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.
466
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13
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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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24
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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 ...
- 1,140
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1
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74
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A problem about Lebesgue integral convergence
Suppose that $\{E_n\}_n$ is a sequence of Lebesgue measurable subset in $\mathbb R$ satisfies $\displaystyle\lim_{n\to \infty}\mu(E_n)=0$. $\{f_n\}_n$ is a Lebesgue measurable function sequence on $\...
- 111
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An exemple that convergence in probability does not imply convergence almost surely
In my lecture notes, there is the following example to study in order to show that convergence in probability does not imply convergence almost surely.
We consider the measurable space $([0,1], \...
- 1,140
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2
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71
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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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1
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31
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Does additivity of probability imply almost-sure pairwise disjointedness?
Suppose we have a probability measure $P$ and a countable collection of sets $\{ E_k\}_{k=1}^{\infty}$ from the corresponding $\sigma$-algebra.
If $$P\left( \bigcup_{k=1}^{\infty} E_k \right) = \sum_{...
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1
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measure theory: $g$ continuous a.e., $f_n \to f$ a.e., does $g \circ f_n \to g \circ f$ a.e.?
Question: Let $(f_n)_{n \geq 1}$, $f$, and $g$ be Borel measurable functions from $\mathbb{R} \to \mathbb{R}$. Suppose $g$ is continuous almost everywhere and $f_n \to f$ almost everywhere, does $g \...
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For a Wiener process, when can one exchange “for all $t$” and “almost surely”?
Certain local properties of the Wiener process $W_t$ are quick to prove at $t = 0$, for instance:
almost surely $W_t$ is monotonous on no interval beginning at $t = 0$;
almost surely $W_t$ is not ...
- 439
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31
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Almost everywhere differentiable functions: what can we do? Integration by parts?
Let $A$ be an open measurable set and $A_0$ is a subset of $A$ which has measure zero.
Let $g$ and $f$ be almost everywhere differentiable function defined on $A$.
What is the impact of "almost ...
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Does separate almost everywhere continuity imply joint almost everywhere continuity?
My question is almost the same as this question.
Let $f(\cdot, \cdot): R^2 \rightarrow R$. Suppose that:
(a) Fix any $y \in R, f(\cdot, y)$ is continuous almost everywhere.
(b) Fix any $x \in R, f(x, \...
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If random variables $X$ and $Y$ are equal in distribution, then there exists a measurable function $f$ such that $X(\omega)=Y(f(\omega))$ a.s.
Let $X$ and $Y$ be two identically distributed random vectors in $\mathbb{R}^{d}$ defined on the same underlying probability space $(\Omega,\mathcal{A},P)$. Suppose that $P$ is non-atomic.
Does there ...
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64
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Showing that a function is nonnegative almost everywhere on (0,1)
Let $f$ be an integrable function in $(0,1)$. Suppose that $$\int_{0}^{1} fg dx \geq 0$$ for any nonnegative continuous function $g:(0,1) \to \mathbb{R}$. Prove that $f \geq 0$ almost everywhere in $(...
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Is convergent sequence in $L^2$ almost surely bounded?
Let $(f_n)_{n\ge 1}$ be a sequence of function that converges to some $f\in L^2(\mathbb{R})$, i.e., $||f_n-f||_{L^2(\mathbb{R})}=0$.
Clearly the sequence of norms $(||f_n||_{L^2(\mathbb{R})})_{n\ge 1}$...
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Conditional probability when the given outcome has probability $0$
Consider two different random variables on $\{0,1\}^{\mathbb N}$, i.e. the set of binary sequences. The first random variable $X_1$ has a distribution defined by letting each of its digits be chosen ...
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Show $\int _{0} ^{\infty} \frac{|f(x)|}{|x-a|^b} dx <\infty$ for a.e. $a\in \mathbb{R}$ if $f$ Lebesgue integrable on $\mathbb{R}$, $b\in (0,1)$
The question: Let $f$ be a Lebesgue integrable function on $\mathbb{R}$ and let $b\in (0,1)$. Prove that $\int _{0} ^{\infty} \frac{|f(x)|}{|x-a|^b} dx <\infty$ for a.e. $a\in \mathbb{R}$.
My ...
