Stochastic (Lecture)

I began following this lecture at the LMU Munich for my B.Sc. Mathematics (LMU) during my 3rd Semester without completing it, I collect here some notes I took during the course.

Lecture Notes

Here is a recap of all definitions and propositions relevant for the exam and presented in the lectures and published material.

• $\Omega$

  • $\Omega$ Raum, $\mathcal{F}$ is a $\sigma$-Al. iff (i) $\Omega \in \mathcal{F}$ (ii) $A\in \mathcal{F}\to A^C\in \mathcal{F}$ (iii) $(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{F}\Rightarrow {\bigcup }_{n\in \mathbb{N}}{A}_n\in \mathcal{F}$
  • $P$ is a w-Maß iff (i) $P(\Omega )=1$ (ii) $P({⨆}_{n\in \mathbb{N}}{A}_n)={\sum }_{n\in \mathbb{N}}P(A_n)$.
  • $m$ Farben, mit Reihenfolge, mit Zurücklegen: ${\Omega }_1^{n,m}:=[m{]}^{[n]}=\{\omega |\omega :[n]\to [m]\}$, $|{\Omega }_1^{[n,m]}|=m^n$
  • $m$ Farben, mit Reihenfolge, ohne Zürucklegen: ${\Omega }_2^{n,m}:=\{\omega \in [m{]}^{[n]}:\omega$ inj.$\}$, ==$|$?$|$==. L.1
  • Multindex Notation: $\alpha \in {\mathbb{N}}^m$, $|\alpha |={\sum }_{i=1}^m{\alpha }_i$, $\alpha !={\alpha }_1!...{\alpha }_m!$. L.2
  • $m$ Farben, mit Reihenfolge innerhalb d. Farbe ${\Omega }_{\alpha }:=\{f:[m{]}^{|\alpha |}|{\forall }_{k\in [m]}|f^{-1}(k)|={\alpha }_k\}$, $|\Omega |=|\alpha |!$.
  • $m$ Farben, ohne Reihenfolge innerhalb d. Farbe ${\tilde{\Omega }}_{\alpha }:={\Omega }_{\alpha }/\sim$, $|{\tilde{\Omega }}_{\alpha }|=\frac{|\alpha |!}{\alpha !}$.

• $\sigma$

  • $\sigma \subset \mathcal{P}(\Omega )$ s.t. (i) $\Omega \in \sigma$, (ii) $A\in \sigma \to A^C\in \sigma$, (iii) ${\bigcup }_{n\le \omega }{A}_n\in \sigma$.
  • Also (iv) $\varnothing \in \sigma$, (v) ${\bigcup }_{n\le \omega }{A}_n\in \sigma$, (vi) $A\cup B,A\cap B,A\setminus B,A\Delta B\in \sigma$
  • For ${\mathcal{A}}_1,{\mathcal{A}}_2$ $\sigma$-al., then ${\mathcal{A}}_1\cap {\mathcal{A}}_2$ is a $\sigma$-al.
  • For $\mathcal{E}\subset \mathcal{P}(\Omega )$, ${\sigma }_{\Omega }(\mathcal{E}):=\bigcap \{\mathcal{A}:\mathcal{A}$ is a $\sigma$-al. $\wedge \mathcal{E}\subset \mathcal{A}\}$, smallest $\sigma$-al. with $\mathcal{E}$.
  • ${\mathcal{B}}_{\mathbb{R}}:={\sigma }_{\mathbb{R}}(\{(a,b):a<b\})$, and in general ${\mathcal{B}}_X:={\sigma }_X(t)$ for $(X,t)$ a topological space L.2 $\Omega$ discrete $\Rightarrow \mathcal{F}=2^{\Omega }$; else: $\Omega =\mathbb{R}\Rightarrow \mathcal{F}=\sigma (\{A\subset \mathbb{R}:A$ open$\})={\mathcal{B}}_{\mathbb{R}}$. Notice: $\sigma (\{(a,b):a<b\wedge a,b\in \mathbb{R}\})=\sigma (\{(-\infty ,q):q\in \mathbb{Q}\})={\mathcal{B}}_{\mathbb{R}}$

