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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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 $\...
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How to show equivalent that uniform integrability and convergent in L1 norm..

Here are the definitions included. Suppose $f_n → f$ a.s.$[µ]$. We say that the collection $\{f_n\}$ is uniformly integrable if given $\epsilon>0$, there exists $M>0$ such that $\sup_n\int_{\{x:|...
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4 votes
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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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2 votes
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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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2 votes
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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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1 vote
1 answer
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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]...
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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 ...
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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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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 ...
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  • 691
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1 answer
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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|...
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2 votes
2 answers
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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 ...
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  • 691
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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 ...
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1 answer
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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: $$ \...
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4 votes
1 answer
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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 $\{ ...
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1 answer
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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 ...
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  • 101
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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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1 vote
1 answer
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Is the process $Y(w)$ uniformly integrable?

I came across this question online and am not completely sure of my answer. On the probability space $([0,1),\mathcal{B}([0,1)),\mathbb{P}=\lambda_1)$, define the sequence of random variables $(Y_n)_{...
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$X_n\stackrel{P}{\to} c$ implies $\mathbb{E}f(X_n){\to} f(c)$ for continuous (but possibly unbounded) $f$

Suppose that $X_n \stackrel{P}{\rightarrow} c$ as $n\to\infty,$ where $X_n, c$ are all positive. Let $f$ be a continuous (but possibly unbounded) function on $(0,\infty)$ such that $$\int_0^\infty |f(...
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  • 1,062
2 votes
1 answer
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Convergence of second moment implies 2-uniform integrability

$(X,d):Polish\ space$ $\mathcal P_2(X):= \{ μ \in \mathcal P (X)|\int_X{d^2(x,x_0)dμ\ \text{for some }x_0\in X} \}$ Definition $\mathcal S \subset \mathcal P_2(X)$ is said 2-uniformly integrable if ...
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0 votes
1 answer
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Uncorrelated Martingales

I'm not sure how to prove that Mn+1 - Mn & Mn are uncorrelated ? If Mn = (Xn)^2 - 2nXn + n(n − 1); Where Xn is a Random walk Xn+1 = Xn +Yn+1 Where Yn - N(1,1) I already know this is a Martingale, ...
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2 votes
1 answer
181 views

Prove Martingale Property

I want to show that $M_n = (X_n)^2 -2nX_n +n(n-1)$ is a Martingale. I know that $\{X_n\}_{n\in\mathbb N}$ is a random walk process such that $X_{n+1} = X_n +Y_n$ and $\{Y_n\}$ is a sequence of i.i.d ...
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$L^p$-convergence of $L^p$-bounded martingale on a countable index set

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $I\subseteq\mathbb R$ be countable, $(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ and $(M_t)_{t\in I}$ be a real-...
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Uniformly integrable sequence: definition and characterization

As far as I know, given a positive measure space, $f\in L^1(\Omega)$ is an uniformly integrable function if for all $\varepsilon>0$, $\delta>0$ exists such that $$\int_A |f| \, d\mu <\...
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  • 2,353
1 vote
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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 }\...
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2 votes
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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 ...
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4 votes
2 answers
102 views

Uniform integrability of $\sqrt{n} Y_n$ for $Y_n \sim \mathcal{N}(0, \frac{\sigma^2}{n})$

Let $Y_n \sim \mathcal{N}(0, \frac{\sigma^2}{n})$. Then the set $\{\sqrt{n} Y_n\}_{n \ge 1}$ is uniformly integrable since $$ \sqrt{n} Y_n \sim \sqrt{n}\mathcal{N}\bigg(0, \frac{\sigma^2}{n}\bigg) = \...
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  • 2,394
2 votes
0 answers
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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 $(*) ...
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  • 195
9 votes
1 answer
329 views

Uniformly integrability of inverse of sample moments

I am interested in the uniform integrability of the set $\{Y_n\}_{n\ge 3}$ where $$ Y_n = \bigg(\frac{1}{n} \sum_{i=1}^n X_i^k\bigg)^{-1}, $$ where the $X_i$'s are i.i.d observations of a continuous ...
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  • 9,333
3 votes
1 answer
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Are the ordinary least squares regression coefficients uniformly integrable?

In my previous question I asked whether a set of asymptotically normal random variables $\{X_n\}_{n \ge 1}$ are uniformly integrable. In the accepted answer the poster showed that this does not hold ...
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  • 9,333
1 vote
1 answer
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Does asymptotic normally imply uniform integrability?

