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Questions tagged [weak-convergence]

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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Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
Zhen Lin's user avatar
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29 votes
1 answer
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weak sequential continuity of linear operators

Suppose I have a weakly sequentially continuous linear operator T between two normed linear spaces X and Y (i.e. $x_n \stackrel {w}{\rightharpoonup} x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel {w}{\...
user1736's user avatar
  • 8,603
7 votes
1 answer
3k views

Pointwise a.e. convergence and weak convergence in Lp

I'm trying to prove the following Theorem: Let $\{f_n\}_{n\in\mathbb N}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \...
KKK's user avatar
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7 votes
3 answers
2k views

If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
saz's user avatar
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4 votes
2 answers
351 views

How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions, $\mathcal C_0(X)$ be the ...
Analyst's user avatar
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17 votes
1 answer
2k views

Is the weak topology sequential on some infinite-dimensional Banach space?

Recall that a topological space is sequential, iff every sequentially closed set is already closed. Is there an infinite-dimensional Banach space on which the weak topology is sequential? I ...
gerw's user avatar
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16 votes
2 answers
18k views

Every bounded sequence has a weakly convergent subsequence in a Hilbert space

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
user avatar
12 votes
1 answer
5k views

Schur's Theorem: In $\ell^1$ weak convergence of $x_n$ is the same as convergence in the norm

I'm having a really hard time with nearly every part of this proof, any help would be appreciated. Schur's Theorem: In $\ell^1$ weak convergence of $x_n$ is the same as convergence in the norm. ...
Dragonite's user avatar
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4 votes
1 answer
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Proving the closed unit ball of a Hilbert space is weakly sequentially compact

I bumped into this statement in Hofer-Zehnder in the middle of proving a Hamiltonian field always has a periodic orbit over a level set of the hamiltonian if that set is a regular compact and strictly ...
MickG's user avatar
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2 votes
2 answers
2k views

Proving $\frac{\sqrt{n}(S_n^2-\sigma^2)}{\sqrt{u_2-\sigma^4}}\rightarrow N(0,1)$ in distribution

Let $X_l, X_2,\ldots$ be i.i.d. with $E(X_1 ) = u$, $\operatorname{Var } (X_1) = \sigma^2$ and $u_2 = E[(X_1 - u)^4]$. Let $S_n^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \bar{X}_n)^2$ where $\bar{X}_n = \...
ghjk's user avatar
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13 votes
3 answers
8k views

A linear operator between Banach spaces is weakly continuous iff norm continuous?

Claim : a linear function $T$ between Banach spaces is weakly continuous iff norm continuous? Okay, So I think I have realised weakly continuous implies norm continuous. As weakly continuous implies ...
user avatar
13 votes
1 answer
14k views

Proving Slutsky's theorem

How do we go about proving the following part of Slutsky's theorem? If $X_n \xrightarrow{d} X,\quad Y_n \xrightarrow{P} c$, then $X_nY_n \xrightarrow{d} Xc$ where $c$ is a degenerate random variable. ...
ChaPi's user avatar
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12 votes
1 answer
1k views

$\ell_\infty$ is a Grothendieck space

The problem I am considering stated formally is this: Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent. We may reduce this to the case where the ...
Keaton's user avatar
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10 votes
1 answer
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Does weak convergence in $L^2$ implies convergence almost everywhere along subsequence?

If I know $\int_{[0,1]} f_{n}(x) g(x) dx \rightarrow \int_{[0,1]} f(x) g(x) dx$ as $n \rightarrow \infty$ for all $g \in L^2([0,1])$ (weak convergence in $L^2$) and $|f_n(x)|_{L^2} <C$ (uniformly ...
mimi's user avatar
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16 votes
2 answers
3k views

Characterization of weak convergence in $\ell_\infty$

Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets? I was only able to come up with a characterization of ...
Martin Sleziak's user avatar
8 votes
4 answers
11k views

Why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$?

