) \d \tau.
\]
If $\Omega=\{\omega_i\}_{i=1}^{n}$, $\F=2^\Omega$ and $\prob[\{\omega\}_i] = p_i$, then
\[
\E[Y] = \sum_{i=1}^{n}p_i g(X(\omega_i)).
\]
See also: law of the unconscious statistician.
\end{enumerate}
\subsection{Decomposition of measures}
Does a density function always exist? The answer is negative, but Lebesgue's decomposition
theorem offers some further insight.
\begin{enumerate}
\item (Singular measures). Let $(\Omega, \F)$ be a measurable space and $\mu$, $\nu$
be two measures defined thereon. These are called \textit{singular} if there are
$A,B\in\F$ so that
\begin{enumerate}[i.]
\item $A\cup B=\Omega$,
\item $A\cap B=\varnothing$,
\item $\mu(B')=0$ for all $B'\in\F$ with $B'\subseteq B$,
\item $\nu(A')=0$ for all $A'\in\F$ with $A'\subseteq A$.
\end{enumerate}
\item (Discrete measure on $\Re$). A measure $\mu$ on $\Re$ equipped with the Lebesgue $\sigma$-algebra,
is said to be discrete if there is a (possibly finite) sequence of elements $\{s_k\}_{k\in\N}$,
so that
\[
\mu(\Re\setminus \bigcup_{k\in\N} \{s_k\}) = 0.
\]
\item (Lebesgue's decomposition Theorem). For every two $\sigma$-finite signed measures $\mu$ and $\nu$
on a measurable space $(\Omega, \F)$, there exist two $\sigma$-finite signed measures $\nu_0$ and $\nu_1$
on $(\Omega, \F)$ such that
\begin{enumerate}[i.]
\item $\nu = \nu_0 + \nu_1$
\item $\nu_0\ll \mu$
\item $\nu_1 {}\bot{} \mu$
\end{enumerate}
and $\nu_0$ and $\nu_1$ are uniquely determined by $\nu$ and $\mu$.
\item (Lebesgue's decomposition Theorem --- Corollary).
Consider the space $(\Re,\B_\Re)$ and let $\mu$ be the Lebesgue measure. Any probability measure $\nu$
on this space can be written as
\[
\nu = \nu_{\text{ac}} + \nu_{\text{sc}} + \nu_{\text{d}},
\]
where $\nu_{\text{ac}} \ll \mu$ (which is easily understood via the
Radon-Nikodym Theorem), $\nu_{\text{sc}}$ is singular continuous (wrt $\mu$) and $\nu_{\text{d}}$
is a discrete measure.
\end{enumerate}
\subsection{$\mathcal{L}^p$ spaces}
\begin{enumerate}
\item ($p$-norm). Let $X$ be a real-valued random variable on $\ofp$. For $p\in[1,\infty)$ define the $p$-norm of $X$ as
\[
\|X\|_p = \E[|X|^p]^{1/p}.
\]
\item ($\mathfrak{L}^p$ spaces). Define $\mathfrak{L}^p\ofp=\{X:\Omega\to\Re,\text{ measurable},
\|X\|_p<\infty\}$ and equip this space with the addition and scalar multiplication
operations $(X+Y)(\omega) = X(\omega) + Y(\omega)$ and $(\alpha X)(\omega) = \alpha X(\omega)$.
This becomes a semi-normed space%
\footnote{$\|X\|=0$ does not imply that $X=0$, but instead
that $X=0$ almost surely. However, $\|\cdot\|_p$ is absolutely
homogeneous, sub-additive and nonnegative}.%
\item ($\mathcal{L}^p$ spaces). Define $\mathcal{N}\ofp = \{X:\Omega\to\Re,
\text{ measurable}, X=0 \text{ a.s.}\}$; this is the kernel of $\|\cdot\|_p$.
Then, define $\mathcal{L}^p\ofp = \mathfrak{L}^p\ofp / \mathcal{N}$.
