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

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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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1answer
35 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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49 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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1answer
42 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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1answer
30 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
48 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
63 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
24 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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1answer
62 views

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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35 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 ...
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75 views

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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20 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} ...
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1answer
52 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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45 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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69 views

Question About Uniform Integrability and $L1$

I have proven successfully that if $X$ is uniformly integrable then $X\in L1$, but is the converse true?
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1answer
74 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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1answer
32 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
35 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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1answer
90 views

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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1answer
74 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
51 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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1answer
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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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1answer
67 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
52 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
25 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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36 views

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
41 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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1answer
102 views

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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1answer
305 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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1answer
37 views

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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1answer
147 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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67 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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1answer
217 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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1answer
88 views

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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1answer
100 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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125 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
283 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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1answer
471 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. ...
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1answer
115 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
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1answer
123 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
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1answer
64 views

Proving a certain limit for uniformly integrable random variables

There is an interesting problem that has been resisting my efforts for a while. Assume that $\{X_n: n = 1, 2, \ldots\} $ is a sequence of uniformly integrable random variables. I would like to show ...
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1answer
100 views

If a family $\{f_n\}$, is uniformly integrable, then also $\{|f_n|\}$ is.

I want to show that if a set of functions $\{f_n\}_n$ is uniformly integrable, then also $\{|f_n|\}_n$ is also uniformly integrable. How can I show this? My guess is to use separate $f_n$ into real ...
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1answer
37 views

Compacting in Uniform Integration

If I have that $f$ is integrable, how show that for all $\varepsilon>0$, there is some $h>0$ for which we have that $$\int_{\{x\in X \colon |f(x)|<h\}} |f(x)|d m<\varepsilon$$ in a ...
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1answer
123 views

$\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$ implies that random variables $X_n$ are uniformly integrable

$X_n$ are uniformly integrable if $\lim_{R \rightarrow \infty} \sup_n E[|X_n|,|X_n| \geq R] = 0$. Show that if $\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$, then $X_n$ are uniformly integrable. ...
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1answer
290 views

Product of sequence of uniform integrable random variables are uniformly integrable?

If $\{X_i\}$ for $i \in I $ and $\{Y_j\}$ for $j \in J$ are uniformly integrable.Then prove that, $\{X_i+Y_j\}$ for $(i,j) \in I \times J$ is uniformly integrable.What about $ \{X_iY_j\} $ for $(i,j) ...
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1answer
141 views

Uniform integrability of a sequence of functions.

I am considering the following sequence of functions. I think it converges pointwise to $0$ because the intervals in the domain in which the nth function is greater than zero eventually shrink and ...
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1answer
210 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
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2answers
460 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
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1answer
239 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...