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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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Convergent subsequences in $L^p(\mathbb R^n)$

This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly ...
person's user avatar
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4 votes
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If $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$

Problem. Prove that if $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$. The problem arises from the following question: Let $(e_i)_{i=1}^\infty $ ...
Robert's user avatar
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-1 votes
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Convergence of scalar product in $L^2$ space. [closed]

Let $ \Omega\subset\mathbb{R}^3$. If $ u^\epsilon\to u ,\text{in}\ (L^2(\Omega))^3 $, as $\epsilon\to 0$, and $ v^\epsilon\to v ,\text{in}\ (L^2(\Omega))^3 $, as $\epsilon\to 0$, then can we ...
Du Xin's user avatar
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1 vote
1 answer
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Weak convergence in $L^2$ equivalence

Problem statement: Denoting by $B_r$ the open ball of $\mathbb{R}^N$ centered at the origin with radius $r$, consider a sequence $f_n \in L^2(B_1)$ which is bounded in the $L^2$ norm. Prove that $f_n$ ...
Mutasim Mim's user avatar
2 votes
1 answer
47 views

Agreement of a pointwise, weak limit and a weak-$\ast$ limit of Bochner-integrable functions

Let $T$ be a positive, real number and $\Omega \subset \mathbb{R}^d$ a bounded, connected, open set. For a given $p \in [2,\infty)$, I have a sequence of functions $(f_n)_{n \in \mathbb{N}} \subset L^\...
squilliam's user avatar
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0 answers
50 views

Any closed subspace of the Schwartz space is dense in its dual?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^n)$ be the continuous dual space. Then, it is well-known that for any $T \in \mathcal{S}'(\mathbb{R}^n)$, we can ...
Keith's user avatar
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3 votes
1 answer
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Sequential weak* compactness of bounded set in $(L^\infty(0,T; H))^*$

Let $H$ be a separable Hilbert space and consider the dual $L_T^* := (L^\infty(0,T; H))^*$ of the Bochner space $L^\infty(0,T; H)$. Let $\{ u_{\varepsilon} : \varepsilon > 0 \}$ denote a uniformly ...
user486506's user avatar
3 votes
1 answer
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Convergence on Borel sets implies weak convergence

Let $X$ be a locally compact space and let $M(X)$ denote all $\mathbb{C}$-valued regular Borel measures on $X$. Let $(\mu_n)_{n\in \mathbb{N}} \subset M(X)$ be a sequence. I want to show that $L(\mu_n)...
fish_monster's user avatar
2 votes
1 answer
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Extracting a subsequence which converges in the $t$-Wasserstein distance?

Posted this to MathOverflow as well. Assume that $\mu_n$ are probability measures on $\mathbb R ^d$ with finite moments of order $t$, and $\mu_n\to\mu$ weakly. Clearly, $\int |x|^t d\mu_n(x)$ is a ...
uniform_on_compacts's user avatar
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Conditions on the weak * convergence to be strong.

I know, thanks to Brezis, that on uniformly convex Banach spaces we have that the strong convergence is equivalent to weak convergence and the convergence of norms. I cannot find anything comparable ...
Lolman's user avatar
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3 votes
1 answer
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Exercise about convex hull and weak convegence

Suposse $X$ is a normed space, $(x_k)_{k \in \mathbb{N}}$ a sequence in $X$, $x \in X$ such that $x_k \underset{weakly}{\rightarrow} x$. Let $co$ denote the convex hull. Show that, there exists $y_k \...
Peter's user avatar
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1 answer
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$X_n \rightarrow_d X$ and UI implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$

Claim: Let $(X_i),X$ be real valued r.v.s. Then $X_n \rightarrow_d X$ and uniformly integrable implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$ How can I prove this claim directly? Here's how I proved ...
ForgeBloyb's user avatar
1 vote
2 answers
31 views

Question about weak and pointwise convergence

I have a question about weak topology. Definition: If $X$ is a LCS, the weak topology on $X$, denoted by "wk" or $\sigma(X,X^*)$, is the topology defined by the family of seminorms $\{p_f : ...
Peter's user avatar
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Interchanging infinite sum and limit in distribution

I'm trying to do a proof for a project and I've run into the following problem. For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
Snildt's user avatar
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0 votes
1 answer
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Weak closure of a subset of the unit sphere of $\ell_1$

It is a well-known and standard fact that for every infinite-dimensional Banach space $E$ the weak closure $\overline{S_E}^w$ of the unit sphere $S_E$ of $E$ is equal to the closed unit ball $B_E$ of $...
Damian Sobota's user avatar
4 votes
0 answers
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Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable

Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
reyna's user avatar
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0 answers
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Best way to think of the space $C^w((0,T); L^p(\mathbb{R})$

Let $f \in C^w((0,T); L^p(\mathbb{R}))$, i.e. $$\{f \mid f:(0,T) \rightarrow L^p(\mathbb{R}) \textrm{ is weakly continuous}\}.$$ Clearly this means for fixed $t$, $f(t,\cdot) \in L^p(\mathbb{R})$. I ...
CBBAM's user avatar
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0 votes
0 answers
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Understanding the definiton of weakly open sest and weak convergence

I am learning about weak and weak* topology. In the book I am reading the following is mentioned Definiton (weak topology) If X is a LCS, the weak topology on X, is the topoloty defined by the family ...
Peter's user avatar
  • 409
1 vote
1 answer
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weak-* convergence does not imply weak convergence [closed]

I wish to show that weak-* convergence does not imply weak convergence with an explicit example. Here are the definitions I adopt: "A sequence $(x_n)_{n=1}^\infty$ converges weakly to $x$ in $X$ ...
Dinoman's user avatar
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2 answers
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How do you prove that compact operators on a Hilbert space are weak to weak continuous?