This is a normed space where for $X\in\mathfrak{L}^p\ofp$ and $[X]=X+\mathcal{N}\in\mathcal{L}^p\ofp$
we have $\|[X]\|_p \dfn \|X\|_p$.
\item ($\infty$-norm, $\mathfrak{L}_\infty$ and $\mathcal{L}_\infty$). The infinity norm is defined as
\[
\|X\|_\infty = \esssup |X| = \inf\{\lambda \in \Re: \prob[|X|> \lambda] = 0\},
\]
or equivalently
\[
\|X\|_\infty = \inf \{\lambda \in \Re: |X|\leq \lambda,\, \prob\text{-a.s.}\}.
\]
The spaces $\mathfrak{L}_\infty\ofp$ and $\mathcal{L}_\infty\ofp$ are defined
similarly.
\item ($\mathcal{L}_\infty\ofp$ as a limit). If there is a $p'\in [1,\infty)$ such
that $X\in\mathcal{L}_\infty\cap \mathcal{L}_{p'}$, then
\[
\|X\|_\infty = \lim_{p\to\infty}\|X\|_p.
\]
\item ($\mathcal{L}_2$ is a Hilbert space). $\mathcal{L}^p\ofp$ is the only Hilbert
$\mathcal{L}^p$ space with inner product
\[
\<X,Y\> = \E[XY].
\]
\end{enumerate}
\subsection{Product spaces}
\begin{enumerate}
\item (Product $\sigma$-algebra). Let $\{X_a\}_{a\in A}$ be an indexed collection of nonempty sets; define
$X=\prod_{a\in A}X_a$ and $\pi_a: X = (x_a)_{a\in A} \mapsto x_a\in X_a$. Let $\F_a$ be a $\sigma$-algebra
on $X_a$. We define the product $\sigma$-algebra as
\[
\bigotimes_{a\in A} \F_a \dfn \sigma\left( \{\pi_a^{-1}(E_a);a\in A, E_a\in \F_a\}\right)
\]
This is the smallest $\sigma$-algebra on the product space which renders all projections measurable
(compare to the definition of the \textit{product topology} which is the smallest topology on
the product space which renders the projections \textit{continuous}).
\item (Measurability of epigraphs). Let $f:(X,\F)\to\barre$ be a measurable proper function. Its epigraph, that is
the set $\epi f \dfn \{(x,\alpha)\in X\times \Re {}\mid{} f(x) \leq \alpha\}$ and its hypo-graph, that is
the set $\hyp f \dfn \{(x,\alpha) \in X\times \Re {}\mid{} f(x) \geq \alpha\}$ are measurable in the product
measure space $(X\times \Re, \F\otimes \B_\Re)$.
\item (Measurability of graph). The graph of a measurable function $f:(X,\F,\mu)\to\Re$ is a Lebesgue-measurable set
with Lebesgue measure zero.
\item (Countable product of $\sigma$-algebras). If $A$ is countable, the product $\sigma$-algebra
is generated by the products of measurable sets $\{\prod_{a\in A}E_a; E_a\in \F_a\}$.
\item (Product measures). Let $(\mathcal{X},\F,\mu)$ and $(\mathcal{Y},\mathcal{G},\nu)$ be two measure spaces.
The product space $\mathcal{X}\times \mathcal{Y}$ becomes a measurable space with the $\sigma$-algebra
$\F\otimes \mathcal{G}$. Let $E_x\in\F$ and $E_y\in\mathcal{G}$; then $E_x\times E_y\in\F\otimes \mathcal{G}$.
We define a measure $\mu\times\nu$ on $(\mathcal{X}\times \mathcal{Y}, \F\otimes\mathcal{G})$ with
\[
(\mu\times \nu)(E_x\times E_y) = \mu(E_x) \nu(E_y).
