\lecture{23}{2023-07-06}{Recap}
\subsection{Recap}

\subsubsection{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}

\subsubsection{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 \item Laws of large numbers: (\autoref{lln})
            \begin{itemize}
                \item WLLN: convergence in probability
                \item SLLN: weak convergence
            \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}

\subsubsubsection{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 Bochner's theorem for positive definite function (\autoref{thm:bochner})
    \item Bochner'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?
        \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}
\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 CLT
    \item Several versions:
        \begin{itemize}
            \item iid (\autoref{clt}),
            \item Lindeberg (\autoref{lindebergclt}),
            \item Lyapanov (\autoref{lyapunovclt})
        \end{itemize}
    \item How to apply this? Exercises!
\end{itemize}

\subsubsection{Conditional expectation}
\begin{itemize}
    \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
        (\autoref{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 Radon-Nikodym Theorem \ref{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 (\autoref{def:martingale})
    \item Doob's convergence theorem (\autoref{doobmartingaleconvergence}),
        Upcrossing inequality (\autoref{lec17l1}, \autoref{lec17l2}, \autoref{lec17l3})
        (downcrossings for submartingales)
    \item Examples of Martingales converging a.s.~but not in $L^1$
        (\autoref{ex:martingale-not-converging-in-l1})
    \item Bounded in $L^2$ $\implies$ convergence in $L^2$
        (\autoref{martingaleconvergencel2}).
    \item Martingale increments are orthogonal in $L^2$!
        (\autoref{martingaleincrementsorthogonal})
    \item Doob's (sub-)martingale inequalities
        (\autoref{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
        (\autoref{ceismartingale}, \autoref{martingaleisce}).
        \begin{itemize}
            \item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu}
            \item This is an important proof.
        \end{itemize}
    \item Uniform integrability (\autoref{def:ui})
    \item What are stopping times? (\autoref{def:stopping-time})
    \item (Non-)examples of stopping times
    \item \textbf{Optional stopping theorem} (\autoref{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}
