Questions tagged [uniform-integrability]
For questions about families of uniformly integrable random variables. Use the tags (measure-theory) or (probablity-theory).
351
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Why is the definition of Uniform Integrability not interesting for infinite measures?
Given $(\Omega, \mathcal F, \mu)$ measure space (In principle, $\mu$ is not necessarily a finite measure.). We know that
$$\int |f| d\mu < \infty \iff \lim_{b \to \infty} \int_{[\,|f|>b\,]} |f| ...
- 341
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How to show following is weakly sequentially compact.
I have defined the integral operator on a finite measure space $(X,\Sigma,\mu)$ Orlicz space $T: L^{\Phi}\to L^{\Psi}$, suppose we have the result that says for any bounded sequence $\{f_n\}$, $\{Tf_n\...
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Convergence in prob. + Uniform integrability => L1 convergence (proof with rate of convergence) [closed]
We know that:
I am wondering if it is possible to be more precise and state that assuming the conditions of the theorem and adding that
$P\left(\left|X_n-X\right| \geq \epsilon\right) \leq C/n$.
...
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43
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Uniform convergence in distribution implies convergence of moments
I am reading a paper in which the author wants to prove the convergence of the moments. He transforms the object of interest $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)$ into
\begin{align*}...
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Details on the proof of: Suppose $X_n \xrightarrow{\mathbb{P}} X$. $X_n$ is uniformly integrable (u.i.) $\implies$ $X_n \xrightarrow{L^1} X$
I am having trouble underestanding the proof of
Suppose $X_n \xrightarrow{\mathbb{P}} X$.
$X_n$ is uniformly integrable (u.i.) $\implies$ $X_n \xrightarrow{L^1} X$
The proof and the definitions ...
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2
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Uniform integrability of $X_n=\alpha^n\mathbb{I}_{(0,2^{-n})}$, $\alpha \in (0,1)$, $\Omega=(0,1)$
Let $((0,1), \mathcal{B}(0,1), \mathcal{Leb}_{(0,1)})$ be a probability space. Let $(X_n)_{n=1}^{\infty}$ be a sequence of random variables such that $X_n=\alpha^n\mathbb{I}_{(0,2^{-n})}$, $n \geq 1$, ...
- 191
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Show that $X$ is finite and $E(|X|)<\infty$
Let $\left(X_{n}\right)_{n \geq 0}$ is a martingale with $X_{0}=0$. Assume
$$\sum_{n=1}^{\infty} E\left(\left(X_{n}-X_{n-1}\right)^{2}\right)<\infty .$$
Show $X=\lim _{n \rightarrow \infty} X_{n}$ (...
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3
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Uniform integrability of a martingale
Let $\left(X_{n}\right)_{n \geq 0}$ is a martingale with $X_{0}=0$. Assume
$$\sum_{n=1}^{\infty} E\left(\left(X_{n}-X_{n-1}\right)^{2}\right)<\infty .$$
Show that the martingale is uniformly ...
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Construction of differentiable function $f$ such that $g(x)=-x\ln f(x)$ is convex.
I am following a proof of the characterization of uniformly integrable functions by test functions, and the proof first proves the following lemma.
Given $f_0:[0,+\infty)\to[0,+\infty)$ to be non ...
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41
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Assume $M^{\tau_n}$ is uniformly integrable. Is $X^{\tau_n}$ uniformly integrable?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$. Let $(\...
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If for any $a$, $g(a, b) \rightarrow 0$ as $b \rightarrow \infty$, then does $\sup_a g(a, b) \rightarrow 0$ when $b \rightarrow \infty$?
Here, $f:A \times \mathbb{R} \rightarrow \mathbb{R}$ is some real-valued function, $A$ is some nonempty set.
motivation: a set of random variables $\{X_\alpha\}$ is uniformaly integrable if $\lim_{K \...
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Show that $\int_{\mathbb R^p} x d\nu = 0$ in a problem related to weak convergence of measures
This question is related to another question that was answered correctly, but not in the way I wanted, as I forgot to state a hypothesis.
Let $(X_{jn})_{1\leq j \leq n}$, $X_{jn} \sim \mu_{jn}$, be a ...
