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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Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. ...
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Weak-topology separates points on $X$.

Let $X$ be a normed vector space and $X^{\ast}$ be its dual consisting of all continuous functionals on $X$. I must show that weak topology separates points on X, i.e. (?) that for any $x,y \in X$ ...
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17 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures ...
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1answer
23 views

Equivalence of norm and weak topologies in finite dimensional space.

I haven't solved a lot of problems on functional analysis, so I don't have intuition whether my idea of proof is correct or not. The weakness of weak-topology is obvious from the definition. To show ...
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3answers
32 views

Every bounded sequence of dual space contains a subsequence which is weak* convergent

I were doing this problem in Functional Analysis of Erwin Kreyszig(part 4.9, problem 10, page 269), but got stuck in the last point to come to the conclusion. Can anyone give me some hint to move on? ...
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1answer
23 views

Prove that a set which satisfies every nonempty subset contains a weak Cauchy sequence must be bounded

I got stuck on this problem about weak convergence in normed space. This problem is exercise 9, page 263 in Functional Analysis of Erwin Kreyszig. So the question is basically on the title, let me ...
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2answers
48 views

Student test statistic and self normalizing sum.

I study the asymptotic distribution of self normalizing sums which are defined as $S_n/V_n$ where $S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$ for some i.i.d RV's $X_i$. Motivation ...
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1answer
77 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm ...
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1answer
41 views

Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
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1answer
26 views

Q: Exercise 3.10 in Brezis' Functional Analysis

I'm trying to solve exercise 3.10 (page 81) from H. Brezis' "Functional Analysis, Sobolev Spaces and PDE". Let be $E$ and $F$ Banach space such that $T\in \mathcal{L}(E,F)$ and ...
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1answer
17 views

Distribution of the radius of a circle uniformly on a square

Suppose U1, U2, . . . , V1, V2, . . . are independent Uniform(1, 1) random variables, so that (U1, V1), (U2, V2), . . . are independent points distributed uniformly over the square $[-1,1]^2$. Let ...
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1answer
42 views

Are probability measures weak-* closed?

Non-duplicates This is in a different setting, and this only deals with compact spaces which is the easy case. Now for the question. Let $X$ be a locally compact Hausdorff space. $\mathcal{C}_0(X)$ ...
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1answer
22 views

Lower-semicontinuity of quadratic functionals and weak convergence weirdly associated

Each random variable (rv) considered here is absolutely continuous. Let $f_{T}$ be a bounded random variable and $T>0$. Suppose we have: $$f_{T}\rightharpoonup f\tag{1}$$ weakly in probability, ...
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0answers
16 views

Finding a random variable

Let $X_{1}, X_{2}, \dots$ be i.i.d Uniform[0,1] random variables. Find a random variable $X$ such that: $$(X_{1}\times\cdots\times X_{n})^{\frac{1}{\sqrt{n}}}e^{\sqrt{n}} \overset{d}{\to} X$$ The ...
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2answers
40 views

Weak and strong convergence in $L^p$

Another practice qual question: Let $X = [-\pi,\pi]$ and consider the Lebesgue measure. Let $p$ be a real number with $1 \leq p < \infty$. Define for each integer $k \geq 1$ that $f_k(x) = ...
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1answer
15 views

A weaker characterization of convergence in distribution

$X_n$ and $X$ are real-valued random variables. If $X_n \to X$ in distribution, then we know that $$P\{X_n \leq a\} \to P\{X \leq a\}$$ for every $a$ at which $x\mapsto P\{X \leq x\}$ is continuous. ...
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1answer
42 views

Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x $ in the ...
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1answer
40 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of ...
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1answer
52 views

Does weak convergence ($X_n\Rightarrow X$) imply weak convergence of the difference to zero ($X_n-X\Rightarrow 0$)?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am wondering if this implies that $X_n-X\Rightarrow 0$ as $n\rightarrow ...
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1answer
48 views

Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty $ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
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1answer
54 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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0answers
28 views

On the completeness of Weak Operator Topology

Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...
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30 views

Weak convergence in probability and functional analysis

Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have ...
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38 views

Continuity in dual space with weak$^*$- topology

Let $X$,$Y$ be locally convex topological vector spaces. Assume now I have an operator $T:Y'\rightarrow X'$ where $Y'$ and $X'$ are equipped with the weak$^*$-topology. Does this imply that $T$ is ...
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1answer
43 views

Does this joint weak convergence result hold?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am trying to determine the most general conditions on a function $f$ ...
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1answer
27 views

(Uniform) continuity of push-forward operator

I am wondering about extension of the the answer given here. Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: ...
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0answers
14 views

Variance converging to zero implies weak convergence to delta measure?

I have the following question: Suppose that I have a sequence of random variables $X_n$ such that all moments exists and are finite. I have that $E[X_n]\to a$, where $a$ is a finite number and also ...
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1answer
46 views

An example of a sequence of functions that is not pointwise convergent

I've been searching for an example of a sequence $f_n(x)$ of functions that is not pointwise convergent, i.e.: $$\lim_{n\rightarrow \infty}\left | f_n(x) - f(x) \right | = 0$$ but I cannot find one. ...
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Notion of conditional weak convergence

I am looking for references or lecture notes which define the notion of conditional weak convergence of a sequence of random variables. In the case of (usual) weak convergence, we say that a sequence ...
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1answer
71 views

Are the weak* and the sequential weak* closures the same?

