Intereting Posts

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The constant distribution

Here is the problem.

I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$.

- Generalized Second Borel-Cantelli lemma
- “Fair” game in Williams
- $P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$
- Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$.
- If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN)
- How can we apply the Borel-Cantelli lemma here?

Is it true? If so, why? If not, what other subsequence can you suggest?

This is from Rosenthal, btw.

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- “Fair” game in Williams
- A variation of Borel Cantelli Lemma
- Showing that $n$ exponential functions are linearly independent.
- independent, identically distributed (IID) random variables
- union of two independent probabilistic event
- Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$.
- Maximum of martingales
- If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN)
- Mutual Independence Definition Clarification

**Hint** It is not generally true that $\sum P(A_{nm})=+\infty$. But it must be true for one of the $m$ following sequences:

\begin{equation*}

(B^{i}_{n})_{n\in\mathbb{N}}=(A_{nm+i})_{n\in\mathbb{N}}, \quad i=0,1..,m-1

\end{equation*}

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