• $\mathbb{P}$

  • For $(\Omega ,\mathcal{A})$, $\mathbb{P}:\mathcal{A}\to {\mathbb{R}}_0^{+}\cup \{+\infty \}$ s.t. (i) $\mathbb{P}(\varnothing )=0$, (ii) $\mathbb{P}({⨆}_{n\le \omega }{A}_n={\sum }_{n\le \omega }\mathbb{P}(A_n)$ L.3
  • If (ii), (iii) $\mathbb{P}(\Omega )=1$ and $\mathbb{P}:\mathcal{A}\to [0,1]$ it is a w-Maß. Consider the set theoretic conseq.
  • For $(A_n{)}_{n\in \mathbb{N}}\in {\mathcal{A}}^{\mathbb{N}}$, and if $A_1\subseteq A_2\subseteq ...$ then ${\lim }_{n\to \infty }\mathbb{P}(A_n)=\mathbb{P}({\bigcup }_{n\in \mathbb{N}}{A}_n)$
  • ${\lambda }_d(Q)={\prod }_{i=1}^d(b_i-a_i)$, or also $A\mapsto \frac{{\lambda }_d(A)}{{\lambda }_d(\Omega )}$ is an equal distribution on a $d$-dim interval. L.4
  • $\varphi (\omega )$ is $\mathbb{P}$-almost sure if $\mathbb{P}(\{\omega \in \Omega :\neg \varphi (\omega )\})=0$.

• Tools

  • On $(\mathbb{R},{\mathcal{B}}_{\mathbb{R}},\mathbb{P})$, $D:\mathbb{R}\to [0,1],x\mapsto \mathbb{P}((-\infty ,\infty ])$ 3.1 (Distribution function)
    • (i) $x\le y\to D(x)\le D(y)$, (ii) $(x_n{)}_{n\le \omega }$ mon.f., ${\lim }_{n\to \infty }{x}_n=x$ then ${\lim }_{n\to \infty }D(x_n)=D(x)$ (iii) $li{m}_{x\to -\infty }D(x)=0\wedge li{m}_{x\to \infty }D(x)=1$. 3.2
  • for $P$, $Q$ meas., for the respective distribution function $D_P$, $D_Q$, holds $D_P=D_Q\Rightarrow P=Q$. 3.3;5
  • $M\subseteq \mathcal{P}(\Omega )$ is $\cap$-st. gen. of $(\Omega ,\mathcal{A})$, if $M$ is $\cap$-st. and ${\sigma }_{\Omega }(M)=\mathcal{A}$
  • $P,Q$ meas. that agree on a $\cap$-st. gen., then they are equal.

• $\mathcal{D}$

  • (i) $\Omega \in \mathcal{D}$, (ii) $A\in \mathcal{D}\to A^C\in \mathcal{D}$, (iii) disj. seq $A_i$ then ${⨆}_{i\le n}{A}_i\in \mathcal{D}$.
  • for $(\Omega ,\mathcal{A})$ and $P,Q$, $D:=\{A\in \mathcal{A}:P(A)=Q(A)\}$
  • $M$ $\cap$-st. and $\mathcal{D}\subseteq \mathcal{P}(\Omega )$ a dyn.sy. on $\Omega$, then $M\subseteq \mathcal{D}\Rightarrow {\sigma }_{\Omega }(M)\subseteq \mathcal{D}$ (Dynkin Lemma)

• $f$

  • $f$ is $\mathcal{A}-{\mathcal{B}}_{\mathbb{R}}-$meas. define for ${\forall }_{B\in {\mathcal{B}}_{\mathbb{R}}}{f}^{-1}(B)\in \mathcal{A}$.
    • A cont. funciton is $\mathcal{A}-{\mathcal{B}}_{\mathbb{R}}-$ meas., also closed under $+$, $\frac{\cdot }{\cdot }$, $\cdot$, $\circ$, $|\cdot |$, $\lim$, $\sup$.
      • from $f\circ g$ meas, consider $+$ as a cont. func on func.
    • for $f:\Omega \to \overline{\mathbb{R}}$ tfae: (i) $f$ is $\mathcal{A}$-${\mathcal{B}}_{\mathbb{R}}$-meas., (ii) ${\forall }_{a\in \mathbb{R}}\{f\le a\}\in \mathcal{A}$
      • $\{f\le a\}:=f^{-1}([-\infty ,a])$
      • same holds for $<$, $>$, $\ge$, or $\{a\le f<b\}$ since they all generate ${\mathcal{B}}_{\mathbb{R}}$.
  • ${\int }_{\Omega }fd\mu :=\sup \{{\sum }_{k=1}^n{\alpha }_k\mu (A_k):n\in \mathbb{N}\wedge {\forall }_k(A_k\in \mathcal{A}\wedge {\alpha }_k\in [0,\infty ]\to {\sum }_{k=1}^n{\alpha }_k{1}_{A_k}\le f)\}$. Simply check what I noted in Analysis III (Lecture).
  • $f_{+}$ just like in Analysis II (Lecture).