If a set of random variables $\{X_n\}_{n \ge 1}$ is asymptotically normal is it also uniformly integrable? Intuitively it seems like this should be true as $\{X_n\}_{n \ge 1}$ becomes 'nicer' as $n$ ...
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  • 9,333
1 vote
1 answer
124 views

Different definition of uniform integrability

I'm reading Tao's An introduction to measure theory.And the definition of uniform integrability in this book is $(X,{\mathfrak {M}},\mu)$ is a measure space(not necessarily finite). A sequence $f_n:X ...
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1 vote
1 answer
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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 ...
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1 vote
1 answer
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Family of RV is uniformly integrable iff there is $\phi$ with $\frac{\phi(x)}{x}\rightarrow\infty$ and $\sup_n E(\phi(\vert X_n\vert))<\infty$

I want to show that a family $(X_n)_{n\in I}$ of real-valued RVs is uniformly integrable if and only if there exists a measurable function $\varphi:[0,\infty)\rightarrow[0,\infty)$ with $\frac{\varphi(...
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1 vote
0 answers
34 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) ...
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  • 21
0 votes
1 answer
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Uniform integrabily of a stopped martingala? [closed]

I'm struggling with an exercise and I really don't know how to solve it, I will apprecciate any help, even if it small, in the direction of helping me solve this problem. I have a discrete time ...
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1 vote
1 answer
78 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 ...
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1 vote
1 answer
62 views

Integrable submartingale

Suppose $X_n$ is a uniformly integrable submartingale and $T$ a stopping time. I wanna show that $X_T$ is integrable. To this end, I write $X_T=\lim_{n\to \infty}\sum_{k=1}^n X_k \mathbb{1}_{\{T=k\}}$....
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  • 663
3 votes
1 answer
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stopped UI discrete time martingale is UI

If $\{X_n\}$ is a uniformly integrable discrete time Martingale, and if $\tau$ is a (possibly $\infty$-valued) stopping time, then $\{X_{\tau \wedge n}\}$ is uniformly integrable (note I don't care ...
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  • 103
0 votes
0 answers
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𝑋𝑡’s are uniformly integrable if 𝑠𝑢𝑝 𝐸|𝑋𝑡|^𝑟 < ∞

I am working on this problem. If $sup_t E|X_t|^r <\infty$ for some $r>1$, then the $X_t$’s are uniformly integrable. Note that $\mu$ is not finite. My attempt: Using Holder's inequality, $ \frac{...
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  • 13
0 votes
1 answer
67 views

If $(X_k)_{k\in\mathbb N}$, show that $\sum_{k=1}^n(X_k-\text E[X_k])$ converges almost surely as $n\to\infty$

Let $\nu$ be a $\sigma$-finite measure on $\mathbb R$ with $$\int_{(-1,\:1)}\nu({\rm d}x)x^2<\infty\tag1$$ and $\nu(\{0\})=0$, $$I_k:=\left(-\frac1k,-\frac1{k+1}\right]\cup\left[\frac1k,\frac1{k+1}\...
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  • 12.8k
3 votes
0 answers
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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 ...
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  • 604
2 votes
1 answer
384 views

Is the exponential martingale of Brownian motion uniformly absolutely continuous?

Let $M = \{M_t\}_{t\ge0}$ be the exponential martingale of Brownian motion $W= \{W_t\}_{t\ge0}$, that is, $$ M_t = \mathcal E(W)_t = \mathrm{exp} \left( W_t - \frac{t}{2}\right). $$ Question: Is $M$ ...
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  • 1,631
1 vote
2 answers
100 views

If $\sup_n$ $E|X_n|^{1+\sigma} \lt \infty$ for $\sigma \gt $0, then $\{|X_n|\}$ is uniformly integrable

If $\sup_n$ $E|X_n|^{1+\sigma} \lt \infty$ for $\sigma \gt $0, then $\{|X_n|\}$ is uniformly integrable. I've seen a similar problem without the exponent on $|X_n|^{1+\sigma}$ and tried to apply it ...
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  • 77
1 vote
1 answer
82 views

Example of a sequence that is not uniformly integrable.

By definition a sequence $\{X_n\}_{n \ge 0}$ is uniformly integrable if $\sup_n E[X_n \cdot \mathbb{I}_{\{X_n > a\}}] \to 0$ and $a \to \infty$. An equivalent definition is that 1) $\sup_n E[|X_n|] ...
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0 votes
1 answer
128 views

Showing $\{X_n\}$ is uniformly integrable

so I want to prove that $\{X_n\}$ is uniformly integrable, given that E[$|X_n|^2$] < C for every n where C is a constant. so using the markov inequality we get P($|X_n|^2$ $\geq$ N) $\leq$ $\frac{(...
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1 vote
0 answers
71 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 ...
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  • 181
1 vote
1 answer
220 views

Show that a stopped process is uniformly integrable

Suppose $(X, \mathscr{F})$ is a martingale. Show that $(X_{\tau \wedge n}, \mathscr{F})$ is a uniformly integrable for any finite stopping time $\tau$ such that $\{X_n\}$ is uniformly integrable. My ...
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2 votes
1 answer
113 views

Continous mapping theorem and uniform convergence of the integrals of a collection of bounded functions

Let $\mu_n$ be a sequence of probability measures on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ such that $\mu_n \to^d \mu$. Let $\{f_a\}_{a\in\mathbb{R}}$ be a colection of bounded real functions, each ...
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