Questions tagged [uniform-integrability]

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

63 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
10
votes
0answers
484 views

What is the origin of the term "Crystal Ball Condition"?

Given $p>0$ and a family of random variables $\{X_{n}\}$, the uniform integrability of $\{|X_{n}|^{p}\}$ follows from the existence of a $\delta>0$ such that $$ \sup_{n} E(|X_{n}|^{p+\delta})&...
5
votes
0answers
388 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable ...
4
votes
0answers
32 views

Showing that $C=\{h \in L^1(\Omega) : u_1(z) \leq h(z) \leq u_2(z)\}$ is weakly compact.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $u_1, u_2 \in L^1(\Omega)$ with $u_1(z) \leq u_2(z)$ almost everywhere in $\Omega$. We let $C$ be the set $C=\{h \in L^1(\Omega) : ...
4
votes
0answers
255 views

Uniform integrability of the maximum of a random walk with negative drift

Given $S_k^{(n)} = X_1^{(n)} + ... + X_k^{(n)}$ for all $k,n\in\mathbb{N}$, where the $X_i^{(n)}$'s are iid with mean $-\gamma$ for some $\gamma > 0$ and unit variance. Let \begin{equation}M^{(n)...
4
votes
2answers
308 views

Uniformly integrable sequence tending to 0 a.s. but with $\mathbb{E}(\sup_n|X_n|) = \infty$

I am trying to find a uniformly integrable sequence of random variables $(X_n: n \in \mathbb{N})$ such that both $X_n \to 0$ almost surely and $\mathbb{E}\left(\sup_n|X_n|\right) = \infty$. I think ...
3
votes
0answers
43 views

If $f_n$ is uniformly integrable, then $\lim_{t\to\infty}\int|f_n|I_{|f_n|>t} = 0$ for all $n$

Show that if $\{f_n\}_{n\geq 1}$ is a uniformly integrable family of functions, then $\lim_{t\to\infty}\int|f_n|I_{|f_n|>t} = 0$ for all $n$. I feel like this should fall right out of the ...
3
votes
0answers
96 views

Product of uniformly integrable functions

Let $S$ be a set of measurable functions $E \to [- \infty, \infty]$. We say that $S$ is uniformly integrable iff for every $\epsilon>0$ there exists a $\delta >0$ such that, for any $f \in S$ ...
3
votes
0answers
158 views

Equivalence of asymptotic uniform integrability and convergence of mean

I was reading van der Vaart's text on asymptotic statistics and found the following theorem. The text only provides a proof for the direction where we assume asymptotic uniform integrability and want ...
3
votes
0answers
255 views

Uniform integrability of continuous function of conditional expectations

Let $X$ be an integrable real valued random variable. Let $\sigma_n$ be a sub-sigma-algebra such that $\sigma(X) = \sigma(\cup_{n\in\mathbb{N}} \sigma_n)$. Suppose $f(X)$ is integrable where function $...
3
votes
0answers
372 views

Asymptotic uniform integrability and moments of Student's $t$

I am working on an exercise where I am trying to show that the moments of a $t$ distribution converge in probability to the moments of a standard normal. I'm in need of help with what I think is the ...
3
votes
0answers
349 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
3
votes
1answer
547 views

Convergence In $L^{1}$ in the Strong Law of Large Numbers

I'm trying to prove that if $(X_n)_{n\geq 1}$ is uniformly integrable, then $X_n$ almost surely converging to $X$ implies $X_n$ converges to $X$ in $L^{1}$. How is this done? Generally speaking: ...
2
votes
0answers
27 views

Are there any positive uniformly integrable martingales which limit to 0?

In a recent exam I was asked to show several different properties about a discrete time martingale $X_n$ which is positive, uniformly integrable, with $\lim_{n\to\infty}X_n = 0$ almost surely. Are ...
2
votes
0answers
16 views

uniform integrability of a martingale related to $E_x(V_A)$ for a hitting time $V_A$

Let $\{X_n\}$ be a Markov chain on a countable state space $S$ s.t. $S-A$ is finite. We denote the hitting time $V_A = \inf \{n \ge 0: X_n \in A\}$. We then consider a function $g$ that satisfies $(*) ...
2
votes
1answer
65 views

Equivalent definitions of uniform integrability

According to Wikipedia, A class $\mathcal{C}$ of random variables is $\textbf{uniformly integrable}$ if given $\epsilon > 0$, there exists $K \in [0. \infty)$ such that $\textbf{E}(|X|I_{|X| \...
2
votes
0answers
89 views