• J, Dichte

  • for $(\Omega ,\mathcal{A},\mu )$, $f\Omega \to [0,\infty ]$ meas. s.t. ${\int }_{\Omega }dd\mu =1$, then $J:\mathcal{A}\to [0,1],A\mapsto {\int }_{A}fd\mu$ is a meas.
  • Examples:
    • Gleichverteilung: $\zeta :[a,b]\to [0,\infty ],x\mapsto \frac{1}{b-a}$, $P_{[a,b]}:A\mapsto {\int }_A{\zeta }_{[a,b]}d\lambda =\frac{\lambda (A)}{b-a}$.
    • Exponentialverteilung: ${\zeta }_a:[0,\infty ],x\mapsto a\cdot e^{-ax}$, $P_a(A)=\int {\zeta }_{a}d\lambda ={\int }_0^{\infty }a{e}^{-ax}dx$
    • Gauß Verteilung: ${\zeta }_{\mu ,\sigma }:x\mapsto \frac{1}{\sqrt{2\pi {\sigma }^2}}\cdot exp(-\frac{1}{2}(\frac{x-\mu }{\sigma }{)}^2)$ for $x\in \mathbb{R}$, then ${\forall }_{A\in {\mathcal{B}}_{\mathbb{R}}}{P}_{\mu ,\sigma }:A\mapsto {\int }_A{\zeta }_{\mu ,\sigma }d\lambda ={\int }_{-\infty }^{\infty }{\zeta }_{\mu ,\sigma }dx$.
    • Dirac measure has no Dichte!

• Quantil

  • $\mu$ prob.meas. on $(\mathbb{R},{\mathcal{B}}_{\mathbb{R}})$ with $D$ dis.f., for $y\in (0,1)$, elem in $\{x\in \mathbb{R}:li{m}_{x^{\prime }\to x}D(x^{\prime })\le y\le D(x)\}$ are $y$-quantils on $D$.
    • $D$ inj. $\Rightarrow$ quantil is unique
  • a “quasi-inverse” $I(y)=sup\{x\in \mathbb{R}:D(x)\le y\}$ a $y$-quantil of $D$, hence is $I$ the quantil function.
  • for $(\tilde{\Omega },\tilde{\mathcal{A}})$ and an index set $I$, ${\forall }_{i\in I}{x}_i:\Omega \to \tilde{\Omega }$, then $\sigma (x_i,i\in I):=\sigma \{x_i^{-1}(\tilde{A}):\tilde{A}\in \tilde{\mathcal{A}}\wedge i\in I\}$ is $\sigma$-al. on $\Omega$. 4.4;9
  • ${⨂}_{i=1}^n{\mathcal{A}}_i=\sigma (x_i,i\in [n])={\sigma }_{\Omega }(\{{\text{bigtimes}}_{i=1}^n:A_i\in {\mathcal{A}}_i\wedge i\in [n]\})$ 4.5;9 (Product $\sigma$-Algebra)
  • (existence and uniqueness of the product measure) 4.6;9

• $({\mathbb{R}}^n,{\mathcal{B}}_{{\mathbb{R}}^n},\mu )$ and $\rho :{\mathbb{R}}^n\to [0,1]$ meas. s.t. ${\forall }_{A\in {\mathcal{B}}_{{\mathbb{R}}^n}}\mu (A)={\int }_A\rho d{\lambda }_{{\mathbb{R}}^n}$, then ${\mathcal{L}}_{\mu }(\pi )$ has the dichte: $J:{\mathbb{R}}^m\to [0,\infty ],x\mapsto {\int }_{{\mathbb{R}}^{n-m}}\rho (x,y)d{\lambda }_{{\mathbb{R}}^{n-m}}(y)$.