I have a question that might be easy for the experts. Let $E$ be a Banach space (non-separable) and let $E'$ be its dual space. Suppose that $X\subset E'$ and assume that $X$ is separable with ...
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2answers
55 views

Characterization of weak convergence in Lp

Weak convergence in $X=L^p(0,1)$ for $1<p<\infty$ can be characterized as following: $f_n\rightharpoonup f$ if and only if $f_n$ is bounded in $X$ and $\int_{(0,t)}\;f_n\;\rightarrow ...
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1answer
65 views

Weak convergence in $W^{1,p}(\mathbb{R})$

Let $\varphi\in\mathcal{C}^\infty_0(\mathbb{R})$ a map with compact support. For all $n\in\mathbb{N}$, I define $u_n$ by $u_n(x)=\varphi (x+n)$. I would like to prove that $u_n$ converges weakly ...
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1answer
35 views

Weak convergence preserver pointwise inequality

The proof of boundedness of Hardy-Littlewood maximal function in Sobolev spaces in Kinnunen's paper has the following argument: "... Hence $(v_k)$ is a bounded sequence in $W^{1,p}(\mathbb{R}^n)$ ...
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0answers
63 views

Weak-* bounded, closed convex set is compact?

Suppose $E$ is a Banach space, and $K\subseteq E^*$ is convex, and is closed and bounded with respect to weak-* topology. Is it true that $K$ is compact? If $E$ is reflexive, then this is the case, ...
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1answer
45 views

If $u_n \rightharpoonup u$ in $H_0^1(\Omega)$, what can we say about $\{\nabla u_n\}$?

I have a bounded sequence $\{u_n\}$ in $H_0^1(\Omega)$. Then, a general result about Hilbert spaces and weak convergence, implies that there exists a subsequence $\{u_{\sigma(n)}\}$ in $H_0^1(\Omega)$ ...
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1answer
139 views

How to prove convergence of process and stopping time

We consider the random function $X^n=(X^n_t)_{t\geq 0}$ with values in the Skorokhod space $\mathcal{D}$ of càdlàg paths, and suppose that it weakly converges (i.e in distribution) to ...
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2answers
28 views

Problem on moment convergence for sufficiently tight measures.

Let $\mu_n$ and $\mu$ Borel probability measures on $\mathbb{R}_+$ such that $\mu_n \Rightarrow \mu$ (converges weakly). Show that if $$ \sup\limits_{n \geq 1} \int x^a d \mu_n(x) <\infty $$ for ...
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1answer
35 views

Characteristic functions of infinite dimensional random elements

I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that ...
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1answer
36 views

Does there exist an unbounded function $f$ such that for all $\mu_n$ converging to some probability measure $\sup_n \int f(z) \mu_n(dz) <\infty$

Does there exist an unbounded positive function $f$ such that for all sequence of probability measures $\mu_n$ converging to some probability measure $$\sup_n \int f(z) \mu_n(dz) <\infty$$
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1answer
33 views

Goldstine theorem

Given the embedding $j:X\to X''$ defined by, $$j=(x\mapsto(\phi\mapsto\phi(x)))\,,$$ according to my interpretation of the wikipedia page, Goldstine theorem says the following: ...
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2answers
63 views

Weak convergence in different $L^p$ spaces

Consider $p \ge \alpha \ge 1.$ If a sequence converges weakly in $L^p,$ say $u_n \rightharpoonup u$, is it true that: $$u_n^{\alpha} \rightharpoonup u^{\alpha} \text{ in $L^{p/ \alpha}$}$$ This ...
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1answer
56 views

Weak * Convergence and Stone-Weierstrass Theorem

I'm trying to solve the first part of the following exercise: Sorry for being lazy, but I am new to using stackexchange and still learning to type LaTeX. I am trying to prove the first part using ...
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1answer
36 views

Weak closure and dual space

Let $X$ be a normed space and let $W\subset X^*$ be a subspace which separates the points in $X$. Let $\psi \in X^*$ such that $\ker \psi $ is $W$-weakly closed. Show that $\psi \in W$. Any ideas?
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Regarding necessity of reflexivity in this problem.

I would like to solve the following problem: Let $L: B \rightarrow R$ be a bounded linear operator between Banach spaces, and $R$ be reflexive. Then $L$ maps weakly convergent sequences to weakly ...
2
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2answers
80 views

Composition of a weakly convergent sequence with a nonlinear function

Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain. Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ ...
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1answer
29 views

Exact definition of weak convergence of empirical processes

Consider the i.i.d. random variables $X_1,...,X_n$ with $X_i:\Omega \rightarrow \mathcal{X}$ $\forall i$ and probability distribution $P$. Consider a class $\mathcal{F}$ of functions ...
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1answer
58 views

On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?

Suppose that $E$ is a Banach space and let $E^*$ denote its dual space with canonical norm $\lVert\bullet\rVert_{E^*}$. Suppose that $\lvert\bullet\rvert_{E^*}$ is an equivalent norm on $E^*$. The ...
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1answer
87 views

Proving that stopped Brownian motions has a limit.

Assume that $(\tau_n)$,$(T_n)$ are finite strictly increasing stopping times and set $\tau=\sup_{n}\tau_n$,$T=\sup_{n}T_n$. Now consider a process $(W_t)_{t<\tau}$, and a Brownian motion $B$, which ...
2
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1answer
65 views

Minimizing a funtional in the Sobolev space $H_0^1$

I am trying to show that, given $f \in H^{-1}(U)$, there exists a unique $u \in H_0^1(U)$ such that: $$\int_U \nabla u\cdot\nabla v \, \mathrm{d}x= \langle f,v \rangle_{H^{-1}} \, , \quad \forall \, v ...
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2answers
71 views

Is the product $f_ng_n$ of weak-star convergence sequences $(f_n)$ and $(g_n)$ in $L^\infty$ also weak-star convergent?

Suppose $X$ is a finite measure space, and $f_n$ is uniformly bounded and converges to $f$ in the weak-star topology of $L^\infty(X)$. This means $\int f_n\phi \to \int f\phi$ for all $\phi\in ...