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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Does $a_n \in H^1(\mathbb R^n)$ and $b_n \rightharpoonup 0$ in $H^1(\mathbb R^n)$ imply $\langle a_n, b_n \rangle \to 0$?

I have a question mainly in functional analysis. Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows: Does $a_n \in H^1(\mathbb R^n)$, $|a_n| ...
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1answer
24 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...
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0answers
8 views

multivariate convergence in distribution for Skorohod topology

If $X_n$ is a $R^d$-valued sequence of stochastic processes and $X$ is a $R^d$-valued the limiting process, all having Cadlag paths. I know what it means by $X_n \Rightarrow X$, i.e. $X_n$ converges ...
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1answer
29 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
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0answers
23 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
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1answer
25 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
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2answers
16 views

A question involving weak and strong convergence

Let E be a Banach space, $K \subset E$ a compact subset in the strong topology and $(x_n)_{n \geq 1} \subset K$, $x_n \rightharpoonup x$ weakly in $\sigma (E, E^*)$. If there is a subseqence ...
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1answer
34 views

Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
3
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1answer
26 views

Dose weak convergence imply tight?

$X$ is a separable metric space, $\{P_n\}_{n=1}^{\infty}$, $P$ are probability measures on $X$, and $P_n$ converges weakly to $P$, can we conclude that $\{P_n\}_{n=1}^{\infty}$ is tight? I know if ...
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0answers
11 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
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1answer
22 views

Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
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0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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1answer
58 views

A multiple of a characteristic function is the weak limit of a sequence of characteristic functions

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...
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1answer
21 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) ...
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1answer
18 views

Weak convergence equivalent to convergence of distribution functions

I'm trying to understand the following proof: I don't see where the Lipschitz property of the $f$ is needed. Why isn't bounded and continuous enough? Can somebody explain it to me? Thanks.
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1answer
24 views

Convergence in Distribution and Exponential Function

There's a well known fact that if a sequence of real numbers, $\{x_{n}\}$ converges to $x$, then: \begin{equation*} \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x_{n}}{n}\right)^{n} ...
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0answers
29 views

Understanding Prohorov's theorem

I've seen Prohorov's theorem in the following formulation (I'll cite only one direction). $\mathcal{M}_{\leq 1}(E)$ is the set of subprobability measures on $(E, \mathcal{B}(E))$: Let $(E, d)$ be ...
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1answer
36 views

Proving the uniqueness of the weak limit

In "A First Look at Rigorous Probability Theory" by J. S. Rosenthal there is the following exercise: Prove that weak limits, if they exist are unique. That is, if $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
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2answers
38 views

Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
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1answer
23 views

Evaluating limit of a characteristic function (Fourier Transform) in $R^k$

I am trying to evaluate this limit: $$\lim_{n \to \infty} \left[(\text{det}\ \Gamma_n)^{-\frac{1}{2}}\exp \left\{ {-\frac{1}{2}(x-m_n)\cdot (\Gamma_n)^{-1}(x-m_n)} \right\} \right]$$ where ...
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1answer
22 views

Weak convergence on $C(K)$ space implies pointwise convergence

I'm wondering a following problem: If $f_n$ weakly converges to $f$ on $C(K)$ space, then can we conclude that $f_n(x)$ converges to $f(x)$ for every $x$ in $K$? Why?
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1answer
54 views

$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
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1answer
38 views

About the weak compactness of a certain set.

Why is the following set weakly compact in $L^1(d\mu)$? $$\left\{-\frac{|x|^2}{2}+O(l)\right\}$$ where $\mu$ is a probability measure in $\mathbb{R}^n$ with finite second order moment: ...
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0answers
23 views

A question involving weak convergence in $\sigma(L^1, L^\infty)$

Exercise 4.15 from H. Brezis - "Functional analysis..." Let I = (0, 1) and $f_n = n e^{-n x}$ a sequence of functions. Show that $$ f_n \;\; \text{does not converge weakly to} \;\; 0 \;\; \text{in} ...
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1answer
95 views

weak convergence lim inf sequence example

I have been stumped by the following problem , I was able to answer the first part of the problem which involved the inequality by using weak convergence. But when it came to looking for such an ...
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1answer
14 views

FDD convergence of subsequences

First, sorry for this probably very stupid question. Let $(X_n)_{n\geq1}$ be a sequence of random variables (e.g. in $\mathbb{R})$ s.t. $X_n\stackrel{d}{\to} X$ in distribution. Now look at the ...
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0answers
31 views

Weak convergence of a sequence of elements of a compact set implies strong convergence

Let E be a Banach space and $K \subset E$ a compact subset in the strong topology. Let $(x_n)_{n \geq 1} \subset K$ such that $x_n \rightharpoonup x$ in $\sigma (E, E^*)$. K is compact, so $(x_n)_n$ ...
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1answer
30 views

Strong convergence of convex combinations of a weakly convergent sequence

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
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1answer
45 views

How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
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1answer
23 views

Continuity sets are neccesary for weak convergence.