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Is $\frac{1}{x^2+1/n}$ uniformly integrable?
For a set of functions $f_n\in F$ my definition of uniform integrability over the range $[-1,1]$ is
$$\forall \epsilon>0, \exists M_{\epsilon}>0 : \sup_{f_n\in F}\int_{\{|f_n|\geq M_{\epsilon}\}}...
- 255
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Can I exchange the Limit and the Inverse DTFT?
I am working on a proof which involves swapping the order of an Inverse DTFT and limit:
$$
\begin{split}
\mathcal{F}^{-1}\left(\lim_{n\rightarrow\infty}{(\hat{f_n})}\right)&=f\\
\text{does this ...
- 255
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1
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72
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A uniformly integrable family induced from taking conditional expectation of a square-integrable random variable
Let $X$ be a square-integrable random variable defined on a probability space and $(\Omega, \mathcal F, \mathbb P)$. Let $(\mathcal F_t, t \ge 0)$ be a filtration and $T>0$. Let $Y_t := \mathbb E [...
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163
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Uniform Integrability and Convergence
I am trying to solve the following problem: A sequence $\{f_n\}_{n \in \mathbb{N}}$ is said to be uniformly integrable in $L(X,d\mu)$ if $$\lim_{t \to \infty}\sup_{n\ge1}\int_{\{|f|>t\}}|f_n| \, d\...
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Uniform integrability and $L^1$ convergence of $(1/X_n)$
Let $X_n \to c > 0$ almost surely, where $c$ is a constant and $X_n > 0$ for all $n$. Also, let $(X_n)$ be uniformly integrable, so in particular $X_n \to c$ in $L^1$.
Question:
Do we have ...
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Stopped cadlag submartingale is integrable
I'm trying to understand, why a stopped submartingale is again a submartingale. In the lecture notes to my lecture this is just stated as a corollary of Doob's Optional Sampling Theorem but I don't ...
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53
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Question about uniform integrabilty and existence of convex function
The following is equivalent.
$(1) \mathscr{F}=\{f_{\alpha},\alpha\in I\}$ is uniformly integrable.
$(2) \lim_{\lambda\to \infty}\int_{[|f_{\alpha}|>\lambda|]}{|f_{\alpha}|d\mu=0},\text{ uniformly ...
- 63
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55
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Question about $L^1$ space and Uniform integrability.
Let $(\Omega,\Sigma,\mu)$ be a finite measure space and let $\mathscr{F}=\{f_{\alpha}:\Omega\to \mathbb{R}:\alpha\in I\}$ be an arbitrary family of measurable real functions on $\Omega$.
$\mathscr{F}$ ...
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33
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Question about $L^1$ Lebesgue space
An $L^1$ functional from a space $X$ to $\mathbb{R}$ is an $\mu$-measurable function such that
$$ \int_{X} |f|\,d\mu < \infty. $$
My question is if suppose I pick one function $f_1$ from $L^1$ so ...
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37
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Difficulty in understanding the following theorem related to convex functions.Please help.
Let $(\Omega,\Sigma,\mu)$ be a finite measure space and let $F=\{f_{\alpha}:\Omega\to \mathbb{R}:\alpha\in I\}$ (where $I$ is some index set) be an arbitrary family of measurable real function on $\...
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140
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Uniformly Integrability and tightness
I have to solve one problem about uniformly integrability and tightness. It is
"Let $f$ be a continuous integrable function on $\mathbb{R}$, and let $g$ be a bounded measurable function on $\...
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147
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Uniformly Integrable Random Variables are Uniformly Bounded Almost Everywhere?
I'm a beginner in stochastic processes(and measure theory). I am trying to prove(or disprove) the fact that, if $\xi_n$ are uniformly integrable random variables, then they are uniformly bounded by ...
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95
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Convergence of a sequence of suprema of expected values
Consider a sequence of stochastic processes
$$((X_f^{(n)})_{f \in F})_{n \in \mathbb{N}}.$$
All the random variables $X_f^{(n)}$ are defined on the same probability space and assume only non-negative ...