\]
\item Let $E\in \F\otimes\mathcal{G}$ and define
$E_x = \{y\in \mathcal{Y}: (x,y)\in E\}$ and $E_y = \{x\in \mathcal{X}: (x,y)\in E\}$.
Then, $E_x\in \F$ for all $x\in\mathcal{X}$, $E_y\in\mathcal{G}$ for all $y\in\mathcal{Y}$.
\item Let $f:\mathcal{X}\times\mathcal{Y}\to \Re$ be an $\F\otimes \mathcal{G}$-measurable function.
Then, $f(x,\cdot)$ is $\mathcal{G}$-measurable for all $x\in\mathcal{X}$ and
$f(\cdot, y)$ is $\F$-measurable for all $y\in\mathcal{Y}$.
\item Let $(\mathcal{X},\F,\mu)$ and $(\mathcal{Y},\mathcal{G},\nu)$ be two $\sigma$-finite measure spaces.
For $E\in\F\otimes\mathcal{G}$, the mappings $\mathcal{X}\ni x\mapsto \nu(E_x) \in \Re$ and
$\mathcal{Y}\ni y\mapsto \mu(E_y)$ are measurable and
\[
(\mu\times \nu)(E) = \int \nu(E_x)\d \mu(x) = \int \mu(E_y)\d \nu(x)
\]
\item (Tonelli's Theorem). Let $h:\mathcal{X}\times \mathcal{Y}\to[0,\infty]$ be an $\F\otimes\mathcal{G}$-measurable
function. Let
\[
f(x) = \int_{\mathcal{Y}} h(x,y) \d\nu(y), \ g(y) = \int_{\mathcal{X}}h(x,y)\d\mu(x).
\]
Then, $f$ and $g$ are measurable and
\[
\int_{\mathcal{X}}f\d\mu = \int_{\mathcal{Y}}g\d\nu = \int_{\mathcal{X}\times\mathcal{Y}}g\d(\mu\times \nu).
\]
\item (\hypertarget{link:fubini}{Fubini's Theorem}).
Let $h:\mathcal{X}\times \mathcal{Y}\to \Re$ be an $\F\otimes\mathcal{G}$-measurable
function and
\[
\int_{\mathcal{X}} \int_{\mathcal{Y}} h(x,y)\d\nu(y) \d\mu(x) < \infty.
\]
Then, $h\in\mathcal{L}_1(\mathcal{X}\times\mathcal{Y}, \F\otimes\mathcal{G}, \mu\times\nu)$ and
\[
\int_{\mathcal{X}} \int_{\mathcal{Y}} h(x,y)\d\nu(y) \d\mu(x) =
\int_{\mathcal{Y}} \int_{\mathcal{X}} h(x,y)\d\mu(x) \d\nu(y) =
\int_{\mathcal{X}\times\mathcal{Y}} h \d(\mu\times \nu)
\]
\item (Consequence of Fubini's theorem). Let $X$ be a nonnegative random variable. Let $E=\{(\omega, x): 0 \leq x\leq X(\omega)\}$.
Then, $X(\omega) = \int_{0}^{\infty}1_{E}(\omega, x)\d x$.
\begin{align*}
\E[X] = \int_{\Omega}X\d\prob &= \int_{\Omega}\int_{0}^{\infty}1_{E}(\omega, x)\d x\d \prob\\
&= \int_{0}^{\infty} \int_{\Omega} 1_{E}(\omega, x)\d \prob \d x\\
&= \int_{0}^{\infty} \prob[X\geq x]\d x.
\end{align*}
\end{enumerate}
\subsection{Transition Kernels}\label{sec:transition_kernel}
\begin{enumerate}
\item (Definition). Let $(\mathcal{X},\F)$, $(\mathcal{Y},\G)$ be two measurable spaces and let $K:\G\times \mathcal{X}\to [0,1]$.
$K$ is called a \textit{(probability) transition kernel} if
\begin{enumerate}[i.]
