# Questions tagged [almost-everywhere]

For questions about the concept of "almost everywhere", that is, questions about properties which holds everywhere, except on a set of measure 0. This is involved in probability theory as well as in the case of infinite measure space. Use the appropriate tags to specify the context.

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### Almost sure convergence to infinity of a series of random variables in a Markov Model

Consider observations $X_1,X_2,\cdots$ from a Markov model (first order) with two states 0 and 1 having transition probabilities from state $i$ to state $j$, $p_{j|i}\neq 0,1/2,1$, for $i,j=0,1$. ...
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### Use Markov inequality to prove that a positive random variable is null a.s

I would like to show that if the expectation of a positive random variable $X$ (integrable) is $0$ then $X$ is null almost surely. I thought on a proof and would like to know if it is correct, it uses ...
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### Measurability and almost sure convergence, going from epsilon to 1/j

I am trying to understand the definition of almost sure convergence. For this one needs to check if there are no measurability issues. By transforming the definition of convergence to set theory ...
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### Condition on Sobolev function on the boundary of a domain

I encountered the following kind of a condition when studying boundary value problems: Let $\Omega \subset \mathbb{R}^2$ be a domain. Let $\varphi \in H^1 (\Omega)$ be such that $\varphi \ge 0$ on ...
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### Defining convergence of a sequence of random variables.

I am trying to understand ideas like Almost sure convergence and convergence of sequences of random variables. I am trying to prove: Given an infinite sequence of random variables $X_i,i\in\mathbb{N}$ ...
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### Is this property of differentiable almost everywhere functions true?

Consider a function $g:\mathcal{I}\to\mathbb{R}$ defined in an open interval $\mathcal{I}$ of the real line and therein differentiable almost everywhere. Is it true that all points in which $g$ is ...
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### Codomain of almost everywhere defined functions

So I am currently wondering about how one can write down the codomain of a function that is only almost everywhere defined. In particular, I am looking at the radon nikodym derivative $f$, which is ...
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### what does almost surely mean in this case

Let $X_1,...,X_n$ be some random variables and $\theta$ be a function of these random variables. I was asked to show that $\mathbb{P}(\theta=\hat{\theta}|X_1,...,X_n)=c$ almost surely. I am able to ...
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### $\sigma$-linearity of the Radon-Nikodym operator.

Let $(X,\mathcal X,\nu)$ be a probability space, let $\{ \mu_n \}_{n\in\mathbb N}$ be a set of finite positive measures on $(X,\mathcal X)$ that are all absolutely continuous with respect to $\nu$. I ...
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1 vote
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### Mill's ratio and convergence almost surely

Let ${X_n}$ be a sequence of independent random variables N[0,1]. Show that: $$\mathcal{P}(\underbrace{lim sup}_{n \longmapsto \infty} \frac{|X_n|}{\sqrt{log \ n}} = \sqrt{2}) = 1$$ I've been asked ...
1 vote
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### How to find to which random variable the sequence converges in probability. Is there almost everywhere convergence?

Given a sequence of random variables $\{\xi_n\}$ \begin{equation*} \xi_n(\omega) = \begin{cases} -1 &\text{with probability $\frac{1}{2n}$}\\ 3 &\text{with probability $\frac{1}{3n}$}\\ ...
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