From 85e63b912b55daa0fc736c845209e64cfd019aeb Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 16 May 2023 17:49:09 +0200 Subject: [PATCH] lecture 12 --- inputs/lecture_11.tex | 1 + inputs/lecture_12.tex | 226 +++++++++++++++++++++++++++++++++++++++++ inputs/lecture_9.tex | 7 +- probability_theory.tex | 1 + 4 files changed, 231 insertions(+), 4 deletions(-) create mode 100644 inputs/lecture_12.tex diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index bc4fdb7..b620f44 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -65,6 +65,7 @@ and $\Var(\frac{S_n - \bE[S_n]}{\sqrt{\Var(S_n)}}) = 1$. \lim_{n \to \infty} \bP\left[\frac{S_n - n \mu}{\sigma \sqrt{n} } \le x\right] = \Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2 \pi} } e^{\frac{-t^2}{2}}dt. \] \end{theorem} +We will abbreviate the central limit theorem by \vocab{CLT}. There exists a special case of this theorem, which was proved earlier: \begin{theorem}[de-Moivre (1730, $p = 0.5$), Laplace (1812, general $p$ )] diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex new file mode 100644 index 0000000..acec514 --- /dev/null +++ b/inputs/lecture_12.tex @@ -0,0 +1,226 @@ +% Lecture 12 2023-05-16 + +We now want to prove \autoref{clt}. +The plan is to do the following: +\begin{enumerate}[1.] + \item Identify the characteristic function of a standard normal + \item Show that the characteristic functions of the $V_n$ converge pointwise + to that of $\cN$. + \item Apply \autoref{levycontinuity} +\end{enumerate} + +First, we need to prove some properties of characteristic functions. + +\begin{lemma} + \label{charfprops} + For every real random variable $X$, we have + \begin{enumerate}[(i)] + \item $\phi_X(0) = 1$ and $|\phi_X(t)| \le 1$ for all $t \in \R$. + \item $\phi_X$ is uniformly continuous. + \item If $\bE[|X|^n] < \infty$ for any $n \in \N$, then $\phi_X$ i + $n$-times continuously differentiable + and $\bE[X^n] = (-\i)^n \phi_X^{(n)}(0)$. + \item For independent random variables $X$ and $Y$, we have + \[ + \phi_{X + Y}(t) = \phi_X(t) \cdot \phi_Y(t). + \] + \end{enumerate} +\end{lemma} +\begin{refproof}{charfprops} + \begin{enumerate}[(i)] + \item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$. + For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE|e^{\i t X}|] = 1$. + + \item Let $t, h \in \R$. + Then + \begin{IEEEeqnarray*}{rCl} + |\phi_X(t+h) - \phi_X(t)| &=& |\bE[e^{\i (t+h) X} - e^{\i t X}]|\\ + &=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\ + &\overset{\text{Jensen}}{\le}& + \bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\ + &=& \bE[|e^{\i h X} - 1|]\\ + \end{IEEEeqnarray*} + + Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$. + We show that $\lim_{h \to 0} g(h) = 0$. + + For all $\omega \in \Omega$, we realize + \[ + \lim_{h \to 0} |e^{\i h X(\omega)}- 1| = 0. + \] + Thus $|e^{\i h X} - 1| \xrightarrow{h \to 0} 0$ almost surely. + Since also for all $h \in \R$ we have $|e^{\i h X} - 1| \le 2$, + it follow that $|e^{\i h X} - 1|$ is dominated for all $h \in \R$. + Thus, we can apply the dominated convergence theorem % TODO REF + and obtain + \[ + \lim_{h \to 0} g(h) = \lim_{h \to 0} \bE[|e^{\i h X} - 1|] + = \bE[\lim_{h \to 0} |e^{\i h X} - 1|] = 0. + \] + It follows that + \[ + \lim_{n \to 0} \sup_{t \in \R} | \phi_X(t + h) - \phi_X(t)| = 0, + \] + which means that $\phi_X$ is uniformly continuous. + \item + \begin{claim} + \label{charfprop:c1} + For $y \in \R$, we have $|e^{\i y} - 1| \le |y|$. + \end{claim} + \begin{subproof} + For $y \ge 0$, we have + \begin{IEEEeqnarray*}{rCl} + |e^{\i y} - 1| &=& |\int_0^y \cos(s) \d s + \i \int_0^y \sin(s) \d s|\\ + &=& |\int_0^y e^{\i s} \d s|\\ + &\overset{\text{Jensen}}{\le}& \int_0^y |e^{\i s}| ds = y. + \end{IEEEeqnarray*} + For $y < 0$, we have $|e^{\i y} - 1| = |e^{-\i y} - 1|$ + and we can apply the above to $-y$. + \end{subproof} + + First, we look at $n = 1$. Then $\bE[|X|] < \infty$. + Consider + \[ + \frac{\phi_X(t + h) - \phi_X(t)}{h} = \bE\left[e^{\i t X} \frac{e^{\i h X} - 1}{h}\right]. + \] + We have $e^z = \sum_{n = 0}^\infty \frac{z^k}{n!}$. + Hence + \begin{IEEEeqnarray*}{rCl} + \lim_{n \to \infty} e^{\i t X} \left( \frac{1 + \i h X + \frac{(\i h X)^2}{2} + o(h^2) - 1}{h} \right) + &=& e^{\i t X} \i X \text{ almost surely.} + \end{IEEEeqnarray*} + + For arbitrary $h \in \R$, we have + \begin{IEEEeqnarray*}{rCl} + |e^{\i t X} \frac{e^{\i h X}}{h}| &\le & \left| \frac{1}{h} \left( e^{\i h X} - 1 \right)\right|\\ + &\overset{\text{\autoref{charfprop:c1}}}{\le }& \left|\frac{1}{h} \i h X\right| = |X|. + \end{IEEEeqnarray*} + Thus the dominated convergence theorem can be applied and we obtain + \[ + \lim_{h \to 0} \frac{\phi_X(t + h) - \phi_X(t)}{h} = \lim_{h \to 0} \bE\left[ e^{\i t X} \left( \frac{e^{\i h X}-1}{h} \right) \right] + = \bE[e^{\i t X} \i X]. + \] + + It follows that $\phi_X$ is differentiable and $\phi_X(t) = \bE[e^{\i t X} \i X]$. + For $t = 0$ we get $\phi'_X(0) = \i \bE[X]$, i.e.~ + -$\i \phi'_X(0) = \bE[X]$. + + Adapting the proof of (ii) gives that + $\phi'_X(t)$ is continuous. + + Adapting the proof of (iii) gives + the statement for arbitrary $n \in \N$. + + \item Easy exercise. + \end{enumerate} +\end{refproof} + +\begin{lemma}\label{lec12_2} % Lemma 2 + For $X \sim \cN(0,1)$, we have $\phi_X(t) = e^{-\frac{t^2}{2}}$ + for all $t \in \R$. +\end{lemma} +\begin{refproof}{lec12_2} + We have + \begin{IEEEeqnarray*}{rCl} + \phi_X(t) &=& \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^\infty e^{\i t x} e^{-\frac{x^2}{2}} \d x\\ + &=& \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^\infty (\cos(tx) + \i \sin(tx)) e^{-\frac{x^2}{2}} \d x\\ + &=& \frac{1}{\sqrt{2 \pi} } \int_{-\infty}^\infty \cos(t x) e^{-\frac{x^2}{2}} \d x,\\ + \end{IEEEeqnarray*} + since, as $x \mapsto \sin(tx)$ is odd and $x \mapsto e^{-\frac{x^2}{2}}$ + is even, their product is odd, wich gives that the integral is $0$. + + \begin{IEEEeqnarray*}{rCl} + \phi'_X(t) &=& \bE[\i X e^{\i t X}] \\ + &=& \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \i x \left( \cos(t x) + \i \sin(tx) \right) e^{-\frac{x^2}{2}} \d x\\ + &=& \frac{1}{\sqrt{2 \pi}} \left( \i \int_{-\infty}^\infty x \cos(tx) \right) e^{-\frac{x^2}{2}} \d x\\ + &=& \frac{1}{\sqrt{2 \pi} } \left(\underbrace{\i \int_{-\infty}^\infty x \cos(tx) e^{-\frac{x^2}{2}} \d x}_{= 0} + \int_{-\infty}^\infty - \sin(t x) e^{-\frac{x^2}{2}} \d x\right)\\ + &=& \int_{-\infty}^\infty \underbrace{\sin(tx)}_{y(x)} \underbrace{ \frac{1}{\sqrt{2 \pi} }(-x) e^{\i\frac{x^2}{2}}}_{f'(x)} \d x\\ + &=& \underbrace{[ \sin(tx) \frac{1}{\sqrt{2 \pi} e^{-\frac{x^2}{2}}}]_{x=-\infty}^\infty}_{=0} + - \int_{-\infty}^\infty t \cos(tx) \frac{1}{\sqrt{2 \pi} } e^{-\frac{x^2}{2}} \d x\\ + &=& -t \phi_X(t) + \end{IEEEeqnarray*} + Thus, for all $t \in \R$ + \[ + (\log(\phi_X(t))' = \frac{\phi'_X(t)}{\phi_X(t)} = -t. + \] + Hence there exists $c \in \R$, such that + \[ + \log(\phi_X(t)) = -\frac{t^2}{2} + c. + \] + Since $\phi_X(0) = 1$, we obtain $c = 0$. + Thus + \[ + \phi_X(t) = e^{-\frac{t^2}{2}}. + \] +\end{refproof} + + +Now, we can finally prove the CLT: +\begin{refproof}{clt} + Let $X_1,X_2,\ldots$ be i.i.d.