\item $f_B(x) \dfn K(B,x)$ is $\F$-measurable for every $B\in\G$,
\item $\mu_x(B) \dfn K(B,x)$ is a measure on $(\mathcal{Y},\G)$ for every $x\in \mathcal{X}$.
\end{enumerate}
\item (Markov kernel). A kernel $K:\G\times \mathcal{X}\to [0,1]$ is called a \textit{Markov kernel} if
$K(\mathcal{Y}, x)=1$ for all $x\in\mathcal{X}$
\item (Existence of transition kernels). Let $\mu$ be a finite measure on
$(\mathcal{X},\F)$ and $k:\mathcal{X}\times \mathcal{Y}\to\Re_+$
be measurable in the product $\sigma$-algebra $\F\otimes \G$ and has the property $\int_{\mathcal{Y}} k(x,y)\nu(\d y)=1$. Then
the mapping $K:\mathcal{X} \times \G \to [0,1]$ given by
\[
K(B,x) = \int_B k(x,y) \mu(\d y),
\]
is a probability transition kernel.
\item (Measure on product space via a kernel).
Let $(\mathcal{X},\F)$, $(\mathcal{Y},\G)$ be two measurable spaces and let $K:\G\times\mathcal{X}\to [0,1]$
be a transition kernel. For $A\in\F$ and $B\in\G$ define
\[
\mu(A\times B) = \int_A K(B,x)\d\prob(x).
\]
This extends to a unique measure on the product space $(\mathcal{X}\times\mathcal{Y}, \F\otimes\G)$.
\end{enumerate}
\subsection{Law invariance}
\begin{enumerate}
\item (Equality in distribution).
Let $X,Y$ be two real-valued random variables on $\ofp$.
We say that $X$ and $Y$ are equal in distribution, and we denote $X\overset{\mathrm{d}}{\sim} Y$,
if $X$ and $Y$ have equal probability distribution functions, that is $F_X(s) = F_Y(s)$ for all $s$.
\item (Equal in distribution, nowhere equal). Let $\Omega = \{-1,1\}$, $\F=2^\Omega$, $\prob[\{\omega_i\}]=\frac{1}{2}$.
Let $X(\omega) = \omega$ and $Y(\omega) = -X(\omega)$. These two variables have the same distribution, but
are nowhere equal.
\item (Equal in distribution, almost nowhere equal). Take $X\sim \mathcal{N}(0,1)$ and $Y=-X$. These
two random variables are almost nowhere equal, but have the same distribution.
\item The following are equivalent:
\begin{enumerate}[i.]
\item $X\overset{\mathrm{d}}{\sim} Y$
\item $\E[e^{-rX}]=\E[e^{-rY}]$ for all $r>0$
\item $\E[f(X)] = \E[f(Y)]$ for all bounded continuous functions
\item $\E[f(X)] = \E[f(Y)]$ for all bounded Borel functions
\item $\E[f(X)] = \E[f(Y)]$ for all positive Borel functions
\end{enumerate}
\end{enumerate}
\subsection{Expectation}
\begin{enumerate}
\item (Definition)
Let $\ofp$ be a probability space and $X$ be a random variable. Then, the expected value of
$X$ is denoted by $\E[X]$ and is defined as the Lebesgue integral
\[
\E[X] = \int_{\Omega} X\d \prob
\]
\item
\label{gx1312}
Because of item~\ref{rv221030} in Sec.~\ref{sec:random_variables}, for $X\geq 0$ nonnegative
\begin{align*}
\E[X] &= \int_{0}^{+\infty} X \d \prob\\
&= \int_{0}^{+\infty} \int_0^{+\infty} 1_{X\geq t}\d t \d \prob\\
&= \int_{0}^{+\infty} \int_0^{+\infty} 1_{X\geq t} \d \prob \d t
\end{align*}
and we use the fact that
\[
\int_{0}^{+\infty} 1_{X>t}\d \prob = \prob[X>t],
\]
so
\[
\E[X] = \int_0^\infty \prob[X>t]\d t.