Characterization of Martingale and Definition of Uniform Integrability

Let $M$ be progressively $\mathbb{F}$-adapted process, where index set is $[0,T]$. $(T<\infty)$ Claim If $E[M_{\tau}] = E[M_0]$ for all stopping times $\tau \in \mathcal{T}(\mathbb{F})$, then $M$...
2
votes
0answers
269 views

Prove that stopped discrete time nonnegative supermartingales are uniformly integrable

I came across this note online. On page 40 theorem 2.2.18, it seems to suggest that any stopped discrete time nonnegative supermartingale is uniformly integrable and I cannot figure out why. To be ...
2
votes
1answer
212 views

When does Moment Convergence imply convergence in $L^p$?

Under which condition does $E[X_n^2]\rightarrow E[X^2]$ impliy that $X_n\xrightarrow{L^2}X$. I think it is sufficient that $X_n$ is uniformly integrable but can't remember a proof at the moment. So is ...
2
votes
0answers
81 views

"A Master Dominated Convergence Theorem" From Gordan Žitkovic's Probability Theory notes

Im looking at "A Master Dominated Convergence Theorem" Proposition 11.11 from Gordan Žitkovic's Probability Theory notes (its on page 145) and the link to his notes can be found here: https://www.ma....
2
votes
1answer
788 views

If $M$ is an $L^2$-bounded continuous local martingale, then $M^2-[M]$ is uniformly integrable

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$ $M$ be a real-valued continuous $L^2(\operatorname P)$-bounded local $\...
2
votes
0answers
86 views

Sum of a stopped random walk and a modified random walk

I'm stuck with the following problem: Let $\{\tilde{X}_n\}_n$ and $\{\tilde{Y}_n\}_n$ be two simple symmetric random walks. Let $\{X_n\}_n$ be $\{\tilde{X}_n\}_n$ stopped when it first hits the level ...
2
votes
0answers
206 views

Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
2
votes
0answers
67 views

Characterization of uniform Integrability

How can show that that $\lim_\limits{k \to \infty} \sup\limits_{i \in J} E[(|Y_i|-k)^+]=0$ implies uniform integrability of the set of r.v's $(Y_i)_{i \in J}$ I have spent quite some time ,...
2
votes
0answers
80 views

Continuity of $t\mapsto \mathbb E X_t$

Is the following statement and its proof correct? Do you know this or related results and where I could read more about such things? Lemma: If $X$ is a right-continuous process, and the collection $...
2
votes
0answers
655 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
2
votes
1answer
148 views

Uniform integrability, book

I search about this theme, in the books is as exercise. But I want some more theory. What book recommend?
1
vote
1answer
42 views

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 ...
1
vote
0answers
32 views

Proof of the fact that $\{\text E[\xi\mid\mathcal F]:\mathcal F\}$ is uniformly integrable in Kallenberg

I don't understand the argument in the proof of Lemma 6.5 of Kallenberg's Foundations of Modern Probability (2nd edition): Let $$\mathfrak A:=\{\mathcal F\subseteq\mathcal A:\mathcal F\text{ is a }\...
1
vote
1answer
36 views

Limits of an Integral: $\lim_{n\to \infty} \int x^2 \sin\left(f_n(x)\right)\mathrm{d}x=0$

I have a problem with the following exercise. I think by the Lebesgue dominated convergence theorem it can be solved but i don't know that $\sin\left(f_n(x)\right) x^2$ dominated by ? It it true to ...
1
vote
0answers
25 views

Uniform integrability and converging process

Let Y = (Y2)n∈N0 such that P({Yo=2})= 1/2 We consider the fY adapted process X = (X2)n∈N0 = (∏j=0nYj)n∈N0 Show that there is X∞ ∈ L1(P) such that Xn -> X∞, but X does not converge to X∞ in L1(P) ...
1
vote
1answer
64 views

Confusion on Uniform Integrability of Random Variables

We have the definition that a random process, $X_n$, is (1st power) uniformly integrable if $$\lim_{M\to\infty}\sup_n\mathbb{E}(|X_n| ; |X_n|\geq M)=0$$. My question is whether the following four ...
1
vote
0answers
51 views

Does Uniform Integrability of a product martingale $M$ imply integrability of $\sup|M_n|$?

Let $(M_n)$ be a non-negative uniformly integrable product martingale - i.e. $M_n=\prod_{j=1}^{n}X_j$ for independent non-negative r.v. $X_n$. The problem asks if there exists a random variable such ...
1
vote
0answers
51 views

$f_\infty(t)=g_\infty(t)$ a.e?

Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that: $$ \qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1 $$ $$ \...
1
vote
1answer
30 views

Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$?

Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
1
vote
0answers
25 views

Why without loss of generality $\{f_n\}$ et $\{g_n\}$ are uniformly integrable over $A$?

Let $(E,\mathcal{A},\mu)$ be a probability space. We have the following lemma: Lemma : (biting lemma) Suppose $\{\phi_n\}$ is a sequence in $\mathcal{L}^1_{\mathbb{R}}$ such that $$ \sup_{n}{\int_{E}{...
1
vote
1answer
281 views

Optional Stopping Theorem and Uniform Integrability

Here's a problem related to the Optional Stopping Theorem: Let $S_n$ be a symmetric simple random walk on $\mathbb{Z}$ (as in, at each step, the random walk moves a distance of 1 in a uniformly ...
1
vote
1answer
72 views

Equivalence condition to uniform integrability

Please help me with this exercise: Let $\mathcal{F}$ be a family of functions, each of which is integrable over $E$. Show that $\mathcal{F}$ is uniformly integrable over $E$ if and only if for each $\...
1
vote
0answers
55 views

Proof: Every supermartingale is locally of class $(D)$

An adapted RCLL process $(X_t)_{t\leq T}$ is said to be of class $(D)$ is the family $$ \{ X_\tau \mid \tau\leq T \text{ is a stopping time}\}$$ is uniformly integrable. I've heard a lemma that ...
1
vote
1answer
76 views

$f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable

If $f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable where $\mu $ is positive measure My Attempt: Uniformly Integrable family: $\{f_n\}_{n\in A}$ is said to be uniformly integrable if $...
1
vote
0answers
54 views

Minimal stopping time of brownian motion

Suppose $W$ is a Brownian motion, let $H_B$ be the hitting of $B \in \mathbb{R}$ and let $\tau$ be another stopping time that is taken to be minimal, i.e $(W_{t\wedge \tau})_{t \geq 0}$ is uniformly ...
1
vote
0answers
74 views

Uniform integrabiliy of submartinales

Let $\mathcal X=(X_i)_{i\in I}$ be a family of integrable, real valued random variables on $(\Omega,\mathcal F,\mathbb P)$. One easily sees that $\mathcal X$ is uniformly integrable iff both $\mathcal ...
1
vote
1answer
151 views

$L^{1}$ bounded but non uniformly integrable family

Let $X_{k}$ be a collection of independent R.V.'s with $E(X_{k})=1$ and $X_{k}>0$ for all $k$. Define $M_{n}=\Pi_{k=1}^{n} X_{k}$ and $a_{k}=E(\sqrt{X_{k}})$. Then 1) If $\Pi_{k=1}^{n}a_{k}\...
1
vote
0answers
346 views

Sufficient Condition for Uniform Integrability $\mathcal{L}^1$-boundedness

I saw the following proof showing necessary and sufficient conditions for Uniform Integrability: And was just wondering does uniform integrability still hold if we ONLY have $\mathcal{L}^1$-...
1
vote
0answers
73 views

Uniform convergence of exp integral

I need to investigate a uniform convergence of the integral $I$ on E, where $$ I = \int_0^{+\infty} x^2 e^{-a x^4} \, \text{d} x, \, E = (0, \infty). $$ It's obvious for me that it converges uniformly ...
1
vote
0answers
169 views

Uniformly Integrability of $M_{t\land \tau}$

I want to show the uniformly integrability of a solution to the Skorokhod embedding Problem. i.e. given a centered probability measure $\mu$ with finite first moment, we want to construct a stopping ...
1
vote
0answers
349 views

Strengthening Central Limit Theorem to Convergence of Mean for Specific Case

The problem is as follows: Given two players, each flips a coin that produces heads with probability $p$. This is repeated $n$ times. What is the asymptotic behavior of the expected absolute total ...
1
vote
0answers
271 views

Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...
1
vote
0answers
152 views

Stochastic processes closed under truncation

I'm currently studying some properties of general stochastic processes, and am having some issue understanding how to prove this (probably simple) example. First, let me introduce the notation & ...
1
vote
0answers
105 views

Uniform integrability of functional of an ergodic Markov Chain?

Suppose $\{X_n\}$ is an ergodic Markov Chain with general state space $\mathcal{X}$ and stationary distribution $\pi$. Suppose $\forall x\in\mathcal{X}, f(x)\geq 0$ and $\mathbb{E}_{\pi}[f(X)]<\...
1
vote
0answers
109 views

Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e. $$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}...