s23-probability-theory/inputs/lecture_2.tex

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\lecture{2}{}{}
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\section{Independence and product measures}
In order to define the notion of independence, we first need to construct
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product measures.
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The finite case of a product is straightforward:
\begin{theorem}{Product measure (finite)}
Let $(\Omega_1, \cF, \bP)$ and $(\Omega_2, \cF_2, \bP_2)$ be probability spaces.
Let $\Omega \coloneqq \Omega_1 \times \Omega_2$
and $R \coloneqq \{A_1 \times A_2 | A_1 \in \cF_1, A_2 \in \cF_2 \}$.
Let $\cF$ be $\sigma(R)$ (the sigma algebra generated by $R$).
Then there exists a unique probability measure $\bP$ on $\Omega$
such that for every rectangle $R = A_1 \times A_2 \in \cR$, $\bP(A_1 \times A_2) = \bP(A_1) \times \bP(A_2)$.
\end{theorem}
\begin{proof}
See Theorem 5.1.1 in the lecture notes on Stochastik.
\end{proof}
We now want to construct a product measure for infinite products.
\begin{definition}[Independence]
A collection $X_1, X_2, \ldots, X_n$ of random variables are called
\vocab{mutually independent} if
\[
\forall a_1,\ldots,a_n \in \R :
\bP[X_1 \le a_1, \ldots, x_n \le a_n]
= \prod_{i=1}^n \bP[X_i \le a_i]
\]
This is equivalent to
\[
\forall B_1, \ldots, B_n \in \cB(\R):
\bP[X_1 \in B_1, \ldots, X_n \in B_n]
= \prod_{i=1}^n \bP[X_i \in B_i]
\]
\end{definition}
\begin{example}
Suppose we throw a dice twice. Let $A \coloneqq \{\text{first throw even}\}$,
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$B \coloneqq \{\text{second throw even}\}$
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and $C \coloneqq \{\text{sum even}\} $.
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It is easy the see, that the random variables are pairwise independent,
but not mutually independent.
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\end{example}
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\begin{definition}
Let $(\Omega, \cF, \bP)$ be a probability space
and $X : ( \Omega, \cF) \to (\R, \cB(\R))$ a random variable.
Then $\Q(\cdot) \coloneqq \bP [ X \in \cdot ]$ is called the \vocab{distribution}
of $X$ under $\bP$.
\end{definition}
Let $X_1, \ldots, X_n$ be random variables and $\Q^{\otimes}(\cdot ) \coloneqq \bP[(X_1,\ldots, X_n) \in \cdot ]$
their \vocab{joint distribution}.
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Then $\Q^{\otimes}$ is a probability measure on $\R^n$.
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The definition of mutual independence can be rephrased as follows:
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\begin{fact}
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$X_1,\ldots, X_n$ are mutually independent iff $\Q^{\otimes} = \Q_1 \otimes \ldots \otimes \Q_n$,
where $\Q_i$ is the distribution of $X_i$.
In this setting, $\Q_i$ is called the \vocab{marginal distribution} of $X_i$.
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\end{fact}
By constructing an infinite product, we can thus extend the notion of independence
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to an infinite number of random variables.
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\begin{goal}
Can we construct infinitely many independent random variables?
\end{goal}
\begin{definition}[Consistent family of random variables]
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\label{def:consistentfamily}
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Let $\bP_n, n \in \N$ be a family of probability measures on $(\R^n, \cB(\R^n))$.
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The family is called \vocab{consistent} if
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$\bP_{n+1}[B_1 \times B_2 \times \ldots \times B_n \times \R] = \bP_n[B_1 \times \ldots \times B_n]$
for all $n \in \N, B_i \in B(\R)$.
\end{definition}
\begin{theorem}[Kolmogorov extension / consistency theorem]
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\label{thm:kolmogorovconsistency}
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Informally:
``Probability measures are determined by finite-dimensional marginals
(as long as these marginals are nice)''
Let $\bP_n, n \in \N$ be probability measures on $(\R^n, \cB(\R^n))$
which are \vocab{consistent},
then there exists a unique probability measure $\bP^{\otimes}$
on $(\R^\infty, B(R^\infty))$ (where $B(R^{\infty}$ has to be defined),
such that
\[
\forall n \in \N, B_1,\ldots, B_n \in B(\R):
\bP^\otimes [\cX : X_i \in B_i \forall 1 \le i \le n]
= \bP_n[B_1 \times \ldots \times B_n]
\]
\end{theorem}
\begin{remark}
Kolmogorov's theorem can be strengthened to the case of arbitrary
index sets. However this requires a different notion of consistency.
\end{remark}
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\begin{example}[A consistent family]
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Let $F_1, \ldots, F_n$ be probability distribution functions
and let $\bP_n$ be the probability measure on $\R^n$ defined
by
\[
\bP_n[(a_1,b_1] \times \ldots (a_n, b_n]]
\coloneqq (F_1(b_1) - F_1(a_1)) \cdot \ldots \cdot (F_n(b_n) - F_n(a_n)).
\]
It is easy to see that each $\bP_n$ is a probability measure.
Define $X_i(\omega) = \omega_i$ where $\omega = (\omega_1, .., \omega_n)$.
Then $X_1, \ldots, X_n$ are mutually independent with $F_i$ being
the distribution function of $X_i$.
In the case of $F_1 = \ldots = F_n$, then $X_1,\ldots, X_n$ are i.i.d.
\end{example}