  • for $\pi :{\mathbb{R}}^n\to {\mathbb{R}}^m$, $(x_1,...,x_m,...,x_n)\mapsto (x_1,...,x_m)$ the proj. of the first $m$ coord.
  • From 13.3 Iterierte Integrale (The Italians) recall: Tonelli, Fubini

• Let (i) $({\mathbb{R}}^n,{\mathcal{B}}_{{\mathbb{R}}^n},\mathbb{P})$, (ii) let $\mathbb{P}$ have a Dichte $f$ respect ${\lambda }_{{\mathbb{R}}^n}$, (iii) $\Phi :U\to V$ ${\mathcal{C}}^1$-Diffeo on $U,V$ op., then: ${\mathcal{L}}_{\mathbb{R}}({\Phi }^{-1})$ Dichte, $g(y)=(f\circ \Phi )(y)\cdot |det(D{\Phi }_y)|$.

  • Recall 13.4 Transformationssatz

• Conditional Probabilities

  • for $(\Omega ,\mathcal{A},\mathbb{P})$, $B\in \mathcal{A}$ a Ereignis s.t. $\mathbb{P}(B)>0$ define $\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$ 4.10;10
    • $\mathbb{P}(\cdot |B):\mathcal{A}\to [0,1],A\mapsto \mathbb{P}(A|B)$ is a prob.meas. on $(\Omega ,\mathcal{A})$
  • for $(A_k{)}_{k\in \mathbb{N}}$ a disj.zer. of $\mathcal{A}$ s.t. ${\forall }_{k\in \mathbb{N}}\mathbb{P}(A_k)>0$ then ${\forall }_{B\in \mathcal{A}}\mathbb{P}(B)={\sum }_{k\in \mathbb{N}}\mathbb{P}(B|A_k)\mathbb{P}(A_k)$
  • for $(A_k{)}_{k\in \mathbb{N}}$ a disj.zer. of $\mathcal{A}$ s.t. ${\forall }_{k\in \mathbb{N}}\mathbb{P}(A_k)>0$, then ${\forall }_{B\in \mathcal{A}}\mathbb{P}(B)\to \mathbb{P}(A_k|B)=\frac{\mathbb{P}(B|A_k)\mathbb{P}(A_k)}{{\sum }_{d\in I}\mathbb{P}(B|A_j)\mathbb{P}(A_j)}$ 4.12;11 (Bayes)