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1
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79
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Weak convergence and almost everywhere convergence on $H_0^1({\mathbb{R}}^N)$
I am seeking a clarification to this question on almost convergence in Sobolev spaces. The essence of the answer there was to prove that if a sequence $(u_n) \rightharpoonup u$ in $H_0^1({\mathbb{R}}^...
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Separable and diffuse probability space is standard
Let $(X,\mathcal{N},\mu)$ be a complete probability space, and assume that the $\sigma$-algebra $\mathcal{N}$ has no atoms, that is, for every set $A \in \mathcal{N}$ with $\mu(A) > 0$ there is a ...
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Show that $\forall\varepsilon>0$, there exists a bounded continuous function $g(x)$ on $E$, such that $m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$.
Let $E$ be a measurable subset of $\mathbb{R}^n$, and $mE<\infty$. $f(x)$ is a bounded measurable function almost everywhere. I want to find such a function $g$.
For $n=1$, i.e., $E\in\mathbb{R}$. $...
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Are $L^p$ functions continuous almost everywhere?
Let $1 \le p < +\infty$ and define the Lebesgue space as the quotient space
$$ L^p(X):= \left\{f\text{: }X\rightarrow \mathbb{R} \mid \int_X \left|f\right|^p d\mu <+\infty\right\}/\sim, $$
where ...
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22
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Almost sure bound for maximal of Gaussian R.V.
The following two questions come from some steps of my thought about solving an exercise. Thanks for help.
Let $X_1, X_2, ..., X_n, ...$ be i.i.d $N(0,1)$ random variables.
Define $M_n:=\max_{1\leq i ...
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47
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Equality of two sets in measure theory, with finite function for almost everywhere
I'm currently solving exercise 10 of chapter 2 of Stein's Real Analysis. The problem is given as follows.
What I'm struggling to is the first part :
$$\bigcup_{k = -\infty}^{\infty}F_k = \{x : f(x) &...
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2
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37
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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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27
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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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1
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58
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Waiting times between record observations
Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. random variables with a continuous CDF.
Let $V_1:=\min \{n\in\mathbb N\, \vert \,X_n>X_1\}$.
Let $V_{r+1}:=\min \{ n\in \mathbb N \,|\, n>V_r \...
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Prove that an announcing sequence converges point wise to a stopping time
In the proof of Theorem 1 in this post of George Lowther's Blog (proof of implication $3\Rightarrow 4$) we have a sequence of stopping times $\{T_k\}$ and a stopping time $T$ and we know that $T_k\leq ...
- 1,234
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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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2
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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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Show that $P($ $\limsup $ $\frac{X_n}{\log n}$ $ =$ $ 1$ $)$ $ =$ $ 1$
QUESTION
Let $X_n$ be independent and exponentially distributed with parameter $1$. Show that $P\left(\limsup \frac{X_n}{\log n} = 1\right) = 1$.
ATTEMPT
What I've tried to do is:
$$
P\left( \frac{...
- 583
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1
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Show that limsup $\frac{X_r}{r} = \infty $ almost surely
Given Xr be independent, non-negtive and iid with infinite mean.
Show that limsup $\frac{X_r}{r} = \infty $
Now the solution assess that:
$$
\sum_{r=1} (\frac{X_r}{r} \geq x) = \sum_{r=1} (\frac{X_r}{...
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0
answers
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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
1
answer
74
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Petrov (1995), Theorem 6.7
Does someone know the proof of the following statement:
Given a sequence of independent random variables $(Y_n)_{n=1}^{\infty}$ and a sequence of positive constants $(a_n)_{n=1}^{\infty}$ such that $...
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2
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100
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Chebishev’s and Markov’s inequalities and convergence a.s.
Let ${X_n}$ be a sequence of i.i.d random variables with mean = 1 and variance $\sigma^2 =3$.
Show if the following converges almost surely:
$$
\frac{1}{n}\sum^n X_i \longmapsto 1
$$
Chebyshev ...
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29
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Uniqueness of the limit and finiteness of the measure space
We know that the convergence almost surely/everywhere implies the convergence in probability/measure.
But is this also true when the measure space is not sigma finite since this can lead to different ...
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Borel Cantelli and convergence almost surely
I was wondering if it's possible to use the Borel Cantelli Theorem in order to ensure that the almost sure convergence DOESN'T exist.
