For questions about families of uniformly integrable random variables. Use the tags [tag: measure-theory] or [tag: probablity-theory].

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Uniform integrability of a sequence of random variables defined by a recursive relation

I have an i.i.d sequence $(u_j)_{j\in \mathbb{Z}_+}$ with zero mean and finite variance, say $\sigma^2$. Furthermore, I have another random variable $X_0$ (defined on the same probability space) which ...
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40 views

Uniform Integrability of Random Variables

$\{X_n\}$ is uniformly integrable if $\lim_{M \rightarrow \infty} (\sup_n \mathbb{E}(|X_n| \chi_{|X_n| > M}) = 0$ I would like to know if $\{X_i\}$ uniformly integrable $\implies \sup_n ...
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37 views

Martingale convergence for UI martingales

I started reading this paper (Lamb, Charles W.. “Shorter Notes: A Short Proof of the Martingale Convergence Theorem”. Proceedings of the American Mathematical Society 38.1 (1973): 215–217) today. In ...
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51 views

Uniform integrability and stopping times

I want to know whether there is any example where $X_n$ is uniformly integrable, $N$ is a stopping time and $E[X_N] =\infty$? Or uniform integrability of $X_n$ implies that $E[X_N]< \infty$?
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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 ...
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62 views

Counterexample for uniform integrability of a stopped process

I want to find an example where $X_n$ is uniformly integrable, $N$ is a stopping time, but $X_n^N = X_{\min\{n,N\}}$ in not uniformly integrable. There is a theorem saying that if $M_n$ is a uniform ...
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convergence of scores of log-concave densities

Given a sequence of log-concave densities $f_n = \exp\phi_n(x)$ converges uniformly in exponential weighting norm to some log-concave density $f_0 = \exp\phi_0(x)$, i.e. $\sup_x\exp ...
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51 views

Counterexamples for expressions regarding uniform integrability

I have two questions regarding uniform integrability. 1- Is there any example such that $E[\sup_{n \geq 1} \vert X_n \vert] < \infty$ but $\sup_{n \geq1} E [\vert X_n \vert^p] =\infty$ for any $p ...
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46 views

Example for uniformly integrable $\mathbb{L}^2$-bounded sequences

How can I construct an example to show that the sequence $\{X_n\}_{n \ge 1}$ can be $\mathbb{L}^2$-bounded, however it has $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Basically I need ...
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40 views

A uniformly integrable sequence that is not uniformly integrable in $\mathbb{L}^p$ for $p>1$

I need to construct an example to show that if for the sequence $\{X_n\}_{n \ge 1}$ we have that $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$, the sequence in not necessarily ...
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1answer
37 views

Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

I need to find an example such that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$ but $\{X_n\}_{n \ge 1}$ are not uniformly integrable. I can show that if $\{X_n\}_{n \ge 1}$ are uniformly ...
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1answer
64 views

Application of the theorem stating that “almost sure convergence plus uniform integrability implies $L^1$ convergence”

I'm confused on an application of the theorem stating that "almost sure convergence plus uniform integrability implies $L^1$ convergence". The example that follows is from van der Vaart "Asymptotic ...
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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 & ...
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55 views

Uniform integrability of “exponent martingale”

Suppose that $X_n$ is an iid sequence of random variables with $P[X_i=1]=p$ and $P[X_i=-1]=q:=(1-p)$. Then, $S_n=X_1+\cdots+X_n$ (with $S_0=0$) is a simple random walk. We can easily check that ...
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67 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 ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
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55 views

Uniform Integrability and relation to $L^p$ for $p>1$

Let $X_n$ be a martingale. Then we know that for $p> 1$ the conditions $\sup_n E[|X_n|^p] < \infty$ and $E[\sup_n |X_n|^p] < \infty$ are equivalent. For $p=1$ this does not hold, because ...
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56 views

Conditions implying uniform integrability

We say that a family of random variables $X_n, n \geq 1$ is uniformly integrable if $$\lim_{M \rightarrow \infty} \sup_{n} E[|X_n| 1_{|X_n|>m}]=0.$$ I am struggling with some proofs and could ...
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2answers
52 views

Monotone convergence theorem without monotonicity

I have a question related to the monotone convergence theorem. Consider a situation in which all assumptions of the monotone convergence theorem are satisfied except the monotonicity, i.e. we have a ...
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1answer
72 views

Exchanging limits and Riemann Integral

Consider a function $f:\Theta \subseteq \mathbb{R}\rightarrow \mathbb{R}$ continuous at $\theta=\theta_0$ such that $f(\theta)\geq 0$ $\forall \theta \in \Theta$. Consider a sequence of real numbers ...
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1answer
32 views

Uniform integrability of reciprocal of random variables

Let $\{X_n, n\geq 0 \}$ be a sequence of positive random variables that are uniformly integrable. Assume that $\frac{1}{X_n}$ is integrable. Then, is it true that $\left\{\frac{1}{X_n}, n \geq ...
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For a martingale $X$ does uniform integrability imply integrability of $\sup |X_{n}|$?

