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2 votes
0 answers
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Conditional Borel-Cantelli lemma [duplicate]

Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. s.}$$...
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6 votes
3 answers
434 views

What is the rate of growth of $M_n := \max_{1 \le i \le n} U_i^{(n)}$, where $U_i^{(n)} \sim \operatorname{Uniform}[0,n]$? Is it "constant"?

On pp. 370-374 (see this previous question) of Cramer's 1946 Mathematical Methods of Statistics, the author shows that for any continuous distribution $P$, if $X_1, \dots, X_n \sim P$ are i.i.d., then ...
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6 votes
1 answer
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Levy's extension of the Borel-Cantelli Lemmas

Following is the statement and proof of Levy's extension of the Borel-Cantelli Lemmas, as given in Williams' "Probability with Martingales" (1991), in section 12.15 on page 124. I understand most of ...
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1 vote
1 answer
591 views

Question regarding Borel-Cantelli lemma

Let $X_1,...X_n$ be a sequence of random variables such that $X_n=1 $ or $0$ and $P(X_1=1) \geq \alpha$ and $P(X_n=1|X_1,...X_{n-1}) \geq \alpha$ for $n=2,3,...$ where $\alpha >0$ I need to show ...
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2 votes
0 answers
381 views

Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. There ...
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2 votes
1 answer
178 views

intuitive reason of the independence assumption behind the second part of the Borel-Cantelli lemma

Is there any intuitive reason of the independence assumption behind the second part of the Borel-Cantelli lemma. or it is just for calculation ?
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0 votes
1 answer
105 views

Limit of sequence of random variables

Suppose that $\{X_n\}_{n=1}^\infty$ is a sequence of random variables such that: $$X_1=\lambda\ , \ X_{n+1}\sim\text{Poi}(X_n)$$ (First, we draw $X_n$, if $X_n=k$ then $X_{n+1}\sim Poi(k)$ I am trying ...
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2 votes
1 answer
93 views

Showing that a martingale $Y_k$ does not converge almost surely.

Let $X_i$ be iid with $$\mathbb{P}(X_i=1)= \mathbb{P}(X_i= -1) = \frac{1}{2i}, \mathbb{P}(X_i=0)=1-\frac{1}{i},$$ where $i=1,2,...$ And define $Y_1=X_1$ and for $k\geq2$ $$Y_k= \begin{cases} X_k, \...
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2 votes
1 answer
75 views

How can I prove this statement about the BC Lemma?

This is what I am thinking: Let $M< N <\infty $ $$P(\bigcap_{n=M}^N A_n^c) = \prod_{n=M}^N(1-P(A_n)) \leq \prod_{n=M}^N e^{(-P(A_n))}$$ $$= e^{\bigg( - \sum_{n=M}^N P(A_n)\bigg)} \...
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0 votes
0 answers
46 views

Show $\sum_{n \geq 1}\mathbb{1}_{A_n}(\omega) = \infty$ iff $\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \infty$

Given a pobability space $(\Omega,\mathcal{A},P)$, a filtration $(\mathcal{F_n})_n$ and a sequence of events $A_n \in \mathcal{F}_n$ I have the martingale $X_n = \sum_{1 \leq j \leq n}(\mathbb{1}_{...
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