• Stochastic Independence

  • $(A_i{)}_{i\in I}\subseteq \mathcal{A}$ sto.ind. for $\mathbb{P}$ if ${\forall }_{E\subseteq I}((|E|<\infty \wedge E\ne \varnothing )\to \mathbb{P}({\bigcap }_{i\in E}{A}_i)={\prod }_{i\in E}\mathbb{P}(A_i))$ 4.12;11
  • $(X_i{)}_{i\in I}$ sto.ind. for $\mathbb{P}$ if: for each $X_i:(\Omega ,\mathcal{A})\to ({\Omega }_i,{\mathcal{A}}_i)$, for all $(A_i{)}_{i\in I}$ s.t. $A_i\in {\mathcal{A}}_i$ holds: $(\{X_i\in A_i\}{)}_{i\in I}$ is sto.ind. for $\mathbb{P}$.
    • $(X_i{)}_{i\in I}$ sto.ind. $\Rightarrow$ $(\sigma (X_i){)}_{i\in I}$ sto.ind.
    • for: $X_i:(\mathbb{R},{\mathcal{B}}_{\mathbb{R}})\to (\mathbb{R},{\mathcal{B}}_{\mathbb{R}})$, $i\in [n]$ and let $D:\mathcal{R}\to [0,1]$ distr. of $\mathbb{P}$, then $(X_i){)}_{i\in I}$ sto.ind. $\iff {\forall }_{(y_i{)}_{i\in [n]}\in {\mathbb{R}}^n}\mathbb{P}(x_i\le y_i,i\in [n])={\prod }_{i\in [n]}{D}_{X_i}(y_i)$
      • in this case also for ${\phi }_k:{\mathbb{R}}^k\to \mathbb{R}$ meas, ${\phi }^k(x_1,...,x_k)$ and $(x_l{)}_{l\le k}$ sto.ind.
      • for $I_k$ pair. disj. subsets. of $I$, s.t. $|I_k|<\infty$, and ${\phi }_k:{\mathbb{R}}^{|I_k|}\to \mathbb{R}$ meas. and for $(X_i{)}_{i\in I}$ sto.ind., then $({\phi }_k\circ (X_i{)}_{i\in I}{)}_{k\in \mathbb{N}}$ sto.ind.
  • $({\mathcal{F}}_i{)}_{i\in I}$ a fam. of $\cap$-st. set.sys. are sto.ind. if for each $\varnothing \ne {\mathcal{F}}_i\subseteq \mathcal{A}$ if for all $(A_i{)}_{i\in I}$ s.t. $A_i\in {\mathcal{F}}_i$ holds $(A_i{)}_{i\in I}$ is sto.ind. for $\mathbb{P}$.
  • for $(\Omega ,\mathcal{A},\mathbb{P})$, and $B\in \mathcal{A}$ then ${\mathcal{D}}_B:=\{A\in \mathcal{A}:(A,B)$ sto.ind. for $\mathbb{P}\}$ is a dyn.sys.

