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\begin{remark*}
  For probability measures, if $X_n$ to $X$
  \gls{vc_wly}, then
  $\EE f(X_n) \to \EE f(X)$ for all $f$ bounded
  continuous.
\end{remark*}

\begin{proof}[Proof of \nameref{levy_thm}]
  Let $g : \RR^d \to \RR$ be compactly supported
  and Lipschitz continuous, i.e.
  \[ |g(x) - g(y)| \le Cg|x - y| \qquad \forall x,
  y \in \RR^d \]
  (any $g \in C_c^\infty$ will do). Enough to show
  $\EE g(X_n) \to \EE g(X)$ (by
  \nameref{lec20_Cc_infty_remark} last lecture).
  Let $Z \sim \normaldist(0, I_d)$ independent of
  $(X_n)$, $X$. Then for fixed $\eps > 0$, choose
  $t$ small enough so that
  \[ Cg\sqrt{t} \EE|Z| \le \frac{\eps}{3} .\]
  Then,
  \begin{align*}
    |\EE g(X_n) - \EE g(X)|
    &\le |\EE g(X_n) - \EE g(X_n + \sqrt{t} Z)|
    + |\EE g(X) - \EE g(X + \sqrt{t} Z)|
    \\
    &~~~~+ |\EE g(X_n + \sqrt{t} Z) - \EE g(X +
    \sqrt{t}Z)| \\
    &\le \EE |g(X_n) - g(X_n + \sqrt{t} Z)| +
    \EE|g(X) - g(X + \sqrt{t} Z)| \\
    &~~~~+ |\EE g(X_n +
    \sqrt{t}Z) - \EE g(X + \sqrt{t} Z)| \\
    &\le \EE|Cg\sqrt{t} Z| + \EE|Cg\sqrt{t} \EE Z|
    + |\EE g(X_n + |sqrt{t} Z) - \EE g(X +
    \sqrt{Z})| \\
    &\le \ub{Cg\sqrt{t} \EE|Z| + Cg\sqrt{t}
    \EE|Z|}_{\le 2 \frac{\eps}{3}} +
    |\EE g(X_n + \sqrt{t} Z) - \EE g(X + \sqrt{t}
    Z)|
  \end{align*}
  $X_n + \sqrt{t} Z$ has density $\mu_{X_n} \conv
  \gt \eqdef f_{t, n}$. Then by \gls{FI},
  \[ f_{t, n}(x) = \frac{1}{(2\pi)^d} \int
  \phi_{X_n}(u) e^{-\frac{t|u|^2}{2}} e^{-i\langle
  u, x\rangle} \dd u \]
  So
  \begin{align*}
    \EE g(X_n + \sqrt{t}Z)
    &= \frac{1}{(2\pi)^d} \iint g(x) \phi_{X_n}(u)
    e^{-\frac{t|u|^2}{2}} e^{-i\langle u,
    x\rangle} \dd u \dd x \\
    &\stackrel[n \to \infty]{\text{\nameref{dct}}}{\longrightarrow}
    \frac{1}{(2\pi)^d} \iint g(x)
    \phi_X(u) e^{-\frac{t|u|^2}{2}} e^{-i\langle
    u, x\rangle} \dd u \dd x \\
    &= \EE (X + \sqrt{t} Z)
  \end{align*}
  (using $g(x) e^{-\frac{t|u|^2}{2}}$ as the
  bounding function to apply \nameref{dct}).
\end{proof}

\begin{flashcard}[clt-thm]
\begin{theorem*}[Central Limit Theorem]
  \label{clt}
  \cloze{
  Let $(X_n)$ be IID random variables on $\RR$
  with $\EE(X_i) = 0$, $\Var(X_i) = 1$ for all
  $i$. Then for $S_n = X_1 + \cdots + X_n$, we
  have, $\frac{S_n}{\sqrt{n}} \to Z \sim
  \normaldist(0, 1)$ \gls{vc_wly} or \emph{in
  distribution}, i.e.
  \[ \PP \left( \frac{S_n}{\sqrt{n}} \le x \right)
  \stackrel{n \to \infty}{\longrightarrow} \PP(Z
  \le x) \qquad \forall x \]
  }
\end{theorem*}

