From 0930ed6c95e993c2280cba1c2aae8d007f39f154 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 23 May 2023 18:02:54 +0200 Subject: [PATCH] lecture 13 part 2 --- inputs/lecture_13.tex | 152 ++++++++++++++++++++++++++++++++++++++++- probability_theory.tex | 1 + 2 files changed, 152 insertions(+), 1 deletion(-) diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 093d992..77dae54 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -40,7 +40,7 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$, The Lyapunov condition implies the Lindeberg condition. (Exercise). \end{remark} -We will not prove the \autoref{linebergclt} or \autoref{lyapunovclt} +We will not prove the \autoref{lindebergclt} or \autoref{lyapunovclt} in this lecture. However, they are quite important. We will now sketch the proof of \autoref{levycontinuity}, @@ -106,4 +106,154 @@ A generalized version of \autoref{levycontinuity} is the following: Exercise: $\phi_{S_n}(t) = e^{-|t|} = \phi_{C_1}(t)$, thus $S_n \sim C$. \end{example} +We will prove \autoref{levycontinuity} assuming +\autoref{lec10_thm1}. +\autoref{lec10_thm1} will be shown in the notes.\todo{TODO} +We will need the following: +\begin{lemma} + \label{lec13_lem1} + Given a sequence $(F_n)_n$ of probability distribution functions, + there is a subsequence $(F_{n_k})_k$ of $F_n$ + and a right continuous, non-decreasing function $F$, + such that $F_{n_k} \to F$ at all continuity points of $F$. + (We do not yet claim, that $F$ is a probability distribution function, + as we ignore $\lim_{x \to \infty} F(x)$ and $\lim_{x \to -\infty} F(x)$ for now). +\end{lemma} +\begin{lemma} + \label{s7e1} + Let $\mu \in M_1(\R)$, $A > 0$ and $\phi$ the characteristic function of $\mu$. + Then $\mu\left( (-A,A) \right) \ge \frac{A}{2} \left| \int_{-\frac{2}{A}}^{\frac{2}{A}} \phi(t) d t \right| - 1$. +\end{lemma} +\begin{refproof}{s7e1} + Exercise.\todo{TODO} +\end{refproof} + + + +\begin{refproof}{levycontinuity} +``$\implies$ '' If $\mu_n \implies \mu$, then +$\int f d \mu_n \to \int f d \mu$ +for all $f \in C_b$ and $x \to e^{\i t x}$ is continuous and bounded. + + +``$ \impliedby$'' + +% Step 1: +\begin{claim} + \label{levyproofc1} + Given $\epsilon > 0$ there exists $A > 0$ such that + $\liminf_n \mu_n\left( (-A,A) \right) \ge 1 - 2 \epsilon$. +\end{claim} +\begin{refproof}{levyproofc1} + If $f$ is continuous, then + \[ + \frac{1}{\eta} \int_{x - \eta}^{x + \eta} f(t) d t \xrightarrow{\eta \downarrow 0} f(x). + \] + Applying this to $\phi$ at $t = 0$, one obtains: + \begin{equation} + \left| \frac{A}{4} \int_{-\frac{2}{A}}^{\frac{2}{A}} \phi(t) dt - 1 \right| < \frac{\epsilon}{2} + \label{levyproofc1eqn1} + \end{equation} + + \begin{claim} + For $n$ large enough, we have + \begin{equation} + \left| \frac{A}{4} \int_{-\frac{2}{A}}^{\frac{2}{A}} \phi_n(t) d t - 1\right| < \epsilon. + \label{levyproofc1eqn2} + \end{equation} + \end{claim} + \begin{subproof} + Apply dominated convergence. + \end{subproof} + So to prove $\mu_n\left( (-A,A) \right) \ge 1 - 2 \epsilon$, + apply \autoref{s7e1}. + It suffices to show that + \[ + \frac{A}{2} \left| \int_{-\frac{2}{A}}^{\frac{2}{A}} \phi_n(t) dt\right| - 