• Theory and Epirie

  • for $(\Omega ,\mathcal{A},\mathbb{P})$ prob.sp., $(\tilde{\Omega },\tilde{\mathcal{A}})$ meas.sp. $X_i:(\Omega ,\mathcal{A})\to (\tilde{\Omega },\tilde{\mathcal{A}})$ for $i\in I$ indep. and identically distributed if $(X_i{)}_{i\in I}$ independent and ${\forall }_{i,j\in I}{\mathcal{L}}_{\mathbb{P}}(X_i)={\mathcal{L}}_{\mathbb{P}}(X_j)$ (i.i.d) 5.1;13
    • Notation: $└x┘=max\{z\in \mathbb{Z}:z\le x\}$
  • for $X:(\Omega ,\mathcal{A})\to (\overline{\mathbb{R}},{\mathcal{B}}_{\overline{\mathbb{R}}})$, ${\mathbb{E}}_{\mathbb{P}}{X}^n:={\int }_{\Omega }{X}^{n}d\mathbb{P}={\int }_{\Omega }x(\omega {)}^{n}d\mathbb{P}(\omega )$, if $|X{|}^n$ int. write $X\in L^n(\Omega ,\mathcal{A},\mathbb{P})$ and say: $X$ has the $n$th. moment in ${\mathbb{E}}_{\mathbb{P}}{X}^n$.
    • check Bsp. 1. at the end of L. 13
  • $L^1(\Omega ,\mathcal{A},\mathbb{P})$ 5.3;14
    1. is a Linear Space ${\forall }_{\alpha ,\beta \in \mathbb{R}}{\forall }_{x,y\in L^1(\Omega ,\mathcal{A},\mathbb{P})}{\mathbb{E}}_{\mathbb{P}}[\alpha x+\beta y]=\alpha {\mathbb{E}}_{\mathbb{P}}[x]+\beta {\mathbb{E}}_{\mathbb{P}}[y]$.
    2. ${\forall }_{x,y{L}^1(\Omega ,\mathcal{A},\mathbb{P})}x\le y\to {\mathbb{E}}_{\mathbb{P}}(x)\le {\mathbb{E}}_{\mathbb{P}}(y)$ (Monotony)
    3. ${\forall }_{A\in \mathcal{A}}{\mathbb{E}}_{\mathbb{P}}(1_A)=\mathbb{P}(A)$
    4. ${\forall }_{x,y\in L^2(\Omega ,\mathcal{A},\mathbb{P})}({\mathbb{E}}_{\mathbb{P}}[xy]{)}^2\le {\mathbb{E}}_{\mathbb{P}}[x^2]{\mathbb{E}}_{\mathbb{P}}[y^2]$ (Cauchy-Schwarz)
  • for $x\in L^2(\Omega ,\mathcal{A},\mathbb{P})$, 5.4;14
    • $va{r}_{\mathbb{P}}(x):={\mathbb{E}}_{\mathbb{P}}(x-{\mathbb{E}}_{\mathbb{P}}(x){)}^2$ (Variance)
      • $va{r}_{\mathbb{P}}(x)={\mathbb{E}}_{\mathbb{P}}(x^2)-({\mathbb{E}}_{\mathbb{P}}(x){)}^2$
      • $va{r}_{\mathbb{P}}(x)\ge 0$
      • $va{r}_{\mathbb{P}}(ax)=a^{2}va{r}_{\mathbb{P}}(x)$
      • $va{r}_{\mathbb{P}}(x+a)=va{r}_{\mathbb{P}}(x)$
    • ${\partial }_{\mathbb{P}}(x):=\sqrt{va{r}_{\mathbb{P}}(x)}$ (Standardabweichung)
      • ${\partial }_{\mathbb{P}}(ax)=|a|{\partial }_{\mathbb{P}}(x)$
    • $co{v}_{\mathbb{P}}(x,y):={\mathbb{E}}_{\mathbb{P}}(x-{\mathbb{E}}_{\mathbb{P}}(x))(y-{\mathbb{E}}_{\mathbb{P}}(y))$ (Covariance)
      • $co{v}_{\mathbb{P}}(x,x)=va{r}_{\mathbb{P}}(x)$
      • $co{v}_{\mathbb{P}}(x,y)=co{v}_{\mathbb{P}}(y,x)$
      • $co{v}_{\mathbb{P}}(x+y,z)=co{v}_{\mathbb{P}}(x,z)+co{v}_{\mathbb{P}}(y,z)$
      • $co{v}_{\mathbb{P}}(\alpha ,y)=\alpha co{v}_{\mathbb{P}}(x,y)$
      • $co{v}_{\mathbb{P}}(x,y)={\mathbb{E}}_{\mathbb{P}}(xy)-{\mathbb{E}}_{\mathbb{P}}(x){\mathbb{E}}_{\mathbb{P}}(y)$
    • $cor{r}_{\mathbb{P}}(x,y):=\frac{co{v}_{\mathbb{P}}(x,y)}{{\partial }_{\mathbb{P}}\cdot {\partial }_{\mathbb{P}}(y)}\in [-1,1]$ (Correlation)
    • ${\mathbb{E}}_{\mathbb{P}}y=0$ then say $y$ is centred
      • ${\mathbb{E}}_{\mathbb{P}}(x-{\mathbb{E}}_{\mathbb{P}}x{)}^n$ central $n$th moment of $x$.
  • for $x,y\in L^1(\Omega ,\mathcal{A},\mathbb{P})$, independent, then: 5.5;14
    • ${\mathbb{E}}_{\mathbb{P}}(xy)={\mathbb{E}}_{\mathbb{P}}(x){\mathbb{E}}_{\mathbb{P}}(y)$
    • $co{v}_{\mathbb{P}}(x,y)=0$ (say those are uncorrelated $\ne$ independent, only one direction. A fam. is uncorr. if pairw. uncorr.)
  • for $(x_i{)}_{i\in \mathbb{N}}\in L^1(\Omega ,\mathcal{A},\mathbb{P})$, ${\mathbb{E}}_{\mathbb{P}}({\sum }_{i=1}^n{x}_i)={\sum }_{i=1}^n{\mathbb{E}}_{\mathbb{P}}(x_i)$
    • for $(x_i{)}_{i\in I}$ uncorr., then $va{r}_{\mathbb{P}}({\sum }_{i=1}^n{x}_i)={\sum }_{i=1}^{n}va{r}_{\mathbb{P}}(x_i)$

• for $x_i\in L^2(\Omega ,\mathcal{A},\mathbb{P})$ uncorr., and ident.distr. then: ${\forall }_{ϵ>0}\mathbb{P}(|\frac{1}{n}{\sum }_{i=1}^n{x}_i-{\mathbb{E}}_{\mathbb{P}}{x}_1|\ge ϵ)\le \frac{va{r}_{\mathbb{P}}(x_1)}{n{ϵ}^2}{\to }^{n\to \infty }0$ 5.6;15 (Weak principle of big numbers)