\]
The function $S(t) = \prob[X>t] = 1-\prob[X\leq t]$ is called the \textit{survival function}
of $X$, or its \textit{tail distribution} or \textit{exceedance}.
\item (Expectation in terms of PDF). Let \(X\) be a real-valued continuous random variable with PDF \(f_X\).
Then,
\[
\E[X] = \int_{-\infty}^{\infty} x f_X(x) \d x.
\]
\item (Expectation in terms of CDF). Let \(X\) be a real-valued random variable. Then,
\[
\E[X] = \int_{-\infty}^{\infty} x \d F(x).
\]
\item Let $\ofp$ be a probability space and $X$ a real-valued random variable thereon. Define
\[
f(\tau) = \int_{\Omega}(X-\tau)^2\d\prob.
\]
Then $\tau = \E[X]$ minimizes $f$ and the minimum value is $\mathrm{Var}[X]$.
\item Let $X$ be a real-valued random variable. Then,
\[
\sum_{n=1}^{\infty}\prob[|X|\geq n] \leq \E[|X|] \leq 1+ \sum_{n=1}^{\infty}\prob[|X|\geq n].
\]
It is $\E[|X|]<\infty$ if and only if the above series converges.
\item If $X$ takes positive integer values, then
\[
\E[X] = \sum_{n=1}^{\infty}\prob[X\geq n]
\]
\item (Finite mean, infinite variance). There are several distributions with finite mean
and infinite variance --- a standard example is the \textit{Pareto distribution}.
A random variable $X$ follows the Pareto distribution with parameters $x_m>0$ and $a$
if it has support $[x_m,\infty)$ and probability distribution
\[
\prob[X\leq x] = \frac{ax_m^a}{x^{a+1}},
\]
for $x\geq x_m$. For $a\leq 1$, $X$ has infinite mean and variance. For $a>1$, its
mean is $\E[X]=\frac{ax_m}{a-1}$ and infinite variance.
\item (Absolutely bounded a.s. $\Leftrightarrow$ Bounded moments)~[\ref{cite:Ambrosio2013}].
Let $X$ be a random variable on $\ofp$. The following are equivalent:
\begin{enumerate}[i.]
\item $X$ is almost surely absolutely bounded (i.e., there is $M\geq 0$ such that $\prob[|X|\leq M]=1$)
\item $\E[|X|^k]\leq M^k$, for all $k\in \N_{\geq 1}$
\end{enumerate}
\item (A useful formula)~[\ref{cite:RLWolpert05}]. For $q>0$
\[
\E[|X|^q] = \int_0^\infty q x^{q-1} \prob[|X|>x]\d x.
\]
\end{enumerate}
\section{Conditioning}
\subsection{Conditional Expectation}
\begin{enumerate}
\item (Conditional Expectation). Let $X$ be a random variable on $\ofp$ and $\HH\subseteq \F$.
A \textit{conditional expectation} of $X$ given $\HH$ is an $\HH$-measurable
random variable, denoted as $\ce{X}$, with
\[
\int_H \ce{X} \d\prob = \int_H X\d\prob,
\]
which equivalently can be written as
\[
\E[X 1_H] = \E[\ce{X}1_H],
\]
for all $H\in\HH$.
\item (Uniqueness). All versions of a conditional expectation, $\ce{X}$, differ only on a
set of measure zero%
\footnote{R. Durrett, ``Probability: Theory and Examples,'' 2013, Available at: \url{https://services.math.duke.edu/~rtd/PTE/PTE4_1.pdf}}.
\item (Equivalent definition). It is equivalent to define the conditional expectation of $X$,
conditioned by a $\sigma$-algebra $\HH$ as a random variable $\ce{X}$ with the property
\[
\E[X Z] = \E[\ce{X}Z],
\]
for all $\HH$-measurable random variables $Z$.
