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