\begin{proof}
  \cloze{
  Set $\phi(u) = \phi_{X_1}(u) = \EE e^{iu X_1}$.
  Then $\phi(0) = 1$ and since $\EE X_1^2 <
  \infty$, we can differentiate under the integral
  sign and get
  \begin{align*}
    \phi'(u)
    &= i\EE X_1 e^{iu X_1} \\
    \phi''(u)
    &= i^2 \EE X_1^2 e^{iu X_1}
  \end{align*}
  (see \es[7.4]{3}), i.e. $\phi'(0) = 0$,
  $\phi''(0) = -1$. By Taylor's Theorem, as $u \to
  0$, $\phi(u) = 1 - \frac{u^2}{2} + o(u^2)$. Let
  $\phi_n$ be the \gls{char_func} of
  $\frac{S_n}{\sqrt{n}}$. Then
  \begin{align*}
    \phi_n(u)
    &= \EE e^{iu \frac{S_n}{\sqrt{n}}} \\
    &= \EE e^{i \frac{u}{\sqrt{n}}(X_1 + \cdots +
    X_n)} \\
    &\stackrel{\text{IID}}{=}
    ( \EE e^{i \frac{u}{\sqrt{n}} X_1} )^n \\
    &= \left( \phi \left( \frac{u}{\sqrt{n}}
    \right) \right)^n \\
    &= \left( 1 - \frac{u^2}{2n} + o \left(
    \frac{u^2}{n} \right) \right)^n \\
    &= \left( 1 - \frac{u^2}{2n} + o \left(
    \frac{1}{n} \right) \right)^n
  \end{align*}
  (as $u$ fixed, $n \to \infty$). The complex
  logarithm satisfies, as $z \to 0$, $\log(1 + z)
  = z + o(z)$. So,
  \begin{align*}
    \log \phi_n(u)
    &= n \log \left( 1 - \frac{u^2}{2n} + o \left(
    \frac{1}{n} \right) \right) \\
    &= n \left( -\frac{u^2}{2n} + o \left(
    \frac{1}{n} \right) \right) \\
    &= -\frac{u^2}{n} + o(1)
  \end{align*}
  So, $\phi_n(u) \to e^{-\frac{u^2}{2}} =
  \phi_Z(u)$. So by \nameref{levy_thm},
  $\frac{S_n}{\sqrt{n}} \to Z$ \gls{vc_wly}.}
\end{proof}
\end{flashcard}

\begin{remark*}
  The \nameref{clt} in $\RR^d$ can be proved similarly
  using the \nameref{Cramer_Wold} and properties
  of multivariate Gaussians (Exercise).
\end{remark*}
\vspace{-1em}

A random variable on $\RR$ is Gaussian
$(\normaldist(\mu, \sigma^2))$ if it has density
$\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x -
\mu)^2}{2\sigma^2}}$ for $\mu \in \RR$, $\sigma >
0$.

\begin{definition*}
  $X$ in $\RR^d$ is Gaussian if
  \[ \langle u, X\rangle = \sum_{i = 1}^d u_i X_i \]
  is Gaussian for all $u \in \RR^d$.
\end{definition*}

\begin{example*}
  If $X_1, \ldots, X_n \iidsim \normaldist(0, 1)$,
  then $X = (X_1, \ldots, X_n)$ is Gaussian in
  $\RR^n$.
\end{example*}

\begin{proposition*}
  Let $X$ be Gaussian in $\RR^n$ and $A$ be an $m
  \times n$ matrix, $b \in \RR^m$. Then
  \begin{enumerate}[(a)]
    \item $AX + b$ is Gaussian in $\RR^m$.
    \item $X \in \Lp[2]$ and $\mu_X$ is determined
      by $\EE(X) = \mu$ and $(\Cov(X_i, X_j))_{i,
      j} = V$.
    \item $\cf_X(u) = e^{i\langle u, \mu\rangle -
      \frac{\langle u, Vu\rangle}{2}}$ for all $u
      \in \RR^n$.
    \item If $V$ is invertile, then $X$ has a pdf
      on $\RR^n$ given by
      \[ f_X(x) = \frac{1}{(2\pi)^{n/2}}
      \frac{1}{|V|^{1/2}} e^{-\langle x - \mu ,
      V^{-1} (x - \mu) \rangle/2} \]
    \item If $X = (X_1, X_2)$ with $X_1 \in
      \RR^{n_1}$, $X_2 \in \RR^{n_2}$, ($n_1 + n_2
      = n$), then $X_1, X_2$ independent if and
      only if $\Cov(X_1, X_2) = 0$.
  \end{enumerate}
\end{proposition*}

\begin{proof}
  Easy. Also see \es{4} and lecturer's online
  notes.
\end{proof}

\subsubsection*{Law of Large Numbers}

Weak Law of Large Numbers:
For $(X_i)$ IID with $\EE X_i = \mu$, $\Var(X_i)
< \infty$. Then $\forall \eps > 0$,
\begin{align*}
  \PP \left( \left| \frac{1}{n} \sum_{i = 1}^n X_i
  - \mu \right| > \eps \right)
  &\le  \frac{1}{n^2 \eps^2} \Var \left( \sum_{i =
  1}^n X_i \right) \\
  &= \frac{\cancel{n} \Var(X_1)}{n^{\cancel{2}}
  \eps^2} \\
  &\to 0
\end{align*}
as $n \to \infty$. So,
\[ \frac{1}{n} \sum_{i = 1}^n X_i \convP \mu \]
Strong Law of Large Numbers: if $(X_i)$ are IID,
$\EE(X_1) = \mu < \infty$, then
\[ \frac{1}{n} \sum_{i = 1}^n X_i \to \mu \]
\gls{al_surely} as $n \to \infty$.