~random variables + with $\bE[X_1] = \mu_1$, $\Var(X_1) = \sigma^2$. + + Let + \[ + Y_i \coloneqq \frac{X_i - \mu}{\sigma} + \] + i.e.~we normalize to $\bE[Y_1] = 0$ and $\Var(Y_1) = 1$. + We need to show that + \[ + V_n \coloneqq \frac{S_n - n \mu}{ \sigma \sqrt{n}} = \frac{Y_1+ \ldots + Y_n}{\sqrt{n}} \xrightarrow{\omega, n\to \infty} \cN(0,1) % TODO + \] + Let $t \in \R$. + Then + \begin{IEEEeqnarray*}{rCl} + \phi_{V_n}(t) = \bE[e^{\i t Y_n}] = \bE[e^{\i t \left( \frac{Y_1 + \ldots + Y_n}{\sqrt{n} } \right) }] \\ + &=& \bE[e^{\i t \frac{Y_1}{\sqrt{n}}}] \cdot \ldots \cdot \bE[e^{\i t \frac{Y_n}{\sqrt{n} }}]\\ + &=& \left( \phi(\frac{t}{\sqrt{n} } \right)^n. + \end{IEEEeqnarray*} + where $\phi(t) \coloneqq \phi_{Y_1}(t)$. + + We have + \begin{IEEEeqnarray*}{rCl} + \phi(s) &=& \phi(0) + \phi'(0) s + \frac{\phi''(0)}{2} s^2 + o(s^2), \text{as $s \to 0$}\\ + &=& 1 - \underbrace{\i \bE[Y_1] s}_{=0} + - \bE[Y_1^2] \frac{s^2}{2} + o(s^2)\\ + &=& 1 - \frac{s^2}{2} + o(s^2), \text{as $s \to $} + \end{IEEEeqnarray*} + + Setting $s \coloneqq \frac{t}{\sqrt{n}}$ we obtain + \[ + \phi\left(\frac{t}{ \sqrt{n} }\right) = 1 - \frac{t^2}{2n} + o\left( \frac{t^2}{n} \right) \text{ as $n \to \infty$} + \] + + + \[ + \phi_{V_n}(t) = \left( \phi\left( \frac{t}{\sqrt{n} } \right) \right)^n = + (1 - \frac{t^2}{2 n } + o\left( \frac{t^2}{n} \right)^n \xrightarrow{n \to \infty} e^{-\frac{t^2}{2}}, + \] + where we have used the following: + + \begin{claim} + For a sequence $a_n, n\in \N$ with $\lim_{n \to \infty} n a_n = \lambda$, + it holds that $\lim_{n \to \infty} (1 + a_n)^n = e^{\lambda}$. + \end{claim} + + + We have shown that + \[ + \phi_n(t) \xrightarrow{n \to \infty} e^{-\frac{t^2}{2}} = \phi_N(t). + \] + Using \autoref{levycontinuity}, we obtain \autoref{clt}. +\end{refproof} + +\begin{remark} + If $X: \Omega \to \R^d$ with distribution $\nu$, + we define + \begin{IEEEeqnarray*}{rCl} + \phi_X: \R^d &\longrightarrow & \C \\ + t &\longmapsto & \bE[e^{\i \langle t , X\rangle}] + \end{IEEEeqnarray*} + where $\langle t, X\rangle \coloneqq \sum_{i = 1}^d t_i X_i$. +\end{remark} +Exercise: Find out, which properties also hold for $d > 1$. + + + diff --git a/inputs/lecture_9.tex b/inputs/lecture_9.tex index 04c26f5..90ab3b8 100644 --- a/inputs/lecture_9.tex +++ b/inputs/lecture_9.tex @@ -43,9 +43,8 @@ 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}. -We will abbreviate the central limit theorem by \vocab{CLT}. +% 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, \[ @@ -79,7 +78,7 @@ This notion of convergence will be defined in terms of characteristic functions \subsection{Characteristic functions and Fourier transform} Consider $(\R, \cB(\R), \bP)$. -For every $t \in \R$ define a function $\phi(t) \coloneqq \phi_\bP(t) \coloneqq \int_{\R} e^{\i t x} \bP(dx)$. +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). diff --git a/probability_theory.tex b/probability_theory.tex index 3a5eefc..5b1acb1 100644 --- a/probability_theory.tex +++ b/probability_theory.tex @@ -35,6 +35,7 @@ \input{inputs/lecture_9.tex} \input{inputs/lecture_10.tex} \input{inputs/lecture_11.tex} +\input{inputs/lecture_12.tex} \cleardoublepage