For questions about weak convergence, which can concern sequences in normed/ topological vectors spaces, or sequences of measures. Please use other tags like (tag: functional-analysis) or (tag: probability-theory).

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1answer
6 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
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0answers
20 views

Exercise on functional analysis

I have a problem with this exercise of functional analysis about weak convergence: Let $U:=\prod_{i}^{d}(a_{i},b_{i})$ a subset of $\mathbb{R}^d$ with $a_{i}<b_{i}$ for each i and let ...
1
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0answers
39 views

Central limit theorem extends to absolutely continuous measures

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ that is absolutely continuous w.r.t. $\mathbb{P}$. Let the ...
2
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2answers
40 views

Limit law of real-valued independent random variables

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp. i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$. Then, are $X$ ...
2
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1answer
32 views

Weak Convergence for a specific example

Let $X=(0,1)$ and $u_v(x) = v^{1/p}1_{(0,1/v)}$, where $1 < p < \infty$. It needs to be shown that $\lim_{v \to \infty} \int_X u_v \phi = 0$ for all $\phi \in \mathcal{L}^{p^\prime}$ (The ...
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0answers
12 views

limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
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3answers
86 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
0
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1answer
28 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
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1answer
32 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
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0answers
32 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
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2answers
33 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
1
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1answer
31 views

$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$. Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty. Can anyone help me with this problem? ...
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0answers
20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
2
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1answer
53 views

Product of weak/strong converging sequences

Let's consider two sequences $u_n$ and $v_n$ such that $$u_n\to u\,,\,\,\,\rm{in}\,\,\,L^\infty(\mathbb{R}^n)$$ and $$v_n\rightharpoonup v\,,\,\,\,\rm{in}\,\,\,L^2(\mathbb{R}^n)$$ What can I say of ...
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0answers
39 views

Sequential compactness in weak topology

When the Banach space $V^*$ is reflexive, we have the unit ball in $V^*$ is weak$^*$ sequentially compact. For a Banach space $V^*$ that might not be reflexive, we have to assume that $V$ ...
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0answers
28 views

Projections as a dense subset of $\ell_p^*$

Okay, it seems I wrote a confusing formulation of my question: $\textbf{(1)}$ I'm dealing with the following problem: prove that if $ \left(\{a_n\}_{n=1}^{\infty}\right)_k \in \ell_p$, then ...
1
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1answer
34 views

Weak convergence in Hilbert space implies strong convergence of averages for some subsequence

Let $H$ Hilbert Space. Show that if $x_n\rightharpoonup x$ then there exists a subsequence $\{x_{nk}\}$ of $\{x_{n}\}$ such that the sequence $\lim_{m\rightarrow \infty } ...
2
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2answers
38 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
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1answer
42 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...
2
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1answer
44 views

Show that a bounded linear transformation is continuous

I am not sure what this question is asking. A linear operator $T$ between normed spaces X and Y is bounded if and only if it is a continuous linear operator. But weak topology is not metrizable. I ...
1
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1answer
45 views

Isn't this just the definition of weakly convergence?

Hi I come across this question when reviewing my notes. But isn't this just the definition of weak convergence? Am I failing to understand the problem? The definition given on my notes is A ...
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1answer
32 views

Find a sequence in the unit ball of L^1([0,1]) that has no convergent subsequence in the weak topology.

This is a question I come across when reading my notes. Find a sequence in the unit ball of $L^1$([0,1]) that has no convergent subsequence in the weak topology. By a corollary of Banach-Alaoglu, ...
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0answers
19 views

A basic question on convergence in Skorohod metric

Consider a sequence of piecewise constant (positive) function in the space $D$ with Skorohod topology such that each of the functions is zero at the origin and any two discontinuities are at least ...
4
votes
2answers
79 views

$e_n \to 0$ weakly in $l^\infty$

Given the the sequence $(e_n)_n$ in $l^\infty$, I want to show that that $e_n$ converges weakly to $0$ in $l^\infty$, i.e. $$e_n\rightharpoonup 0 \text{ as } n\to \infty.$$ By $e_n\in l^\infty$, ...
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0answers
21 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
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1answer
50 views

Weak Convergence and Weak Topology

In discussing weak topology of a normed space $X$, a lemma is given as follows. If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded. I understand the proof of this ...
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0answers
18 views

Existence of a sequence of random variables, provided weak convergence

I'm trying to prove the following statement: Let $ X_n, X_0 $ be such R.V.'s that $ X_n $ converge to $ X_0 $ in distribution (weakly). Prove that there exist $ Y_n, Y_0 $ on the probabilistic space ...
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1answer
36 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
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1answer
34 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
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0answers
19 views