All is in the title: if $(X_{n})$ is a uniformly integrable martingale is it true that $\sup_{n\in \mathbb{N}} |X_{n}|$ is an integrable variable ? If I had to take a guess I'd say the answer is no, ...
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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 ...
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A Probabilistic Approach to Stirling's Formula

I am working on the following problem: Suppose $X_1, X_2,\dots$ are i.i.d Poisson$(1)$ random variables, and let $S_n=X_1+\dots+X_n$. a)Compute $E\left[\left( \frac{S_n-n}{\sqrt{n}} ...
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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} ...
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1answer
57 views

Showing $|S_n/\sqrt{n}|^\alpha$ is uniformly integrable given certain conditions

Let $X_1, X_2, \dots$ be i.i.d. with $E(X_1)=0$ and $E(X_1^4)<\infty.$ Let $S_n=X_1+\dots +X_n$. Show that $|S_n/\sqrt{n}|^\alpha$ are uniformly integrable for any $0<\alpha<4$. I am ...
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49 views

Characterization of uniform Integrability

How can show that that $\lim_{k \to \infty} \sup_{i \in J} E[(|Y_i|-k)^+]=0$ implies uniform integrability of the set of r.v's $(Y_i)_{i \in J}$ I have spent quite some time ,unsuccessfully trying to ...
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1answer
82 views

$\sup\limits_{X \in \mathcal X, Y\in \mathcal Y}E[|Y| \mathbb 1_{\{ |X| \ge K\}}]\to0$ for $\mathcal X$, $\mathcal Y$ uniformly integrable

I want to show that $\lim_{k \to \infty } \sup_{X \in \mathcal{X}, Y\in \mathcal{Y}}E[|Y| \mathbb{1}_{\{ |X| \geq K\}}]=0$ It is given that the set of random variables $\mathcal{X}$ and ...
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33 views

$L^2$ convergence from convergence in distribution and uniform integrability

Is it true that if $X_n \to X$ in distribution and $\{X_n^2\}$ are uniformly integrable then $X_n \to X$ in $L^2$
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1answer
37 views

An example of a non trivial weakly compact in $L^1$

Let us consider the space $L^1$ of measurable functions associated to a probability space. I would like to see non trivial examples of weakly-compact sets. Thank you.
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Uniformly integrability and convergence

Question 1: $X_n$'s are non-negative, uniformly integrable. Then $E\left[\dfrac{\max_{1\leq k \leq n} X_k}{n}\right]\rightarrow 0$. Question 2: If u.i. is dropped then above the may fail. My ...
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84 views

Characterization of uniform integrability of random variables

Let $\{X_n \}$ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P)$. Then, $\{X_n\}$ is uniformly integrable if $$\lim_{M \to \infty} \sup_n \int_{|X_n| > M } |X_n| = ...
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2answers
58 views

Uniform integrability of stopped martingale

Let $(M_t,\mathcal{F}_t)_{t\geq 0}$ be a martingale with continuous paths and $(\tau_k)_{k\geq 0}$ stopping times. Hence we know that $M_{t\wedge\tau_k}=\mathbb{E}[M_t|\ \mathcal{F}_{t\wedge\tau_k}]$. ...
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$E(f(|X_n|))$ property implies uniform integrability?

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
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74 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
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1answer
62 views

Dominated convergence martingale and uniform integrability

For a fixed $t\in [0,1]$ I have a sequence $(X^t_n)_{n\geq 1}$ of normal distributed random variables which is a martingale and bounded in $L^2$. So by the martingale convergence theorem there exists ...
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1answer
26 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
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Is this modified Dirichlet function integrable and/or uniformly integrable?

I was considering the following functions $X_n(x)$ , similar to the Dirichlet function: \begin{Bmatrix} q\: \; \textrm{when} \; x\:=\frac{p}{q} \in \mathbb{Q} \cap [0, \frac{1}{n}]& \; p \in ...
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1answer
50 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
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Family of functions that are bounded in $L^1$ but *NOT* Uniformly Integrable

I'm having a difficult time constructing a counter example to this. My intuition (sloppily) is to construct a family of functions {$X_n$} that have Dirac pulses at $n$ and $-n$. Such that $\sup_n \Bbb ...
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Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
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340 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
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Uniform integrability of $\cos ^n\left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$

I am trying to show uniform integrability of $\cos^n \left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$, where $c$ is some positive constant. I was able to show that $\cos^n ...
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158 views

Uniform Integrbility

I need to show that if $|f_n|\leq g$, and $g$ is integrable, then $\{f_n\}$ is uniformly integrable, i.e., $\underset{a\rightarrow\infty}\lim\sup_n\int_{[|f_n|\geq a]} |f_n| d\mu=0$. Here is how I ...
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69 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 ...
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292 views

Tightness and Uniform Integrability

I'm trying to develop some intuition for the concepts of tightness and uniform integrability, in a probabilistic context. (i) Does uniform integrability imply tightness? (ii) If not, is ...
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Prove that process is uniformly integrable

Let $(X_t)_{t\ge 0}$ be a stochastic process, and let $Y$ be an integrable random variable, such that $|X_t|\le Y$ for $t\ge0$. Prove that $(X_t)_{t\ge 0}$ is uniformly integrable. From ...
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104 views

Product of two uniformly square integrable random variables

Take two classes of uniformly square integrable random variables $\lbrace X_t:t\in T\rbrace$ and $\lbrace Y_t:t\in T\rbrace$. Is the class $\lbrace X_tY_t :t\in T\rbrace$ uniformly integrable? My ...
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130 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
351 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
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482 views

Convergence in $L^1$ and uniform integrability

Suppose that $X_n\leq Y_n\leq Z_n$ where $X_n\to X$, $Y_n\to Y$, $Z_n\to Z$ in probability. If $E(X_n)\to E(X)$ and $E(Z_n)\to E(Z)$, show that $E(Y_n)\to E(Y)$. I want to solve the task above. ...