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31
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Question about uniform integrability of a collection of measurable fucntions
I am reading uniform integrability and it is defined as below
Let $(\Omega,\Sigma,\nu)$ be a finite measurable space and $F$=$f_{\alpha}\colon\Omega \to R$ (measurable) where $\alpha\in I$} then $F$ ...
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Is this set compact in $L^1$?
Let $(f_n)_{n\in \mathbb{N}}$ be a bounded sequence in $\mathcal{M}(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ with $p>1$, where $\mathcal{M}(\mathbb{R}^n)$ denotes the set of all finite measures ...
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Limit and conditional expectation commute in a uniformly integrable sequence
I am thinking of the next proposition:
Proposition.
Let $(\Omega, \mathcal{F}, P)$ a probability space, and $\{X_n\}_{n=1,\cdots}$ a uniformly integrable r.v. sequence s.t. $X_n \rightarrow X ~ \text{...
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Uniform integrability of conditional distributions
Let $X_n = (X_1^n, X_2^n) \to (X_1, X_2)$ in distribution ($\mathbb{R}^2$-valued) and suppose that $(X^n)$ is uniformly integrable, i. e.
$$
\lim_{a \to \infty} \sup_n E[1_{\|X_n\| \geq a} \|X_n\|] = ...
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2
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Does uniform integrability imply almost sure convergence of a subsequence?
True or false? If $X_n$ is uniformly integrable then there exists a subsequence $X_{n_k}$ such that $X_{n_k}$ converges a.s. to a random variable $X$ as $k→\infty$.
My attempt so far:
Since $X_n$ is ...
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Proof of a martingale not being uniformly integrable
Fix $1/2<p<1$. Let $X_1,X_2,\dots$ be independent identically distributed random variables with $\mathbb{P}(X_1=1)=p$ and $\mathbb{P}(X_1=-1)=1-p$. Let $S_0=0$ and $S_n=X_1+\dots+X_n$ and $\...
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Uniform integrability of conditional quantile functions
Let $Z^n$ be $\mathbb{R}$-valued random variables which are uniformly integrable, i.e.
$$
\lim_{a \to \infty} \sup_{n} E[1_{\{|Z^n| \geq a\}} |Z^n|] = 0.
$$
Let $X^n \to N(0,1)$ in distribution, and $...
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59
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Some variant of Law of Large Numbers for any continuous function?
Let $\{X_i\}$ be iid random variables on $\mathbb R$ $X_i\sim X$, let $f: \mathbb R\to \mathbb R$ be a continuous function s.t. expectations $\mathrm EX$ and $\mathrm Ef(X)$ are finite.
Does it ...
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Stopped version of UI martingale is UI
Suppose $(X_n)_{n\geq0}$ is a uniformly integrable martingale. Let $T$ be any stopping time. Show that the stopped process $(X_n^T)_{n\geq0}=(X_{n\wedge T})_{n\geq0}$ is uniformly integrable.
My only ...
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Tightness vs Equi-integrability, Prokorov
I am a bit confused by this Theorem in the book 'Optimal Transport
for Applied Mathematicians' of Santambrogio. This question concerns absolutely continuous (w.r.t Lebesgue) probability measures on $\...
4
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Martingale is Uniformly Integrable
Let $(Z_n)_{n\ge0}$ be a sequence of i.i.d. r.v.'s with $P(Z_1=1)=P(Z_1=-1)=\frac{1}{2}$. Let $S_0=0$ and $S_n=Z_1+\dots+Z_n$. Let $\mathcal{F}_0=\{\emptyset,\Omega\}$ and $\mathcal{F}_n=\sigma(Z_1,\...
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How to show that $B_{t\land T_a}$ is uniformly integrable?
For a Brownian motion $B_t$ started from $0$ (it is also a martingale), denote by $T_a=\inf\{t\ge 0: B_t=a\}$ the stopping time. Consider the stopped process $B_{t\land T_a}$ (this is still a ...
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112
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Checking uniform integrability of random series
Let $(X_n)_{n\geq 1}$ be a sequence of iid random variables such that $\mathbb{E}[X_n]=0$ and $\operatorname{\mathbb{V}ar}[X_n]=\sigma^2<\infty$, define $$ Y_n = \frac{X_1+\dots+X_n}{\sqrt{n}}$$
...
