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Questions tagged [uniform-integrability]

For questions about families of uniformly integrable random variables. Use the tags (measure-theory) or (probablity-theory).

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Show $M^2$ is of class D if $M$ is a martingale bounded in $L^2$

Let $M$ be a martingale null at $0$ and bounded in $L^2$. Then $M^2$ is a submartingale (just apply Jensen's inequality), and by Doob's $L^2$ inequality, we have $$E\left[\sup_t M_t^2\right]\leq 4E\...
Mingzhou Liu's user avatar
4 votes
1 answer
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$X_n \rightarrow_d X$ and UI implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$

Claim: Let $(X_i),X$ be real valued r.v.s. Then $X_n \rightarrow_d X$ and uniformly integrable implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$ How can I prove this claim directly? Here's how I proved ...
ForgeBloyb's user avatar
4 votes
1 answer
53 views

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable?

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable? This statement is used, without proof, in another statement. I haven't been able to prove it. Is ...
math_undergrad_questions's user avatar
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If $C_1$ and $C_2$ are uniformly integrable then $\{f_1+f_2:f_1\in C_1, f_2\in C_2\}$ is?

Let $C_1$ and $C_2$ be uniformly integrable collections of functions on a measure space $(\Omega,\mathscr{F},\mu)$, and let $C$ be the collection defined as \begin{equation*} C = \{f_1 + f_2:f_1\in ...
Bart's user avatar
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1 answer
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Show that $\mathbb E(X_n) \xrightarrow{n \to \infty} \mathbb E(X)$ using the following decomposition.

Suppose that each $X_n$ and $X$ are non-negative random variables and $X_n \stackrel d \to X.$ Assume that $\{X_n \}_{n \geq 1}$ is uniformly integrable. Prove the following decomposition $$\mathbb E(...
Akiro Kurosawa's user avatar
1 vote
2 answers
105 views

Convergence in $\mathscr{L}^1$ implies uniform integrability?

Quoting from Rogers and Williams, p. 116, Theorem 21.2 (A necessary and sufficient condition for $\mathscr{L}^1$ convergence): Let $(X_n)$ be a sequence in $\mathscr{L}^1$, and let $X\in\mathscr{L}^1$....
Bart's user avatar
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KL-divergence, total variation distance and weak convergence against unbounded functions

Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence: $$ D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$ As a ...
tituf's user avatar
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weak convergence imply expectation $\le$ lower limit of sequence expectation

$(X_{n})$ are uniformly integrable stochastic process ,$X_{n}\xrightarrow{d}X$ Proof $E[\left | X \right |]\le \liminf_{n\to\infty}E[\left | X_{n} \right |]$ I see a way that $f_{m}(x)=\begin{cases}\...
Yu GongLian's user avatar
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111 views

Let $(X, \mathcal M, \mu)$ be a finite measure space. If $f_n, f$ are $L^1$ functions, $f_n$ unif. integrable, $f_n \to f$, then $f_n \to f$ in $L^1$

Let $(X, \mathcal M, \mu)$ be a measure space with $\mu(X)$ finite, and $f,f_1,f_2,\dots$ be $L^1$ functions. Show that if $\{ f_n \}$ is uniformly integrable and $f_n \to f$ for a.e. $x \in X$, then $...
Squirrel-Power's user avatar
1 vote
1 answer
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In a finite measure space, uniform integrability implies $L^p$ bounded?

In the probability ambient we work with a measure space $(\Omega, \mathscr{F}, \mathbb{P})$ that is of finite measure (i.e. $\mathbb{P}(\Omega)<+\infty$). I'm interest to understad all the relation ...
Manuel Bonanno's user avatar
5 votes
1 answer
128 views

Uniform Integrability and relative compactness

I am trying to proof relative compactness in L2(0,1) for a specific set of functions $(\phi_n)_{n \in \mathbb{N}}$ with following properties: $\int_0^1 \phi(x) dx = 0 $ $||\phi_n^2||_{L1(0,1)} = 1 $ $...
Alphacache's user avatar
2 votes
0 answers
162 views

Central limit theorem: negative moments

Let $X_{1},\ldots,X_{N}$ be i.i.d. random variables with mean 0 and variance 1. Assume that $X_{i}$ are continuous random variables with all finite moments and a nice density function. Let \begin{...
Ele's user avatar
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Check my work: Find almost sure limit of product of uniform distributions

Suppose $\{\xi_n\}$ are i.i.d uniform random variables on $[0,1]$, and $$\eta_n = 2^n\Pi_{i=1}^n\xi_i$$ then: Show that $\{\eta_n\}$ is a martingale, Find the almost sure limit of $\{\eta_n\}$ ...
喵喵露's user avatar
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1 answer
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If $\{f_n\}$ is uniformly integrable, then there's a subsequence $(f_{k_n})$ of $(f_n)$ such that $\big(\int _Ef_{k_n}d\mu \big)$ is a Cauchy sequence

The page 20 of the book "Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications" (written by Ben Amar and O'Regan) has the following lemma. Lemma: Let $(X,...
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Redundancy In Definition of Uniform Integrability?

