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
36 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 ...
0
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0answers
24 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
0
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1answer
37 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 ...
2
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1answer
50 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 ...
1
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1answer
46 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 ...
2
votes
1answer
48 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 ...
1
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0answers
23 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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0answers
26 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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0answers
32 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 ...
1
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1answer
42 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 $f$ ...
0
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1answer
25 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_*: ...
1
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0answers
13 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 ...
0
votes
1answer
43 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. ...
1
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0answers
45 views

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 ...
3
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1answer
61 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 ...
1
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2answers
41 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 ...
3
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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 ...
1
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1answer
29 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 $W^{1,p}(\mathbb{R}^n)$ ...
2
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0answers
45 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, ...
1
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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 $H_0^1(\Omega)$ ...
2
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1answer
138 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 ...
1
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2answers
27 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 ...
2
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1answer
34 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 ...
0
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1answer
35 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 $$\sup_n \int f(z) \mu_n(dz) <\infty$$
0
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1answer
32 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: ...
5
votes
1answer
56 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 ...
2
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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 ...
1
vote
1answer
34 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?
2
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0answers
19 views

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
76 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 $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ ...
0
votes
1answer
28 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 ...
2
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1answer
50 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 ...
2
votes
1answer
86 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
votes
1answer
63 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 ...
2
votes
2answers
69 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 ...
4
votes
2answers
57 views

Pointwise convergence of series $\sum_{n=1}^{\infty}\frac{1}{n}e^{-nx}$

For my math course I have to show that the series $$\sum_{n=1}^{\infty}\frac{1}{n}e^{-nx}$$ converges pointwise on $I=(0,\infty)$ and find the limit. I think I should prove that $f_n$ is a Cauchy ...
3
votes
1answer
32 views

Functional analysis, weak* convergence

in lecture notes in the Internet i found the following example for weak and weak* convergence. The unit vectors $\{e_n\}_{n\in \mathbb{N}} $ converges in $l_1$ weak* towards $0 $ but not weak to ...
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0answers
48 views

Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S ...
1
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0answers
45 views

Prove or disprove weak convergence for random variable

Suppose $X_1,X_2,\dots$ be independent random variables such that $P(X_k=ke^k)=e^{-2k}, \ P(X_k=-ke^k)=e^{-2k}$, $P(X_k=k)=\frac{1}{2}-e^{-2k},\ P(X_k=-k)=\frac{1}{2}-e^{-2k}$. ...
2
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0answers
27 views

Weak Convergence Inequality

Let $X$ be a Banach space and $X^*$ it's dual space. a) If $\left\lbrace x_n\right\rbrace$ converges weakly to $x$ in $X$, then $sup_n \|x_n\| < \infty$ and $\liminf_n \|x_n\| \geq \|x\|$ ...
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0answers
33 views

Can $\sup_n \int f d\mu_n <\infty$ for a discontinuous $f$ where $\mu_n \implies \mu$?

Let $\mu_n$ be a sequence of measures converging to $\mu$ in $P(S)$ where $S$ is a compact metric space. I am interested in knowing conditions under which a discontinuous unbounded positive-valued ...
6
votes
2answers
80 views

Weak convergence problem

Let $\Omega$ ∈ $\mathbb{R}^d$ be a bounded set and $d\ge$ 1. Consider function sequence $f_n ∈ L_3(\Omega)$ such that, $$f_n \to f\mbox{ weakly in } L_2(\Omega)\mbox{ and } \|f_n\|_{L_3(\Omega)} ...
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0answers
19 views

Application of continuous mapping theorem

Consider a sequence of random vectors $\Delta_n$ taking values in $\mathbb{R}^l$ and converging in distribution to a random vector $\Delta$ distributed as a $N(0_l,E)$, i.e. $\Delta_n\rightarrow_d ...
1
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1answer
30 views

Closed subspaces and weak topology

Let $B$ be a Banach space, and $E$ a closed subspace of $B$. I have a sequence $x_n \in E$ and some $x \in E$. Is it true that $x_n \to x$ in the weak topology of $E$ if and only if $x_n \to x$ in ...
0
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0answers
15 views

Show that the point of minimum of a random function is bounded in probability

Consider a sequence of random variables $X_n:\Omega \rightarrow \mathbb{R}^k$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Consider a parameter $\theta \in \Theta \subseteq ...
0
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0answers
21 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
5
votes
1answer
95 views

Weak convergence and convergence of moments

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ such that $X:\Omega\rightarrow \mathbb{R}$. Suppose that $X\sim N(\mu, \sigma^2)$. Consider a random ...
1
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1answer
53 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...
2
votes
0answers
28 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
1
vote
1answer
43 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...