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Almost-sure convergence of a series of log-normal random variables
Let $(X_n)_{n\geq 1}$ be a sequence of independent and identically
distributed standard normal random variables and define for $n\geq 1$,
$$ W_n = \exp\left\{\sum_{i=1}^nX_i-\frac{n}{2}\right\} $$ ...
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113
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Showing uniform integrability to prove the limit of expectation
Let $(X_n)_{n \geq 1}$ be a sequence of random variables. We know that $\lim_{n \to \infty} X_n = X$ (a.s.), where $X$ is a deterministic value.
I want to show that $\lim_{n \to \infty} \mathbb{E}[X_n]...
- 462
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132
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Two questions about uniform integrability
I have two questions about the uniform integrability.
The definition I am using is that a class of random variables $\mathbb{\chi}$ is uniformly integrable if given an $\epsilon >0$, there exists a ...
- 462
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1
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35
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Integrability of a family of functions, parameterized by the points of the compact
For any $\theta \in \Theta$, let $g(t;\theta)$ be integrable in $t$ over, say, the half line $\mathbb{R}_+$. Then suppose $\Theta$ is a compact set, then can we say $\sup_{\Theta}g(t;\theta)$ is also ...
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171
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How to prove that $(UM_t)_{t \geq 0}$ is a Continuous Local Martingale?
Consider a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)$ and $(M_t)_{t \geq 0}$ a continuous local martingale (CLM) w.r.t $(\mathcal{F}_t)_{t \geq 0}$.
Let $U$ denote ...
- 1,031
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1
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78
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If, for $r>1$, $\operatorname{sup}_n |X_n|^r < \infty$, then $(X_n)$ is uniformly integrable
Consider a sequence $(X_n)$ of random variables satisfying $\operatorname{sup}_n E|X_n|^r< \infty$, for $r>1$. How do I show that $(X_n)$ is uniformly integrable? The answer I found is
$$E(|X_n|...
- 570
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2
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Show $(\mathbb{E}[Z\mid\mathcal{F}_t],\mathcal{F_t})_{t \geq0}$ is an *uniformly integrable* martingale.
Suppose $Z \in \mathcal{L}^1(P)$. I want to show that $(\mathbb{E}[Z\mid\mathcal{F}_t],\mathcal{F_t})_{t \geq0}$ is an uniformly integrable martingale.
I have managed to show to it is a martingale but ...
- 1,031
2
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1
answer
154
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Uniform integrability of rescaled sample mean
Assume that $X_1, X_2, ...$ are independent and identically distributed random variables with mean $\mu$ and variance $1$, then let $\bar{X}_n=n^{-1}\sum_{i=1}^n X_i$ be the sample mean. We all know ...
- 1,696
2
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1
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236
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Almost sure convergence and uniform integrability of a product of i.i.d random variables with density
I'm struggling with the following problem:
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(X_n)_{n \in \mathbb{N}} \subseteq \mathcal{L}^0(\mathbb{P};\mathbb{R})$ i.i.d with:
$$
\...
- 23
4
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136
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If the positive part of submartingale is uniformly integrable, it is closable
I am kind of confused in reading the next proof of a theorem in "Stochastic Analysis: Itô and Malliavin Calculus" in Tandem by Matsumoto & Taniguchi;
Here, a closable submartingale $\{ ...
- 163
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1
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98
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Uniform Integrability and Exponential Tightness
I would like to know if the following idea (for which i didn't find references) is correct or not.
Let $\mu_n$ be an exponentially tight sequence of non negative measure i.e. such that there exists a ...
- 121
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1
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99
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Every martingale that converges in $L^p$ converges in $L^1$? If not, what is a conterexample?
We know that, for a martingale $\{X_n\}_n$ converge in $L^1$ it is necessary that
$$\lim_{M \to \infty}\left(\sup_iE[|X_i|;|X_i|>M]\right)=0$$
that is, $\{X_n\}_n$ is uniformly integrable.
On the ...
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