# 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 appropriated tags to specify the context.

256 questions
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### Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
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### Is this sequence almost sure convergent?

Consider the sequence of independent random variables $\{X_n\}$ such that \begin{align} P(X_n = 1) &= 1/n \\P(X_n = 0) &= 1 - 1/n \end{align} I saw this as an example of convergent in ...
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### How to show $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$

Let $(g_{n})_{n}$ be bounded and measurable where $g_{n}\xrightarrow{n \to \infty} g, \mu-$a.e. and $f_{n} \xrightarrow{L^{p}}f$. I need to show that $g_{n}f_{n} \xrightarrow{L^{1}}gf$, and my proof ...
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### Brezis 4.15, $f_n(x)=ne^{−nx}$

Let $\Omega=(0,1)$ and $f_n(x)=ne^{-nx}$ prove that (i) $f_n\rightarrow0$ a.e. (ii) $f_n$ is bounded in $L_1(\Omega)$ I know that for $f_n$ to converge a.e. to zero, the set of points in which this ...
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### For which x the following sequences converges

If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges? $$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$ I guess that for no $x$ the sequence converges. I tried to ...
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### Product of Uniform Distribution

I know that there exists some discussions related to my question, however, I couldn't find an explanation for my question. I hope it is not a duplicate. Let $X_n$ be sequence of i.i.d. uniform ...
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### Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?

Let $\{X_n\}_{n=0}^\infty$ be an absorbing Markov chain. It is well-known that $$P(\text{chain gets absorbed}|X_0=i)=1.$$ My question is how this is interpreted in practice. We have that for almost ...
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### Simple example illustrating “almost sure convergence”

I'm trying to check my knowledge on th enotion of almost sure convergence. I've cooked up this example: Let $X_1$ be a Bernoulli random variable with paramter $\frac{1}{2}$. Now consider the sequence ...
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### Checking almost sure converge of a random variable experimentally

Suppose $X_1, X_2, X_3, \ldots$ is a sequence of random variables that converges almost surely to a random variable $X$. How could I check for this experimentally? I don't need this to be some ...
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### Proving almost sure convergence (help understanding a step in a proof)

We have $$|V_{n+1}-V^\prime_{n+1}|=\prod_{m=1}^n|W_m||V_1-V^\prime_1| \tag{1}\label{1}$$ where $P(|W|=1)\ne 1$ (with $W\in[-1,1]$) and $W_n$s are iid and independent of $V$s and $V^\prime$s. Show ...
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### Does $\operatorname EX_n\to0$ as $n\to\infty$?

Suppose that $X_1,X_2,\ldots$ are non-negative random variables defined on a probability space $(\Omega,\mathcal F,P)$ with $\operatorname E|X_n|^p<\infty$ with some $p>2$ for each $n\ge1$. ...