We know that:
Let $A_n$ be a sequence of events in a probability ...
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0
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55
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Show that $f$ is 1-1
Let $f$ holomorphic, $f:\mathbb{D}\rightarrow\mathbb{D}$.
If $w\in\mathbb{D}$ we denote as $n_f(w)$ the number of zeros of the function $f(z)-w$. If $n_f=1$ almost everywhere in $\mathbb{D}$, show ...
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How to prove this almost surely convergence in $L^{\infty}$ norm?
Let $X_{n}, n = 1, 2...$ be i.i.d. random variables, with continuous cumulative distribution function $F$. Let
$F_{n}(x) = \frac{\sum_{i = 1}^{n} \chi_{X_{i} \leq x}(\omega)}{n}$.
Show $||F_{n} - F ||...
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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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1
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52
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Almost sure convergence of variable
Let $(X_1, Y_1),...$ be iid distributed $\mathbb{R}^2$ RV's.
Assume $X_i$ and $Y_i$ have first moment $1$ and second moment 2.
Further assume $E X_i Y_i = 1-p$
Let $T_n = \frac{\sum_{i=1}^n X_i - Y_i}...
- 15
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1
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79
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Bartle's theorem 7.6: "If a sequence $(f_n)$ converges in measure to $f$, then some subsequence converges almost everywhere to $f$"
Why in the theorem taken a Cauchy in measure sequence and not any sequence that converges in measure? Before state the theorem, the text says "we shall now prove a result due to F. Riesz that ...
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56
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Question about whether a series converges almost surely
We are given that $\{X_n\}_{n\in\mathbb N}$ are iid $\text{Bernoulli}(1/2)$. Then let $Y_n=X_n/n^{\theta},\ \theta>0$. The question is to say whether $S_n=\sum_{i=1}^nY_i$ converges almost surely ...
- 1,670
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23
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Example of a "Poisson process" with no piecewise constant trajectories almost surely
Let $T_n$ be an independent sequence of exponential random variables with rate parameter $\lambda$ and $W_n = T_1 + \cdots +T_n$.
Define the random variable $N_t(\omega) = \sum_{j = 1}^{\infty} 1_{\{...
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1
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122
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Positivity of a sequence of rv's converging in probability to an almost surely strictly positive random variable
Suppose $X_n$ is a sequence of random variables such that $X_n\stackrel{p}{\longrightarrow}X$ with $\mathbb{P}\left[X>0\right]=1$. I want to prove that, for all $\varepsilon>0$, it holds $\...
- 1,234
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0
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71
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Reference request: almost sure weak convergence
I've encountered the term "almost sure weak convergence" of empirical measures in several places but haven't been able to find a textbook reference. Could anyone point me to a good reference?...
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Proof:$X_n\overset{a.s}{\rightarrow}X,Y_n\overset{a.s}{\rightarrow}Y \Rightarrow X_n+Y_n\overset{a.s.}{\rightarrow}X+Y$
I have a simple question but i didn't find a full solution to it anywhere.
I would like to know if my demonstration of the following theorem is correct.
Property:
Let $X_n\left ( \omega \right )\...
- 1,056
2
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1
answer
127
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Convergence almost surely of the sample mean
In a probability textbook I have been working through, I came across the following exercise involving almost sure convergence for the sample mean of a given sequence of random variables and was unsure ...
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56
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Show that a Function is Almost Everywhere Finite
I state the problem here:
Let $q_1, q_2, \cdots$ be an enumeration of rational numbers, and define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = \sum_{k = 1} ^\infty \frac{e^{-|x - q_k|}}{2^k \sqrt{|x - ...
- 1,096
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1
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Cauchy in Measure Implies Subsequential Almost Uniform Convergence Confusion
I am confused when reading the proof for the statement from my lecture note: "Cauchy in Measure implying subsequential almost uniform convergence".
Definitions:
Let $(X, \mathcal{F}, \mu)$ ...
- 1,096
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0
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28
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Help understanding definition of the ($\mu$-)null set.
Let $\left(X,\mathscr{A},\mu\right)$ be a measure space. In René Schilling's Measures, Integrals and Martingales, Definition 11.1 reads:
A $\left(\mu-\right)$null set $N\in\mathscr{N}_\mu$ is a ...
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