In his notes here on modes of convergence for measurable functions $f_n : X \to \mathbb{C}$, Terry Tao provides the following definition of uniform integrability: Let $(X,\mathcal{B},\mu)$ be a ...
Nick A.'s user avatar
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Uniform integrability with respect to a family of measures

Let $(\mu_n)_{n \in \mathbb N}$ be a sequence of Levy measures such that: $$\lim_{n \to \infty}\mu_n(E) = \mu(E), \quad (\forall\,\, E \,\, \mu-\hbox{Continuity set}, 0 \notin \overline{E})$$ where $\...
PSE's user avatar
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2 answers
183 views

Sufficient condition for $L^1$ convergence using uniformly integrabllity

I would like to prove the following result : let $(Xn)_n$ be a sequence in $L^{1}(\Omega,\mathcal{F}_t, \mathbb{P})$ that converges almost surely to $X\in L^1$. Then if $(X_n)_n$ is uniformly ...
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4 votes
2 answers
236 views

Show that a family is uniformly integrable

I consider a family of sub sigma algebra $(\mathcal{F}_s)_{s\in S}$ on ($\Omega,\mathcal{A}, \mathbb{P}$) and $X\in L^1(\Omega,\mathcal{A}, \mathbb{P})$. I want to show that $Y_s =\mathbb{E}(X | \...
G2MWF's user avatar
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1 vote
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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| ...
André Goulart's user avatar
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0 answers
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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\...
Andyale's user avatar
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1 answer
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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 $P\left(\left|X_n-X\right| \geq \epsilon\right) \leq C/n$. ...
Eryna's user avatar
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2 votes
0 answers
76 views

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*}...
SafariPark's user avatar
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1 answer
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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 ...
some_math_guy's user avatar
2 votes
1 answer
44 views

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 $X_n=\alpha^n\mathbb{I}_{(0,2^{-n})}$, $n \geq 1$, ...
Fran712's user avatar
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4 votes
2 answers
183 views

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 $X=\lim _{n \rightarrow \infty} X_{n}$ (...
math math's user avatar
1 vote
3 answers
393 views

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 ...
math math's user avatar
3 votes
0 answers
49 views

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 $(\...
Akira's user avatar
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0 votes
1 answer
27 views

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 \...
zlbi's user avatar
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4 votes
1 answer
61 views

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}\}}...
Daniel Williams's user avatar
1 vote
1 answer
160 views

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 [...
Analyst's user avatar
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3 votes
3 answers
567 views

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\...
ALNS's user avatar
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-1 votes
1 answer
93 views

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 ...
nien4351's user avatar
2 votes
0 answers
109 views

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 ...
Florian Ente's user avatar
1 vote
1 answer
92 views

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 ...
Andyale's user avatar
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3 votes
1 answer
89 views

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 $\Omega$. $\mathscr{F}$ ...
Andyale's user avatar
  • 191
0 votes
0 answers
34 views

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 ...
Andyale's user avatar
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1 vote
0 answers
39 views

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 $\...
Andyale's user avatar
  • 191
1 vote
1 answer
244 views

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 $\...
FactorY's user avatar
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2 votes
0 answers
323 views

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 ...
Ssay's user avatar
  • 488
4 votes
1 answer
98 views

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 ...
Marcin Wnuk's user avatar
1 vote
0 answers
36 views

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 $F$ ...
Exa Ready's user avatar
1 vote
1 answer
221 views

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 ...
user99432's user avatar
  • 890
6 votes
1 answer
333 views

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{...
mathmrk's user avatar
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1 vote
0 answers
103 views

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\|] = ...
nien4351's user avatar
1 vote
2 answers
419 views

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 ...
Tars's user avatar
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4 votes
1 answer
172 views

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 $\...
verygoodbloke's user avatar
2 votes
1 answer
66 views

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 $...
nien4351's user avatar
2 votes
1 answer
76 views

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 ...
Andrew's user avatar
  • 12.1k
2 votes
1 answer
68 views

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 ...
verygoodbloke's user avatar
0 votes
0 answers
58 views

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 $\...
non parratimo's user avatar

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