% =================================================================
%  Lecture 10
%  Source: handwritten notes (Mathpix-converted) + Kashlak STAT 571
%
%  Note: the modes of convergence (a.s., probability, L^p, distribution)
%  and the Portmanteau theorem belong to Lecture 9; here we focus on
%  the two Borel--Cantelli lemmas and Prohorov's theorem.
% =================================================================
\section[Lecture 10 -- Borel-Cantelli; Prohorov]{Lecture 10 \textemdash{} Borel--Cantelli Lemmas; Prohorov's Theorem}
\label{sec:lec10}

Building on the modes of convergence from Lecture~\ref{sec:lec09}, we
record two indispensable tools: the Borel--Cantelli lemmas, which give
almost-sure conclusions from summability of probabilities, and
Prohorov's theorem, which is the sequential-compactness statement
underlying weak convergence of probability measures.

\subsection{Borel--Cantelli lemmas}

Let \((\Omega,\Fcal,\mu)\) be a probability space. For
\(\{A_i\}_{i=1}^{\infty}\subseteq\Fcal\), define
\[
\limsup_i A_i \;=\; \bigcap_{i=1}^{\infty}\bigcup_{j>i}A_j,
\qquad
\liminf_i A_i \;=\; \bigcup_{i=1}^{\infty}\bigcap_{j>i}A_j.
\]

\begin{remark}
The set \(\limsup_i A_i\) is also called ``\(A_i\) \emph{infinitely
often}'' (\(A_i\) i.o.), since \(\omega\in\limsup_i A_i\) means that
for every \(N\in\N\) there exists \(n>N\) with \(\omega\in A_n\).
Symmetrically, \(\liminf_i A_i\) is ``\(A_i\) \emph{eventually}''
(\(A_i\) ev.): there exists \(N\) such that \(\omega\in A_n\) for all
\(n\ge N\).
\end{remark}

\begin{lemma}{First Borel--Cantelli}{borel-cantelli-1}
Let \(\{A_i\}_{i=1}^{\infty}\) with \(A_i\in\Fcal\). If
\(\sum_{i=1}^{\infty}\mu(A_i)<\infty\), then
\[
\mu\bigl(\limsup_i A_i\bigr) \;=\; 0,
\]
i.e.\ the set of \(\omega\) lying in infinitely many \(A_i\) has
probability zero.
\end{lemma}

\begin{lemma}{Second Borel--Cantelli}{borel-cantelli-2}
Let \(\{A_i\}_{i=1}^{\infty}\) be an \emph{independent} collection with
\(A_i\in\Fcal\). If \(\sum_{i=1}^{\infty}\mu(A_i)=\infty\), then
\[
\mu\bigl(\limsup_i A_i\bigr) \;=\; 1,
\]
i.e.\ the set of \(\omega\) lying in infinitely many \(A_i\) has
probability one.
\end{lemma}

\begin{remark}
The two lemmas together produce a sharp \(0\)-\(1\) dichotomy for
independent events: \(\mu(\limsup_i A_i)\) is \(0\) or \(1\) according
as \(\sum\mu(A_i)\) converges or diverges. This is the fundamental tool
for almost-sure statements; it is used in the proof of the strong law
of large numbers in the next lecture.
\end{remark}

\subsection{Prohorov's theorem}

We close with sequential compactness for weak convergence: when does a
sequence of probability measures admit a weakly convergent
subsequence? The condition is \emph{tightness}, the measure-theoretic
analogue of boundedness in the Bolzano--Weierstrass theorem.

\begin{definition}{Uniform tightness}{uniform-tightness}
A collection \(\{\mu_i\}_{i\in I}\) of probability measures on a metric
space \(S\) is \emph{uniformly tight} if for every \(\varepsilon>0\)
there exists a compact set \(K_\varepsilon\subseteq S\) such that
\[
\mu_i(K_\varepsilon) \;>\; 1-\varepsilon
\qquad \text{for all } i\in I.
\]
\end{definition}

\begin{theorem}{Prohorov's theorem}{prohorov}
Let \(\{\mu_i\}_{i=1}^{\infty}\) be a sequence of probability measures
on a metric space \(S\). If \(\{\mu_i\}\) is uniformly tight, then it
is \emph{relatively (sequentially) compact} for weak convergence: every
subsequence \(\mu_{i_k}\) admits a further subsequence \(\mu_{i_{k_r}}\)
and a probability measure \(\mu\) (depending on the subsequence) with
\(\mu_{i_{k_r}}\Rightarrow\mu\).
\end{theorem}

The next proposition is the standard ``subsubsequence'' upgrade: if
every subsequence has a further subsequence converging to the
\emph{same} limit, then the whole sequence converges.

\begin{proposition}{Subsubsequence criterion}{subsubsequence}
Let \(\{\mu_i\}_{i=1}^{\infty}\) and \(\mu\) be probability measures on
\(S\). If for every subsequence \(\mu_{i_k}\) there exists a further
subsequence \(\mu_{i_{k_r}}\Rightarrow\mu\), then
\(\mu_i\Rightarrow\mu\).
\end{proposition}