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
Theorem's user avatar
  • 7,989
4 votes
1 answer
3k views

Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $ X_n \Rightarrow 0 $ (convergence in distribution) if and only if $ (\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1 $, where $\phi_n(t)$ is ...
benny's user avatar
  • 1,231
1 vote
2 answers
174 views

Let $X$ be locally compact separable and $\mu_n\overset{*}{\rightharpoonup}\mu$. Then $|\mu|(O)\le\liminf_n|\mu_n|(O)$ for all open subsets $O$ of $X$

Let $X$ be a metric space, $\mathcal C_b(X)$ the space of real-valued bounded continuous functions on $X$, $\mathcal C_c(X)$ the space of real-valued continuous functions on $X$ with compact supports,...
Analyst's user avatar
  • 5,677
19 votes
1 answer
16k views

Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$. Context This problem comes from a question in my exam paper; ...
Ricardo Gomes's user avatar
15 votes
3 answers
27k views

Convergence in probability implies convergence in distribution

A sequence of random variables $\{X_n\}$ converges to $X$ in probability if for any $\varepsilon > 0$, $$P(|X_n-X| \geq \varepsilon) \rightarrow 0$$ They converge in distribution if $$F_{X_n} \...
Hawii's user avatar
  • 1,289
15 votes
2 answers
15k views

Proof for convergence in distribution implying convergence in probability for constants

I'm trying to understand this proof (also in the image below) that proves if $X_{n}$ converges to some constant $c$ in distribution, then this implies it converges in probability too. Specifically, my ...
A user's user avatar
  • 966
10 votes
2 answers
734 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
user119148's user avatar
10 votes
1 answer
2k views

Weak convergence of random variables

Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures $(X_n)_*(\mathbb{P})$ ...
Mizar's user avatar
  • 5,911
6 votes
1 answer
3k views

In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.

Assume that $H$ is a $\mathbb K$-Hilbert space, $(x_n)_{n \ge 1}$ a sequence in $H$ and $x ∈ H$. Show that $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$. I'm ...
user372904's user avatar
5 votes
2 answers
182 views

Is it necessary to consider the case $p=1$ separately?

Let $p \in [1, \infty]$. Let $f \in L^p_{\text{loc}} (\mathbb R)$ be $T$-periodic, i.e., $f(x+T) = f(x)$ a.e. $x \in \mathbb R$. Let $$ \bar f := \frac{1}{T} \int_0^T f (t) \, dt. $$ We define a ...
Analyst's user avatar
  • 5,677
4 votes
1 answer
567 views

$\overline{B_1(0)}$ is not weakly* sequentially compact in $(l^\infty)$'

I know the following theorem from the lecture: Let $X$ be a seperable Banachspace. Then $\overline{B(0)}$ is weakly* sequentially compact in $X'$. Since it is specified that $X$ has to be ...
Yasuduck's user avatar
  • 719
2 votes
1 answer
827 views

$C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
passenger's user avatar
  • 3,823
24 votes
1 answer
7k views

A closed subspace of a reflexive Banach space is reflexive

Let $X$ be a reflexive Banach Space. Let $Y$ be a closed subspace of it.I need to show that $Y$ is reflexive as well. So as usual I consider the inclusion map $$J: Y \to Y'', J(y)=j_{y}, j_{y}(y')=y'(...
tattwamasi amrutam's user avatar
17 votes
0 answers
724 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
blindman's user avatar
  • 3,117
17 votes
2 answers
12k views

Intuitive explanation of Lyapunov condition for CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = \...
Mathrobot's user avatar
  • 211
15 votes
1 answer
5k views

Uniform convergence and weak convergence

Assume $F_{n},F$ are distribution functions of r.v.$X_{n}$ and $X$, $F_{n}$ weakly converge to $F$. If $F$ is pointwise continuous in the interval $[a,b]\subset\mathbb{R}$, show that $$\sup_{x\in[a,b]}...
Jacky Zhang's user avatar
12 votes
2 answers
5k 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 $\phi_n$ ...
Shine's user avatar
  • 3,035
10 votes
2 answers
2k 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 $e_n\in l^\infty$, ...
Peter's user avatar
  • 3,393
8 votes
3 answers
4k views

Closed $\iff$ weakly closed subspace

On this link http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst_2004;task=show_msg;msg=1414.0001 is the argument that a linear subspace in a normed space is closed w.r.t. norm iff it is weakly ...
user35953's user avatar
  • 475
8 votes
2 answers
2k views

When weak convergence implies moment convergence?