1 \ge 1 - 2\epsilon + \] + or + \[ + 1 - \frac{A}{4} \left|\int_{-\frac{2}{A}}^{\frac{2}{A}} \phi_n(t) dt \right| \le \epsilon, + \] + which follows from \autoref{levyproofc1eqn2}. +\end{refproof} + +% Step 2 +By \autoref{lec13_lem1} +there exists a right continuous, non-decreasing $F $ +and a subsequence $(F_{n_k})_k$ of $(F_n)_n$ where $F_n$ is +the probability distribution function of $\mu_n$, +such that $F_{n_k}(x) \to F(x)$ for all $x$ where $F$ is continuous. +\begin{claim} + \[ + \lim_{n \to -\infty} F(x) = 0 + \] + and + \[ + \lim_{n \to \infty} F(x) = 1, + \] + i.e.~$F$ is a probability distribution function.\footnote{This does not hold in general!} +\end{claim} +\begin{subproof} + We have + \[ + \mu_{n_k}\left( (- \infty, x] \right) = F_{n_k}(x) \to F(x). + \] + Again, given $\epsilon > 0$, there exists $A > 0$, such that + $\mu_{n_k}\left( (-A,A) \right) > 1 - 2 \epsilon$ (\autoref{levyproofc1}). + + Hence $F(x) \ge 1 - 2 \epsilon$ for $x > A $ + and $F(x) \le 2\epsilon$ for $x < -A$. + This proves the claim. +\end{subproof} + +Since $F$ is a probability distribution function, there exists +a probability measure $\nu$ on $\R$ such that $F$ is the distribution +function of $\nu$. +Since $F_{n_k}(x) \to F_n(x)$ at all continuity points $x$ of $F$. +By \autoref{lec10_thm1} we obtain that +$\mu_{n_k} \overset{k \to \infty}{\implies} \nu$. +Hence +$\phi_{\mu_{n_k}}(t) \to \phi_\nu(t)$, by the other direction of that theorem. +But by assumption, +$\phi_{\mu_{n_k}}(\cdot ) \to \phi_n(\cdot )$ so $\phi_{\mu}(\cdot) = \phi_{\nu}(\cdot )$. +By \autoref{charfuncuniqueness}, we get $\mu = \nu$. + +We have shown, that $\mu_{n_k} \implies \mu$ along a subsequence. +We still need to show that $\mu_n \implies \mu$. +\begin{fact} + Suppose $a_n$ is a bounded sequence in $\R$, + such that any subsequence converges to $a \in \R$. + Then $a_n \to a$. +\end{fact} +\begin{subproof} + \todo{in the notes} +\end{subproof} +Assume $\mu_n$ does not converge to $\mu$. +By \autoref{lec10_thm1}, pick a continuity point $x_0$ of $F$, +such that $F_n(x_0) \not\to F(x_0)$. +Pick $\delta > 0$ and a subsequence $F_{n_1}(x_0), F_{n_2}(x_0), \ldots$ +which are all outside $(F(x_0) - \delta, F(x_0) + \delta)$. +Then $\phi_{n_1}, \phi_{n_2}, \ldots \to \phi$. +Now, there exists a further subsequence $G_1, G_2, \ldots$ of $F_{n_i}$, +which converges. +$G_1, G_2, \ldots$ is a subsequence of $F_1, F_2,\ldots$. +However $G_1, G_2, \ldots$ is not converging to $F$, +as this would fail at $x_0$. This is a contradiction. + +\end{refproof} + + + +% IID is over now +\subsection{Summary} +What did we learn: + +\begin{itemize} + \item How to construct product measures + \item WLLN and SLLN + \item Kolmogorov's three series theorem + \item Fourier transform, weak convergence and CLT +\end{itemize} + diff --git a/probability_theory.tex b/probability_theory.tex index 5b1acb1..9ab7263 100644 --- a/probability_theory.tex +++ b/probability_theory.tex @@ -36,6 +36,7 @@ \input{inputs/lecture_10.tex} \input{inputs/lecture_11.tex} \input{inputs/lecture_12.tex} +\input{inputs/lecture_13.tex} \cleardoublepage