Portmanteau theorem In particular if $\mu_n \to \mu$ weakly then $\mu_n(A)\to \mu(A)$ for each continuity set. I want an example to show that the hypothesis of $\mu(\partial(A))=0$ is neccesary. ...
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1answer
29 views

Weak and vague convergence of normal distribution

Let $\mu_n = \mathcal{N}(0,n)$ be the normal distribution with mean $0$ and variance $n$ on $\mathbb{R}$, $\nu$ the zero-measure (which is defined by $\nu(A) = 0$ for any ...
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0answers
30 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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22 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
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1answer
43 views

sequence in $L^1$ converging pointwise a.e., but not weakly.

Find a $(X,\Sigma,\mu)$, a $\sigma$-finite measure space and a norm-bounded sequence $\{f_n\}$ in $L^1(X)$ that converges almost everywhere to $f$ but does not converge weakly to $f$. Can you help me ...
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1answer
50 views

Weak convergence and lim inf and lim sup of the sequence of norms

Assume $x_n$ is a sequence in a Banach space that converges weakly to $x$. Then we know that $\|x\| \leq \lim \inf \|x_n\|$. 1)But can we say that $\lim \inf \|x_n\| < \infty$ or is this in ...
2
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1answer
47 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
3
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2answers
100 views

Weakly convergent in different spaces

Given $\Omega \subset \mathbb{R}^n$ open connected and $k\geq 0$. Let $f_n, f$ be distributions such that $$f_n \rightharpoonup f \in \mathcal{D}'(\Omega)$$ as $n\to \infty$. Assume that $f_n \in ...
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0answers
61 views

weakly converges distribution sense implies uniform convergence

Given a set of equicontinuous and uniformly bounded functions $\{f_n, f\}$ defined over an open connected $\Omega \in \mathbb{R}^n$. Suppose $(f_n)_n$ weakly converges to some $f$ in distribution ...
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1answer
40 views

Convergence in $L^2$ norm given inner product converges

Suppose $(f_n)$ and $f$ are in $L^2$ and $\int_E f_ng \rightarrow \int_E fg,\,$ for $g \in L^2(E)$. If $\|\,f_n\|_2 \rightarrow \|\,f\|_2$, then show that $f_n \rightarrow f$ in $L^2$. (All functions ...
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86 views

Galerkin method for Poisson's equation

This is problem 3 from chapter 7 of Evans book: Suppose $f\in L^2(U)$ and assume that $u_m=\sum_{k=1}^md_m^kw_k$ solves $\int_UDu_m\cdot Dw_k=\int_Uf\cdot w_kdx$ for k=1,...,m. Show that a subsequence ...
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0answers
142 views

Complicated convergence of nonlinear term

Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in ...
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20 views

Weak Convergence in Metric Space proof

I have been reading Billingsleys book where I came across this theorem and proof. I am having difficulty understanding the theorem/proof. I feel there is a better, more complete way to prove it. Does ...
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1answer
39 views

If the sequence of distribution functions weakly converge, the sequence of corresponding subprobability measures converges weakly, too

Let $\mu,\mu_n$ be subprobability measures on $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$ $F,F_n$ be the distribution functions of $\mu,\mu_n$ with ...
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1answer
164 views

Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and strongly converge to $u$ in $L^p(\Omega).$ We define a function $f:\Omega\times ...
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42 views

The weak convergence between $L^2$ and $H^1$

If the sequence $u_n\in H^1(R^n)$, $n\geq 3$ and $u_n$ weak converges to $u$ in $H^1(R^n)$, if we can get $u_n$ weak converges to $u$ in $L^2(R^n)$? Furthermore, if we can have $\nabla u_n$ weak ...
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1answer
49 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
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0answers
20 views

Improving weak convergence of random variables

Assume you know $X_n\Rightarrow X$ in a (nice, e.g. Polish) space $(E,d)$, that is $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for all $f\in\mathcal{C}(E,\mathbb{R})$. Now the topology on $E$ determines ...
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2answers
49 views

Show that there is a probability such that $P_n$ converges weakly/in distribution as $n \to \infty$.

Suppose that $P_n$ $n \ge 1$ is a sequence of probabilities concentrated on $[a,b]$. Suppose that one may show for each positive integer $r$ that $\int_{[a,b]}x^rP_n(dx) \to m_r \in R$ as $n \to ...
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0answers
28 views

Seeking help understanding steps in a proof, in “Convergence of probability measures by P.Billingsley”.

To not waste anyones time: this question is directed at people who either have the mentioned book in their possesion, or have at some point read it. I am reading through Patrick Billingsley's book: ...
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2answers
42 views

Convergence in distribution (weak convergence) of sum of real-valued random variables

Suppose that $\{(X_n,Y_n)\}^\infty_{n=1}$ is a sequence of pairs of real-valued random variables that converge in distribution to $(X,Y)$. Show that $X_n + Y_n$ converges in distribution to X+Y. ...