Weak convergence of probability measure on $\mathscr{C[0,\infty]}$

What is the reason for considering the spaces of probability measures on the space of all continuous functions and then considering weak convergence there ? Is it that we can then use Skorohod's ...
0
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1answer
36 views

A basic question on weak convergence of measures

Why do we need separability of the space to talk about weak convergence of measures ?
2
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1answer
53 views

Weak convergence in $\mathcal{l}_p$ and coordinatewise convergence

Let $x^n=(x^n_1, x^n_2,...)$ be a bounded sequence in $\mathcal{l}_p$ for $1<p<\infty$ and such that $x^n_i$ converges to $x_i$ for all $i\in\mathbb{N}$. I'm trying to prove that ...
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1answer
89 views

Almost everywhere convergence of a bounded sequence in $H_0^1(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement (e.g. $\Omega = \mathbb{R}^N \setminus \overline{B(0;1)}$), and $(u_n)$ be a bounded sequence in $H^1_0(\Omega) = ...
2
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0answers
50 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
3
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0answers
21 views

“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ...
0
votes
1answer
39 views

Weak convergence on L^p

Let Let $X=[0,1]$ with the Lebesgue measure, find a sequence $\{f_n\}$ of measurable functions $f_n:X \rightarrow{ \mathbb{R} } $ such that: $f_n(x)\rightarrow{0}$ almost everywhere $x∈[0,1]$ ...
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0answers
31 views

monotone convergence of $f_n$ plus weak convergence of $\mu_n$ implies convergence?

I would like to know if somebody is aware of some result that looks like the following. Let us consider the space $C_b(X)$ of continuous bounded function over a measurable space $X$. Suppose that: ...
0
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0answers
43 views

Weak topology in Banach spaces

Let $X$ be a Banach space and suppose that $\{x_n\}$ is a sequence in X that is bounded, and suppose that $f(x_n) \longrightarrow f(x)$ $\forall f \in A$ where $A$ is subset dense of $X^*$. Show ...
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0answers
38 views

Weak convergence with the condition of Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
2
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0answers
21 views

Relation between a.s. convergence and weak convergence in in $L^p$ spaces

Is there a relation between almost-sure convergence and weak convergence of $f_n\in L^p(\mathbb{R})$? (i.e. convergence if tested with $L^{p'}$) I know that none implies the other (see masses ...
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1answer
33 views

Question on Outer Expectation unequal to Expectation of minimal measurable majorant

This is a problem given by Aad W. van der Vaart and Jon A. Wellner in their "weak convergence and stochastic process". It aims at constructing an example of "$E^{\ast}T$ is not always $ET^{\ast}$". ...
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1answer
63 views

Application of Slutsky's Theorem

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$. Let ...
1
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1answer
25 views

weak convergence of a pair of random elements

Consider two sequences $(X_n)$ and $(Y_n)$ of random elements in some nice (e.g. Polish) space s.t. $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ ("$\Rightarrow$" denotes weak convergence). Then we know ...
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votes
0answers
45 views

Weak convergence and convergence in distribution

Is convergence in distribution related to weak convergence in Banach theory? Where by weak convergence I mean: for every functional f the sequence $\langle f,x_n\rangle \overset{n}{\rightarrow} ...
1
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1answer
67 views

uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
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0answers
32 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
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vote
1answer
37 views

Weak convergence of random variables implies $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$

Proof that, if $X_n \rightarrow X$ weakly, then $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$. I know, that I should use Fatou's lemma but I don't know what can I do first.
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1answer
39 views

Show a bounded linear operator is weakly sequentially continuous

Let $X$ and $Y$ be normed linear spaces, $T \in B(X,Y)$, and $\{x_{n}\}_{n=1}^{\infty}\subset X$. If $x_n \rightharpoonup x$, I need to prove that $T x_{n} \rightharpoonup T x$ in $Y$, where ...
1
vote
1answer
29 views

Portmanteaus Theorem and Lipschitz functions

I was looking at some basics in the theory of convergence of distributions, e.g. Portmanteaus Thm. One part of it asserts that it suffices to check $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for bounded ...
0
votes
1answer
63 views

If a sequence converges weakly in a closed subspace $M$ of a Banach space, then the (strong) limit point is in $M$.

Let $M$ be a closed subspace of the Banach space $X$ and let $x_{n}\in M$ converge weakly to $x$. Show that $x \in M$. We use the following definition $x_n\rightharpoonup x$ in $X$, $x_n$ ...