  • ${\forall }_{i\in [n]}{\mathcal{L}}_{\mathbb{P}}(x_i)={\mathcal{L}}_{\mathbb{P}}(x_1)$ (ident.distr.)
  • ${\mathbb{E}}_{\mathbb{P}}{X}_1={\mathbb{E}}_{\mathbb{P}}\frac{1}{n}{\sum }_{i=1}^n{x}_i$
  • ${\hat{x}}_n:=\frac{1}{n}{\sum }_{i=1}^n{x}_i$

• for $A\in \mathcal{A}$, $c\in {\mathbb{R}}_0^{+}$ and $x\ge 0$ zufallv. s.t. ${\forall }_{\omega \in A}x(\omega )\ge c$, then ${\mathbb{E}}_{\mathbb{P}}x\ge c\mathbb{P}(A)$ (Tschebyschaff Inequality)

• $x$ zufallv., $m>0$, $a>0$ then: $\mathbb{P}(|x|\ge a)\le a^{-m}{\mathbb{E}}_{\mathbb{P}}|x{|}^m$

• $(x_n{)}_{n\in \mathbb{N}}$ zufallv. and $x\in \mathbb{R}$, $x_n{⟶}_{n\to \infty }^{\mathbb{P}}x\iff {\forall }_{ϵ>0}li{m}_{n\to \infty }\mathbb{P}(|x_n-x|\ge ϵ)=0$ (conv. in prob.)

• $x$ zufallv., def. $L_x:{\mathbb{R}}_0^{+}\to \mathbb{R},s\mapsto {\mathbb{E}}_{\mathbb{P}}[e^{sx}]$ (Laplace-Transformation/Momenterzwegende Funktion) 5.10;16

• $x$ zufallv. with real values, ${\forall }_{a\in \mathbb{R}}{\forall }_{s\ge 0}{\mathbb{E}}_{\mathbb{P}}(e^{sx})\ge e^{sa}\mathbb{P}(x\ge a)$ and $\mathbb{P}(x\ge a)\le in{f}_{s\ge 0}(e^{-sa}{\mathbb{E}}_{\mathbb{P}}(e^{s}x))$ (exp. Tschebyschaff Inequality)

• for $(A_n{)}_{n\in \mathbb{N}}$ ind. s.t. ${\forall }_{n\in \mathbb{N}}\mathbb{P}(A_n)=p$ then $\frac{1}{n}{\sum }_{i=1}^n{1}_{A_i}{⟶}_{n\to \infty }^{\mathbb{P}}p$

• for $(x_n{)}_{n\in \mathbb{N}}$ iid zufallv., $\mu \in \mathbb{R}$, $\partial \in {\mathbb{R}}_0^{+}$ s.t. ${\forall }_{n\in \mathbb{N}}{\mathbb{E}}_{\mathbb{P}}{x}_n=\mu$, $va{r}_{\mathbb{P}}(x_n)={\partial }^2$, then $\frac{1}{n}{\sum }_{i=1}^n{x}_i=:{\hat{x}}_n{⟶}_{n\to \infty }^{\mathbb{P}-f.s.}\mu$ ($\mathbb{P}$-fast sicher), 5.12;16 (Strong principle of big numbers)

  • $\mathbb{P}(\widehat{(}x{)}_n{⟶}^{n\to \infty }\mu )=\{\omega \in \Omega :li{m}_{n\to \infty }{\hat{x}}_n(\omega )=\mu \}=1$

• $(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{A}$ and $A^{\ast }:=limsu{p}_{n\to \infty }(A_n)$, then (i) ${\sum }_{n\in \mathbb{N}}\mathbb{P}(A_n)<\infty \Rightarrow \mathbb{P}(A^{\ast })=0$, (ii) ${\sum }_{n\in \mathbb{N}}\mathbb{P}(A_n)=\infty$ and $(A_n{)}_{n\in \mathbb{N}}$ ind. $\Rightarrow \mathbb{P}(A^{\ast })=1$ (Borel-Cantelli Lemma)

