josia-notes/s23-probability-theory
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Josia Pietsch 2023-07-12 15:27:39 +02:00
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inputs/lecture_07.tex
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@ -43,8 +43,8 @@ For the proof we'll need a slight generalization of \autoref{thm2}:
By \autoref{thm4} it follows that $\sum_{n \ge 1} Y_n < \infty$ By \autoref{thm4} it follows that $\sum_{n \ge 1} Y_n < \infty$
almost surely. almost surely.
Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$. Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
Since the first series $\sum_{n \ge 1} \bP(A_n) < \infty$, Since $\sum_{n \ge 1} \bP(A_n) < \infty$ by assumption,
by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occur}] = 0$. Borel-Cantelli yields $\bP[\text{infinitely many $A_n$ occur}] = 0$.
For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$ For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$