some facts
This commit is contained in:
parent
f097e9adb9
commit
bf808be83e
3 changed files with 60 additions and 5 deletions
|
@ -83,6 +83,35 @@ This will be the weakest notion of convergence, hence it is called
|
|||
This notion of convergence will be defined in terms of
|
||||
characteristic functions of Fourier transforms.
|
||||
|
||||
\subsection{Convolutions${}^\dagger$}
|
||||
|
||||
\begin{definition}+[Convolution]
|
||||
Let $\mu$ and $\nu$ be probability measures on $\R^d$
|
||||
with Lebesgue densities $f_\mu$ and $f_\nu$.
|
||||
Then the \vocab{convolution} of $\mu$ and $\nu$,
|
||||
$\mu \ast \nu$,
|
||||
is the probability measure on $\R^d$
|
||||
with Lebesgue density
|
||||
\[
|
||||
f_{\mu \ast \nu}(x) \coloneqq
|
||||
\int_{\R^d} f_\mu(x - t) f\nu(t) \lambda^d(\dif t)
|
||||
\]
|
||||
|
||||
\end{definition}
|
||||
|
||||
\begin{fact}+[Exercise 6.1]
|
||||
If $X_1,X_2,\ldots$ are independent with
|
||||
distributions $X_1 \sim \mu_1$,
|
||||
$X_2 \sim \mu_2, \ldots$,
|
||||
then $X_1 + \ldots + X_n$
|
||||
has distribution
|
||||
\[
|
||||
\mu_1 \ast \mu_2 \ast \ldots \ast \mu_n.
|
||||
\]
|
||||
\end{fact}
|
||||
\todo{TODO}
|
||||
|
||||
|
||||
\subsection{Characteristic Functions and Fourier Transform}
|
||||
|
||||
\begin{definition}
|
||||
|
@ -106,7 +135,30 @@ We have
|
|||
\item We have $\phi(0) = 1$.
|
||||
\item $|\phi(t)| \le \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
|
||||
\end{itemize}
|
||||
\todo{Properties of characteristic functions}
|
||||
|
||||
\begin{fact}+
|
||||
Let $X$, $Y$ be independent random variables
|
||||
and $a,b \in \R$.
|
||||
Then
|
||||
\begin{itemize}
|
||||
\item $\phi_{a X + b}(t) = e^{\i t b} \phi_X(\frac{t}{a})$,
|
||||
\item $\phi_{X + Y}(t) = \phi_X(t) + \phi_Y(t)$.
|
||||
\end{itemize}
|
||||
\end{fact}
|
||||
\begin{proof}
|
||||
We have
|
||||
\begin{IEEEeqnarray*}{rCl}
|
||||
\phi_{a X + b}(t) &=& \bE[e^{\i t (aX + b)}}]\\
|
||||
&=& e^{\i t b} \bE[e^{\i t a X}]\\
|
||||
&=& e^{\i t b} \phi_X(\frac{t}{a}).
|
||||
\end{IEEEeqnarray*}
|
||||
Furthermore
|
||||
\begin{IEEEeqnarray*}{rCl}
|
||||
\phi_{X + Y}(t) &=& \bE[e^{\i t (X + Y)}]\\
|
||||
&=& \bE[e^{\i t X}] \bE[e^{\i t Y}]\\
|
||||
&=& \phi_X(t) \phi_Y(t).
|
||||
\end{IEEEeqnarray*}
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and
|
||||
|
|
|
@ -1,7 +1,5 @@
|
|||
\lecture{10}{2023-05-09}{}
|
||||
|
||||
% RECAP
|
||||
|
||||
First, we will prove some of the most important facts about Fourier transforms.
|
||||
|
||||
We consider $(\R, \cB(\R))$.
|
||||
|
|
|
@ -93,7 +93,13 @@ In this lecture we recall the most important point from the lecture.
|
|||
|
||||
\item Non-examples: $(\delta_n)_n$
|
||||
\item How does one prove weak convergence? How does one write this down in a clear way?
|
||||
% TODO
|
||||
\begin{itemize}
|
||||
\item \autoref{lec10_thm1},
|
||||
\item Levy's continuity theorem
|
||||
\ref{levycontinuity},
|
||||
\item Generalization of Levy's continuity theorem
|
||||
\ref{genlevycontinuity}
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
|
||||
\paragraph{Convolution}
|
||||
|
@ -104,7 +110,6 @@ In this lecture we recall the most important point from the lecture.
|
|||
\end{itemize}
|
||||
|
||||
|
||||
|
||||
\subsubsubsection{CLT}
|
||||
\begin{itemize}
|
||||
\item Statement of the CLT
|
||||
|
|
Loading…
Reference in a new issue