Given a sequence $(\mu_n)_n$ of probability measures on $\mathbb R$, which converges weakly to a probability measure $\mu$, when do we have $$ \tag{1} \lim_{n}\int x^kd\mu_n(x)=\int x^k d\mu(x) \qquad ...
Student's user avatar
  • 3,526
7 votes
4 answers
4k views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
batman's user avatar
  • 2,065
6 votes
1 answer
1k views

Lévy's metric on $\mathbb{R}^d$

I know that a sequence of measures on $\mathbb{R}$ converges in distribution if and only if the corresponding Lévy's metric converges (Relationship to weak toplogy (Lévy metric)). According to ...
mathex's user avatar
  • 642
5 votes
1 answer
8k views

Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert space,...
Cristyan Pinheiro's user avatar
5 votes
1 answer
1k views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$ Another way to ...
user avatar
4 votes
2 answers
583 views

Closed convex hull of weak convergent sequence: How to show that $x$ is the only element in $\bigcap K_n$?

Le $H$ be a Hilbert space and Suppose $x_n$ converges weakly to $x$ in $H$. Let $K_n$ be the closed convex hull $\bar{co}\{x_k:k\geq n\}$. I would like to show $\bigcap K_n=\{x\}$. What I know so far ...
user136592's user avatar
  • 1,754
4 votes
1 answer
530 views

definition of "weak convergence in $L^1$"

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set $...
s_2's user avatar
  • 485
4 votes
0 answers
694 views

Convergence of generalized inverses

I copy-paste the post from Math Overflow - maybe someone can give me a tip. During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation ...
user2280549's user avatar
4 votes
1 answer
680 views

Calculation problem with Central limit theorem

Let $X_1,X_2,\dots\,$ i.i.d random variables with mean zero and variance $1$. Let $S_n=\sum_{i=1}^n X_i\,,n\in \mathbb N.$ Compute the weak limes $\lim_{n\to\infty} \frac1n \sum_{i=1}^n \frac{S_i}{\...
user avatar
4 votes
1 answer
1k views

Does the sequence $(\sqrt{n} \cdot 1_{[0, 1/n]})_n$ converge weakly in $L^2$?

Let $f_n (x) = \sqrt n 1_{[0,1/n ] } (x) \in L^2 (\mathbb R)$. Does $f_n$ converges weakly to $0$ in $L^2$? I tried to prove it by using Hölder's inequality or the Lebesgue differentiation theorem, ...
user112564's user avatar
  • 3,562
3 votes
1 answer
2k views

Convexity and strong lower semicontinuity imply weak lower semicontinuity

I have seen that if a set $K$ on an Hilbert space $H$ is convex and strongly sequentially-closed, it is weakly closed. The teacher said that if you take a convex and weakly lower semicontinuous ...
tommy1996q's user avatar
  • 3,364
2 votes
2 answers
553 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n \...
anonymous's user avatar
  • 1,019
0 votes
1 answer
83 views

References for Prokhorov's theorem for finite signed measures on a completely regular space

Let $X$ be a topological space and $\mathcal M(X)$ the space of all finite signed Borel measures on $X$. For $\nu \in \mathcal M(X)$, let $(\nu^+, \nu^-)$ be its Jordan decomposition, and $|\nu| := \...
Analyst's user avatar
  • 5,677
23 votes
2 answers
8k views

Weak convergence in probability implies uniform convergence in distribution functions

Exercise 1: Let $\mu_n$, $\mu$ be probability measures on $\left(\mathbb{R}, \mathcal{B}\left(\mathbb{R}\right)\right)$ with distribution functions $F_n$, $F$. Show: If $\left(\mu_n\right)$ converges ...
Keith's user avatar
  • 7,715
17 votes
1 answer
3k views

Confusion with the narrow and weak* convergence of measures

Think of a LCH space $X.$ Consider the spaces $C_{0}(X)$ of continuous functions "vanishing at infinity" and the space $BC(X)$ of bounded continuous functions. Consider as well the space of Radon (...
Qwertuy's user avatar
  • 1,139
10 votes
3 answers
6k views

Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence i.e....
Theorem's user avatar
  • 7,989

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