s23-probability-theory/inputs/lecture_23.tex

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2023-07-07 17:42:38 +02:00
\lecture{23}{2023-07-06}{}
\section{Recap}
In this lecture we will recall the most important point from the lecture.
\subsection{Construction of iid random variables.}
\begin{itemize}
\item Definition of a consistent family (\autoref{def:consistentfamily})
\item Important construction:
Consider a distribution function $F$ and define
\[
\prod_{i=1}^n (F(b_i) - F(a_i)) \text{\reflectbox{$\coloneqq$}}
\mu_n \left( (a_1,b_1] \times x \ldots \times x (a_n, b_n] \right).
\]
\item Examples of consistent and inconsistent families
\todo{Exercises}
\item Kolmogorov's consistency theorem
(\autoref{thm:kolmogorovconsistency})
\end{itemize}
\subsection{Limit theorems}
\begin{itemize}
\item Work with iid.~random variables.
\item Notions of convergence (\autoref{def:convergence})
\item Implications between different notions of convergence (very important) and counter examples.
(\autoref{thm:convergenceimplications})
\item \begin{itemize}
\item Laws of large numbers: (\autoref{lln})
\begin{itemize}
\item WLLN: convergence in probability
\item SLLN: weak convergence
\end{itemize}
\end{itemize}
\item \autoref{thm2} (building block for SLLN):
Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$,
then $ \sum X_n $ converges a.s.
\begin{itemize}
\item Counter examples showing that $\impliedby$ does not hold in general are important
\item $\impliedby$ holds for iid.~uniformly bounded random variables
\item Application:
$\sum_{i=1}^{\infty} \frac{(\pm_1)}{n^{\frac{1}{2} + \epsilon}}$ converges a.s.~for all $\epsilon > 0$.
$\sum \frac{\pm 1}{ n^{\frac{1}{2} -\epsilon}}$ does not converge a.s.~for any $\epsilon > 0$.
\end{itemize}
\item Kolmogorov's inequality (\autoref{thm:kolmogorovineq})
\item Kolmogorov's $0-1$-law. (\autoref{kolmogorov01})
In particular, a series of independent random variables converges with probability $0$ or $1$.
\item Kolmogorov's 3 series theorem. (\autoref{thm:kolmogorovthreeseries})
\begin{itemize}
\item What are those $3$ series?
\item Applications
\end{itemize}
\end{itemize}
\subsubsection{Fourier transform / characteristic functions / weak convegence}
\begin{itemize}
\item Definition of Fourier transform
(\autoref{def:characteristicfunction})
\item The Fourier transform uniquely determines the probability distribution.
It is bounded, so many theorems are easily applicable.
\item Uniqueness theorem (\autoref{charfuncuniqueness}),
inversion formula (\autoref{inversionformula}), ...
\item Levy's continuity theorem (\autoref{levycontinuity}),
(\autoref{genlevycontinuity})
\item Bockner's theorem for positive definite function % TODO REF
\item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
\item Related notions
\todo{TODO}
\begin{itemize}
\item Laplace transforms $\bE[e^{-\lambda X}]$ for some $\lambda > 0$
(not done in the lecture, but still useful).
\item Moments $\bE[X^k]$ (not done in the lecture, but still useful)
All moments together uniquely determine the distribution.
\end{itemize}
\end{itemize}
\paragraph{Weak convergence}
\begin{itemize}
\item Definition of weak convergence % ( test against continuous, bounded functions).
(\autoref{def:weakconvergence})
\item Examples:
\begin{itemize}
\item $(\delta_{\frac{1}{n}})_n$,
\item $(\frac{1}{2} \delta_{-\frac{1}{n}} + \frac{1}{2} \delta_{\frac{1}{n}}$,
\item $(\cN(0, \frac{1}{n}))_n$,
\item $(\frac{1}{n} \delta_n + (1- \frac{1}{n}) \delta_{\frac{1}{n}})_n$.
\end{itemize}
\item Non-examples: $(\delta_n)_n$
\item How does one prove weak convergence? How does one write this down in a clear way?
% TODO
\end{itemize}
\paragraph{Convolution}
\begin{itemize}
\item Definition of convolution.
\todo{Copy from exercise sheet and write a section about this}
\item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$.
\end{itemize}
\subsubsection{CLT}
\begin{itemize}
\item Statement of the CLT
\item Several versions:
\begin{itemize}
\item iid (\autoref{clt}),
\item Lindeberg (\autoref{lindebergclt}),
\item Luyapanov (\autoref{lyapunovclt})
\end{itemize}
\item How to apply this? Exercises!
\end{itemize}
\subsection{Conditional expectation}
\begin{itemize}
\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
\item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
is the (unique) projection on the closed subspace $L^2(\Omega, \cG, \bP)$.
Why is this a closed subspace? Why is the projection orthogonal?
\item Radon-Nikodym Theorem (Proof not relevant for the exam)
\item (Non-)examples of mutually absolutely continuous measures
Singularity in this context? % TODO
\end{itemize}
\subsection{Martingales}
\begin{itemize}
\item Definition of Martingales
\item Doob's convergence theorem, Upcrossing inequality
(downcrossings for submartingales)
\item Examples of Martingales converging a.s.~but not in $L^1$
\item Bounded in $L^2$ $\implies$ convergence in $L^2$.
\item Martingale increments are orthogonal in $L^2$!
\item Doob's (sub-)martingale inequalities
\item $\bP[\sup_{k \le n} M_k \ge x]$ $\leadsto$ Look at martingale inequalities!
Estimates might come from Doob's inequalities if $(M_k)_k$ is a (sub-)martingale.
\item Doob's $L^p$ convergence theorem.
\begin{itemize}
\item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu}
\item This is an important proof.
\end{itemize}
\item Uniform integrability % TODO
\item What are stopping times?
\item (Non-)examples of stopping times
\item \textbf{Optional stopping theorem} - be really comfortable with this.
\end{itemize}
\subsection{Markov Chains}
\begin{itemize}
\item What are Markov chains?
\item State space, initial distribution
\item Important examples
\item \textbf{What is the relation between Martingales and Markov chains?}
$u$ \vocab{harmonic} $\iff Lu = 0$.
(sub-/super-) harmonic $u$ $\iff$ for a Markov chain $(X_n)$,
$u(X_n)$ is a (sub-/super-)martingale
\item Dirichlet problem
(Not done in the lecture)
\item ... (more in Probability Theory II)
\end{itemize}