s23-probability-theory/inputs/lecture_23.tex
2023-07-28 03:45:37 +02:00

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\lecture{23}{2023-07-06}{Recap}
\subsection{Recap}
\subsubsection{Construction of iid random variables.}
\begin{itemize}
\item Definition of a consistent family (\yaref{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
(\yaref{thm:kolmogorovconsistency})
\end{itemize}
\subsubsection{Limit theorems}
\begin{itemize}
\item Work with iid.~random variables.
\item Notions of convergence (\yaref{def:convergence})
\item Implications between different notions of convergence (very important) and counter examples.
(\yaref{thm:convergenceimplications})
\item Laws of large numbers: (\yaref{lln})
\begin{itemize}
\item WLLN: convergence in probability
\item SLLN: weak convergence
\end{itemize}
\item \yaref{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 \yaref{thm:kolmogorovineq}
\item \yaref{kolmogorov01}
In particular, a series of independent random variables converges with probability $0$ or $1$.
\item \yaref{thm:kolmogorovthreeseries}
\begin{itemize}
\item What are those $3$ series?
\item Applications
\end{itemize}
\end{itemize}
\subsubsubsection{Fourier transform / characteristic functions / weak convegence}
\begin{itemize}
\item Definition of Fourier transform
(\yaref{def:characteristicfunction})
\item The Fourier transform uniquely determines the probability distribution.
It is bounded, so many theorems are easily applicable.
\item \yaref{charfuncuniqueness},
\yaref{inversionformula}, ...
\item \yaref{levycontinuity},
\yaref{genlevycontinuity}
\item \yaref{thm:bochner}
\item \yaref{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).
(\yaref{def:weakconvergence})
\item Examples:
\begin{itemize}
\item $(\delta_{\frac{1}{n}})_n$,
\item $(\frac{1}{2} \delta_{-\frac{1}{n}} + \frac{1}{2} \delta_{\frac{1}{n}})_n$,
\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?
\begin{itemize}
\item \yaref{lec10_thm1},
\item \yaref{levycontinuity},
\item Generalization of Levy's continuity theorem
\ref{genlevycontinuity}
\end{itemize}
\end{itemize}
\paragraph{Convolution}
\begin{itemize}
\item Definition of convolution.
\todo{Copy from exercise sheet and write a subsection 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}
\subsubsubsection{CLT}
\begin{itemize}
\item Statement of the \yaref{clt}
\item Several versions:
\begin{itemize}
\item iid,
\item \yaref{lindebergclt},
\item \yaref{lyapunovclt}
\end{itemize}
\item How to apply this? Exercises!
\end{itemize}
\subsubsection{Conditional expectation}
\begin{itemize}
\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
(\yaref{conditionalexpectation})
\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 \yaref{radonnikodym}
(Proof not relevant for the exam)
\item (Non-)examples of mutually absolutely continuous measures
Singularity in this context? % TODO
\end{itemize}
\subsubsection{Martingales}
\begin{itemize}
\item Definition of Martingales (\yaref{def:martingale})
\item Doob's convergence theorem (\yaref{doobmartingaleconvergence}),
Upcrossing inequality (\yaref{lec17l1}, \yaref{lec17l2}, \yaref{lec17l3})
(downcrossings for submartingales)
\item Examples of Martingales converging a.s.~but not in $L^1$
(\yaref{ex:martingale-not-converging-in-l1})
\item Bounded in $L^2$ $\implies$ convergence in $L^2$
(\yaref{martingaleconvergencel2}).
\item Martingale increments are orthogonal in $L^2$!
(\yaref{martingaleincrementsorthogonal})
\item Doob's (sub-)martingale inequalities
(\yaref{dooblp}),
\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
(\yaref{ceismartingale}, \yaref{martingaleisce}).
\begin{itemize}
\item Why is $p > 1$ important? \textbf{Role of \yaref{banachalaoglu}}
\item This is an important proof.
\end{itemize}
\item Uniform integrability (\yaref{def:ui})
\item What are stopping times? (\yaref{def:stopping-time})
\item (Non-)examples of stopping times
\item \textbf{\yaref{optionalstopping}}
- be really comfortable with this.
\end{itemize}
\subsubsection{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}