\item (Best estimator). Assuming $\E[Y^2]<\infty$, the best estimator of $Y$ given $X$ is $\E[Y{}\mid{}X]$
\item (Radon-Nikodym definition). The conditional expectation as introduced above, is the Radon-Nikodym
derivative
\[
\ce{X} = \frac{\d \mu^X_{\HH}}{\d \prob_{\HH}},
\]
where $\mu^X_{\HH}:\HH\to [0,\infty]$ is the measure induced by $X$
restricted on $\HH$, that is $\mu^X_{\HH}:H\mapsto \int_H X\d\prob$.
This is absolutely continuous with respect to $\prob$. The measure $\prob_{\HH}$
is the restriction of $\prob$ on $\HH$.
\item (Conditional expectation wrt random variable). Let $X,Y$ be random variables on $\ofp$.
The conditional expectation of $X$ given $Y$ is $\E[X\mid Y]\dfn \E[X\mid \sigma(Y)]$,
where $\sigma(Y)$ is the $\sigma$-algebra generated by $Y$, that is
$\sigma(Y) = Y^{-1}(\F) = \{Y^{-1}(B); B\in\F\}$.
\item (Conditional expectation using the push-forward $Y_*\prob$).
Let $X$ be an integrable random variable on $\ofp$. Then, there is a $Y_*\prob$-unique
random variable $\E[X\mid Y]$
\[
\int_{Y^{-1}(B)} X\d \prob = \int_B \E[X{}\mid{}Y]\d(Y_*\prob).
\]
\item (Conditioning by an event). The conditional expectation $\E[X\mid H]$, conditioned
by an event $H\in\F$ is given by
\[
\E[X\mid H] = \frac{1}{\prob[H]}\int_H X\d\prob = \frac{1}{\prob[H]}\E[X1_H].
\]
\item (Properties of conditional expectations).
The conditional expectation has the following properties:
\begin{enumerate}[i.]
\item \label{tp01} (Monotonicity). $X\leq Y \Rightarrow \ce{X} \leq \ce{Y}$
\item (Positivity). $X\geq 0 \Rightarrow \ce{X} \geq 0$ [Set $Y=0$ in~\ref{tp01}].
\item (Linearity). For $a,b\in\Re$, $\ce{aX+bY}=a\ce{X} + b \ce{Y}$
\item (Monotone convergence). $X_n\geq 0$, $X_n \uparrow X$ implies $\ce{X_n}\uparrow \ce{X}$
\item (Fatou's lemma). For $X_n\geq 0$, $\ce{\liminf_n X_n}\leq \liminf_n \ce{X_n}$
\item (Reverse Fatou's lemma).
\item (Dominated convergence theorem). $X_n\to X$ (point-wise) and $|X_n|\leq Y$ $\prob$-a.s. where $Y$ is
integrable. Then, $\ce{X}$ is integrable and
\[
\ce{X_n} \to \ce{X}.
\]
\item (Jensen's inequality). Let $X\in\mathcal{L}_1\ofp$, $f:\Re\to\Re$ convex. Then
\[
f(\ce{X})\leq \ce{f(X)}.
\]
\item (Law of total expectation). For any $\sigma$-algebra $\HH \subseteq \F$,
\[
\E[\ce{X}] = \E[X].
\]
\item (Tower property). For two $\sigma$-algebras $\HH_1$ and $\HH_2$ with $\HH_1\subseteq \HH_2$,
\[
\E[\E[X {}\mid{} \HH_1] {}\mid{} \HH_2] = \E[\E[X {}\mid{} \HH_2] {}\mid{} \HH_1] = \E[X {}\mid{} \HH_1].
\]
\item (Tower property with $X$ being $\HH_i$-measurable). Let $\HH_1\subseteq \HH_2$ be two $\sigma$-algebras.
If $X$ is $\HH_1$-measurable, then it is also $\HH_2$-measurable.
\item If $X$ is $\HH$-measurable then