\subsubsection{Application: Percolation}


We will now discuss another application of Kolmogorov's $0-1$-law, percolation.

\begin{definition}[\vocab{Percolation}]
    Consider the graph with nodes $\Z^d$, $d \ge 2$, where edges from the lattice are added with probability $p$. The added edges are called \vocab[Percolation!Edge!open]{open};
    all other edges are called
    \vocab[Percolation!Edge!closed]{closed}.

    More formally, we consider
    \begin{itemize}
        \item $\Omega = \{0,1\}^{\bE_d}$, where $\bE_d$ are all edges in $\Z^d$,
        \item $\cF \coloneqq  \text{product $\sigma$-algebra}$,
        \item $\bP \coloneqq  \left(p \underbrace{\delta_{\{1\} }}_{\text{edge is open}} + (1-p) \underbrace{\delta_{\{0\} }}_{\text{edge is absent closed}}\right)^{\otimes \bE_d}$.
    \end{itemize}
\end{definition}
\begin{question}
    Starting at the origin, what is the probability, that there exists
    an infinite path (without moving backwards)?
\end{question}
\begin{definition}
    An \vocab{infinite path} consists of an infinite sequence of distinct points
    $x_0, x_1, \ldots$
    such that $x_n$ is connected to $x_{n+1}$, i.e.~the edge $\{x_n, x_{n+1}\}$ is open.
\end{definition}

Let $C_\infty \coloneqq  \{\omega | \text{an infinite path exists}\}$.

\begin{exercise}
    Show that changing the presence / absence of finitely many edges
    does not change the existence of an infinite path.
    Therefore $C_\infty$ is an element of the tail $\sigma$-algebra.
    Hence $\bP(C_\infty) \in  \{0,1\}$.
\end{exercise}
Obviously, $\bP(C_\infty)$ is monotonic with respect to $p$.
For $d = 2$ it is known that $p = \frac{1}{2}$ is the critical value.
For $d > 2$ this is unknown.

% TODO: more in the notes
We'll get back to percolation later.


\section{Characteristic functions, weak convergence and the central limit theorem}

% Characteristic functions are also known as the \vocab{Fourier transform}.
%Weak convergence is also known as \vocab{convergence in distribution} / \vocab{convergence in law}.

So far we have dealt with the average behaviour,
\[
\frac{\overbrace{X_1 + \ldots + X_n}^{\text{i.i.d.}}}{n} \to  \bE(X_1).
\]
We now want to understand \vocab{fluctuations} from the average behaviour,
i.e.\[
X_1 + \ldots + X_n - n \cdot \bE(X_1).
\]
% TODO improve
The question is, what happens on other timescales than $n$?
An example is
\[
\frac{X_1 + \ldots + X_n - n \bE(X_1)}{\sqrt{n} } \xrightarrow{n \to \infty} hv \cN(0, \Var(X_i)) (\ast)
\]
Why is $\sqrt{n}$ the right order? (Handwavey argument)

Suppose $X_1, X_2,\ldots$ are i.i.d. $\cN(0,1)$.
The mean of the l.h.s.~is $0$ and for the variance we get
\[
\Var(\frac{X_1 + \ldots + X_n - n \bE(X_1)}{\sqrt{n} }) = \Var\left( \frac{X_1+ \ldots + X_n}{\sqrt{n} } \right) = \frac{1}{n} \left( \Var(X_1) + \ldots + \Var(X_n) \right) = 1
\]
For the r.h.s.~we get a mean of  $0$ and a variance of $1$.
So, to determine what $(\ast)$ could mean, it is necessary that $\sqrt{n}$
is the right scaling.
To define $(\ast)$ we need another notion of convergence.
This will be the weakest notion of convergence, hence it is called
\vocab{weak convergence}.
This notion of convergence will be defined in terms of characteristic functions of Fourier transforms.

\subsection{Characteristic functions and Fourier transform}

Consider $(\R, \cB(\R), \bP)$.
For every $t \in \R$ define a function $\phi(t) \coloneqq \phi_\bP(t) \coloneqq \int_{\R} e^{\i t x} \bP(dx)$.
We have
\[
\phi(t) = \int_{\R} \cos(tx) \bP(dx) + \i \int_{\R} \sin(tx) \bP(dx).
\]
\begin{itemize}
    \item Since $|e^{\i t x}| \le  1$ the function $\phi(\cdot )$ is always defined.
    \item We have $\phi(0) = 1$.
    \item $|\phi(t)| \le  \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
\end{itemize}
We call $\phi_{\bP}$ the \vocab{characteristic function} of $\bP$.

\begin{remark}
    Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and
    $X: (\Omega, \cF) \to  (\R, \cB(\R))$ is a random variable.
    Then we can define
    \[
    \phi_X(t) \coloneqq  \bE[e^{\i t x}] = \int e^{\i t X(\omega)} \bP(d \omega) = \int_{\R} e^{\i t x} \mu(dx) = \phi_\mu(t)
    \]
    where $\mu = \bP x^{-1}$.
\end{remark}

\begin{theorem}[Inversion formula] % thm1
    \label{inversionformula}
    Let $(\Omega, \cB(\R), \bP)$ be a probability space.
    Let $F$ be the distribution function of  $\bP$
    (i.e.~$F(x) = \bP((-\infty, x])$ for all $x \in \R$ ).
    Then for every $a < b$ we have
    \begin{eqnarray}
    \frac{F(b) + F(b-)}{2} - \frac{F(a) + F(a-)}{2} = \lim_{T \to \infty} \frac{1}{2 \pi} \int_{-T}^T \frac{e^{-\i t b} - e^{- \i t a}}{- \i t} \phi(t) dt
    \label{invf}
    \end{eqnarray}
    where $F(b-)$ is the left limit.
\end{theorem}
% TODO!
We will prove this later.

\begin{theorem}[Uniqueness theorem] % thm2
    \label{charfuncuniqueness}
    Let $\bP$ and $\Q$ be two probability measures on $(\R, \cB(\R))$.
    Then $\phi_\bP = \phi_\Q \implies \bP = \Q$.

    Therefore, probability measures are uniquely determined by their characteristic functions.
    Moreover, \eqref{invf} gives a representation of $\bP$ (via $F$)
    from $\phi$.
\end{theorem}
\begin{refproof}{charfuncuniqueness}
    Assume that we have already shown \autoref{inversionformula}.
    Suppose that $F$ and $G$ are the distribution functions of $\bP$ and $\Q$.
    Let $a,b \in \R$ with $a < b$.
    Assume that $a $ and $b$ are continuity points of both $F$ and $G$.
    By \autoref{inversionformula} we have
    \begin{IEEEeqnarray*}{rCl}
        F(b) - F(a) = G(b) - G(a) \label{eq:charfuncuniquefg}
    \end{IEEEeqnarray*}

    Since $F$ and $G$ are monotonic, \autoref{eq:charfuncuniquefg}
    holds for all $a < b$ outside a countable set.

    Take $a_n$ outside this countable set, such that $a_n \ssearrow -\infty$.
    Then, \autoref{eq:charfuncuniquefg} implies that
    $F(b) - F(a_n) = G(b) - G(a_n)$ hence  $F(b) = G(b)$.
    Since $F$ and $G$ are right-continuous, it follows that $F = G$.
\end{refproof}