92 lines
3.7 KiB
TeX
92 lines
3.7 KiB
TeX
% TODO \begin{goal}
|
|
% TODO We want to drop our assumptions on finite mean or variance
|
|
% TODO and say something about the behaviour of $ \sum_{n \ge 1} X_n$
|
|
% TODO when the $X_n$ are independent.
|
|
% TODO \end{goal}
|
|
\begin{theorem}[Theorem 3, Kolmogorov's three-series theorem] % Theorem 3
|
|
\label{thm3}
|
|
Let $X_n$ be a family of independent random variables.
|
|
\begin{enumerate}[(a)]
|
|
\item Suppose for some $C \ge 0$, the following three series
|
|
of numbers converge:
|
|
\begin{itemize}
|
|
\item $\sum_{n \ge 1} \bP(|X_n| > C)$,
|
|
\item $\sum_{n \ge 1} \underbrace{\int_{|X_n| \le C} X_n d\bP}_{\text{\vocab{truncated mean}}}$,
|
|
\item $\sum_{n \ge 1} \underbrace{\int_{|X_n| \le C} X_n^2 d\bP - \left( \int_{|X_n| \le C} X_n d\bP \right)^2}_{\text{\vocab{truncated variance} }}$.
|
|
\end{itemize}
|
|
Then $\sum_{n \ge 1} X_n$ converges almost surely.
|
|
\item Suppose $\sum_{n \ge 1} X_n$ converges almost surely.
|
|
Then all three series above converge for every $C > 0$.
|
|
\end{enumerate}
|
|
\end{theorem}
|
|
For the proof we'll need a slight generalization of \autoref{thm2}:
|
|
\begin{theorem}[Theorem 4] % Theorem 4
|
|
\label{thm4}
|
|
Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded}
|
|
(i.e. $\exists M < \infty : \sup_n \sup_\omega |X_n(\omega)| \le M$).
|
|
Then $\sum_{n \ge 1} X_n$ converges almost surely
|
|
$\iff$ $\sum_{n \ge 1} \bE(X_n)$ and $\sum_{n \ge 1} \Var(X_n)$
|
|
converge.
|
|
\end{theorem}
|
|
\begin{refproof}{thm3}
|
|
Assume, that we have already proved \autoref{thm4}.
|
|
We prove part (a) first.
|
|
Put $Y_n = X_n \cdot \One_{\{|X_n| \le C\}}$.
|
|
Since the $X_n$ are independent, the $Y_n$ are independent as well.
|
|
Furthermore, the $Y_n$ are uniformly bounded.
|
|
By our assumption, the series
|
|
$\sum_{n \ge 1} \int_{|X_n| \le C} X_n d\bP = \sum_{n \ge 1} \bE[Y_n]$
|
|
and $\sum_{n \ge 1} \int_{|X_n| \le C} X_n^2 d\bP - \left( \int_{|X_n| \le C} X_n d\bP \right)^2 = \sum_{n \ge 1} \Var(Y_n)$
|
|
converges.
|
|
By \autoref{thm4} it follows that $\sum_{n \ge 1} Y_n < \infty$
|
|
almost surely.
|
|
Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
|
|
Since the first series $\sum_{n \ge 1} \bP(A_n) < \infty$,
|
|
by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occcur}] = 0$.
|
|
|
|
|
|
For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$
|
|
for almost every $\omega$.
|
|
Fix an arbitrary $C > 0$.
|
|
Define
|
|
\[
|
|
Y_n(\omega) \coloneqq \begin{cases}
|
|
X_n(\omega) & \text{if} |X_n(\omega)| \le C,\\
|
|
C &\text{if } |X_n(\omega)| > C.
|
|
\end{cases}
|
|
\]
|
|
Then the $Y_n$ are independent and $\sum_{n \ge 1} Y_n(\omega) < \infty$
|
|
almost surely and the $Y_n$ are uniformly bounded.
|
|
By \autoref{thm4} $\sum_{n \ge 1} \bE[Y_n]$ and $\sum_{n \ge 1} \Var(Y_n)$
|
|
converge.
|
|
Define
|
|
\[
|
|
Z_n(\omega) \coloneqq \begin{cases}
|
|
X_n(\omega) &\text{if } |X_n| \le C,\\
|
|
-C &\text{if } |X_n| > C.
|
|
\end{cases}
|
|
\]
|
|
Then the $Z_n$ are independent, uniformly bounded and $\sum_{n \ge 1} Z_n(\omega) < \infty$
|
|
almost surely.
|
|
By \autoref{thm4} we have
|
|
$\sums_{n \ge 1} \bE(Z_n) < \infty$
|
|
and $\sums_{n \ge 1} \Var(Z_n) < \infty$.
|
|
|
|
We have
|
|
\[
|
|
\bE(Y_n) &=& \int_{|X_n| \le C} X_n d \bP + C \bP(|X_n| \ge C)\\
|
|
\bE(Z_n) &=& \int_{|X_n| \le C} X_n d \bP - C \bP(|X_n| \ge C)\\
|
|
\]
|
|
Since $\bE(Y_n) + \bE(Z_n) = 2 \int_{|X_n| \le C} X_n d\bP$
|
|
the second series converges,
|
|
and since
|
|
$\bE(Y_n) - \bE(Z_n)$ converges, the first series converges.
|
|
For the third series, we look at
|
|
$\sum_{n \ge 1} \Var(Y_n)$ and
|
|
$\sum_{n \ge 1} \Var(Z_n)$ to conclude that this series converges
|
|
as well.
|
|
\end{refproof}
|
|
|
|
|
|
|