  • consider $lim$ $sup$ and $lim$ $inf$ as in Analysis III (Lecture).
  • also: $(x_n{⟶}_{n\to \infty }^{\mathbb{P}-f.s.}x)\Rightarrow (x_n{⟶}_{n\to \infty }^{\mathbb{P}}x)$ (see exercise)

• ${\tilde{x}}_n:=\frac{1}{\sqrt{n}}{\sum }_{i=1}^n(x_i-{\mathbb{E}}_{\mathbb{P}}(x_i))$ (Skalierung)

• ${\Phi }_x:\mathbb{R}\to \mathbb{C},y\mapsto L_x(i\cdot y)={\mathbb{E}}_{\mathbb{P}}[e^{iyx}]$ (Characteristical Function)

  • ${\Phi }_x(0)=1$
  • $|{\Phi }_x(y)|\le 1$
  • $y\mapsto {\Phi }_x(y)$ glm.stet.
  • ${\Phi }_x(y)=\overline{{\Phi }_x(-y)}$
  • for $x_1,x_2$ zufallv. ind.: ${\Phi }_{x_1+x_2}(y)={\Phi }_{x_1}(y)\cdot {\Phi }_{x_2}(y)$
  • (3), (4);18

• for $(x_i{)}_{i\in \mathbb{N}}$ real zufallv. s.t. ${\mathbb{E}}_{\mathbb{P}}{x}_i=0$ and ${\mathbb{E}}_{\mathbb{P}}{x}^2<\infty$ then $\Phi {\tilde{x}}_n(y){⟶}^{n\to \infty }exp(-\frac{1}{2}{y}^{2}va{r}_{\mathbb{P}}(x_1))$ 5.15;18

• for $(x_i{)}_{i\in \mathbb{N}}$ real zufallv. uniformely distr. and ind., for $\mu \in \mathbb{R}$, $\sigma \in {\mathbb{R}}_0^{+}$ s.t. ${\mathbb{E}}_{\mathbb{P}}{x}_i=\mu$ and $\sigma ={\sigma }_{\mathbb{P}}(x_i)$, then ${\forall }_{a,b\in \mathbb{R}}a\le b\to {\lim }_{n\to \infty }\mathbb{P}({\tilde{x}}_n\in [a,b])=\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_a^b{e}^{-\frac{1}{2}(\frac{x}{\sigma }{)}^2}dx$ 5.16;18 (Zentrale Grenzwertsatz)

  • ${\tilde{x}}_n:=\frac{1}{n}{\sum }_{i=1}^n(x_i-{\mathbb{E}}_{\mathbb{P}}(x_i))$

• $x$ real zufallv. and distr. ${\mathcal{L}}_{\mathbb{P}}(x)$ with Dicht e $\delta (y)=\frac{1}{\sqrt{2\pi {\sigma }^2}}exp(-\frac{1}{2}(\frac{y-\mu }{\sigma }{)}^2)$, then $x\sim \mathcal{N}(\mu ,\sigma )$ 5.17;18 (Normal Distribution)

  • Standard Normal Distribution if $\mu =0$ and $\sigma =1$.
  • for $\mu ={\mathbb{E}}_{\mathbb{P}}(x_i)$ and $\sigma ={\sigma }_{\mathbb{P}}(x_i)$
    • ${\mathcal{L}}_{\mathbb{P}}(\frac{1}{\sqrt{n}}{\sum }_{i=1}^n(x_i-\mu )){⟶}^{n\to \infty }\mathcal{N}(0,\sigma )$
    • ${\mathcal{L}}_{\mathbb{P}}(\frac{1}{\sqrt{n}\sigma }{\sum }_{i=1}^n(x_i-\mu )){⟶}^{n\to \infty }\mathcal{N}(0,1)$
    • $\mathbb{P}(\frac{\sigma }{\sqrt{n}}a+\mu \le \frac{1}{n}{\sum }_{i=1}^m{x}_i\le \mu +\frac{\sigma }{\sqrt{n}}b){⟶}^{n\to \infty }\frac{1}{\sqrt{2\pi }}{\int }_a^b{e}^{-\frac{1}{2}{x}^2}dx)$