I'm actually trying to understand the proof for the statement that Let $T \in K(H,H)$ for Hilbert space $H$, and $(x_n) \subset H, x_n > \rightharpoonup 0$ weakly. Then $\|Tx_n\| \to 0$ strongly. ...
plshelp's user avatar
  • 129
0 votes
1 answer
61 views

In this case does convergence of marginal distribution imply joint convergence in distribution

If $(S_n)_{n\geq 1}$, $A$ and $B$ (here we assume $A$ and $B$ are defined on the same probability space) are well-defined random variables, and $f$ and $g$ are two measurable functions. Then if we ...
Randomwandering's user avatar
2 votes
1 answer
25 views

Weakly, but not strongly convergenct sequence in $(C([0,1]),||\cdot||_{\infty})$

I'm looking for a sequence $(f_n)_n$ and some $f$ in $C([0,1])$ such that $f_n\rightharpoonup f$, but $||f-f_n||_{\infty}\nrightarrow 0$. I know that $f_n\rightharpoonup f$ is equivalent to $f_n\to f$ ...
math-jl's user avatar
  • 223
3 votes
1 answer
48 views

weak closedness of a set with bounded functions

Let $A$ be a compact topological space equipped with the Borel $\sigma$-algebra, and $X=B_b(A)$ be the vector space of bounded measurable functions. Let $Y=\mathcal M(A)$ be the vector space of ...
John's user avatar
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$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Suppose that $\Omega \subset \mathbb{R}^d$ is bounded with a Lipschitz boundary and $f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$. Show that, for a subsequence, $\nabla(...
Mr. Proof's user avatar
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0 votes
1 answer
57 views

Show that $(X_n, Y_n) \stackrel d \to (X,Y).$

Let $\{X_n \}_{n \geq 1}$ and $\{Y_n \}_{n \geq 1}$ be independent random variables such that $X_n \stackrel d \to X$ and $Y_n \stackrel d \to Y.$ Suppose that $X$ and $Y$ are also independent. Then ...
Akiro Kurosawa's user avatar
0 votes
1 answer
47 views

Rate of convergence of mollified exponential on $[0, \infty)$

Consider the function $$ f(t)=e^{-at} $$ with $a>0$, $t \in [0, \infty)$ and let $\rho$ a smooth function on $[0, \infty)$ such that $$ \int_0^\infty \rho(t)=1 $$ and let $\rho_\delta(t):= \frac{1}{...
Marco's user avatar
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1 vote
2 answers
52 views

Does convergence in distribution implies imply this inequity?

Consider a sequence of random variables $\{X_n\}_n$. Given that $$X_n - \sqrt{n} c \to N(0,1)$$ in distribution ($c>0$ is a constant number bounded away from zero). Show that for any constant $t>...
Mingzhou Liu's user avatar
3 votes
1 answer
43 views

Sufficient conditions for finitely supported measures being dense

Let $(X,\mathcal{B})$ be a Hausdorff topological space with its Borel $\sigma$-algebra. What are some general conditions we could impose on $X$ so that finitely supported measures (i.e. finite affine ...
Saúl RM's user avatar
  • 3,535
2 votes
1 answer
99 views

Show that there exists $x \in \mathbb R$ such that $\mathbb P (Y = x ) = 1.$

Let $\{x_n \}_{n \geq 1}$ be a sequence of real numbers and $\{X_n \}_{n \geq 1}$ be a sequence of random variables such that $\mathbb P (X_n = x_n) = 1,\ n \geq 1.$ Let $Y$ be a random variable such ...
Anacardium's user avatar
  • 2,460
1 vote
1 answer
42 views

Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets?

Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$). ...
Michael's user avatar
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1 vote
0 answers
37 views

Approximating a weakly convergent sequence "uniformly" by a dense subspace

For a fixed $p \in (1,\infty)$, consider $L^p(\Omega)$ for a bounded domain $\Omega$ in the Euclidean space and a sequence $\{ f_{n} \} \subset L^p(\Omega)$ converging weakly. That is, for each $g \in ...
Keith's user avatar
  • 7,715
0 votes
1 answer
31 views

The convergence in sense of distribution associated with Lebesgue integrable function but not differential.

Let $f \in L^1(\mathbb{R}^n)$ and $T$ is a distribution. Show that $T_{D_j^hf} \rightarrow \partial x_j T_f$ as $h \rightarrow 0$ in the sense of distributions, where $$D_j^hf(x) := \dfrac{f(x + e_jh) ...
Lilili123's user avatar
  • 139
0 votes
1 answer
63 views

Understanding example of weak convergence does not imply strong convergence

Consider the sequence $\left\{e_n\right\}_{n \in \mathbb{N}} \subset \ell^p(\mathbb{F}), 1 \leq p \leq \infty$, of unit vectors. Then $\left\|e_n\right\|_p=1$ for all $n \in \mathbb{N}$, and $\left\{...
juan19.99's user avatar
0 votes
1 answer
51 views

Equivalence of strong operator and weak operator convergence for a sequence of orthoprojectors

I want to prove the equivalence of strong operator convergence and weak operator convergence for a sequence of orthoprojectors {$P_n$}. These orthoprojectors act from one Hilbert space H into ...
Tom Sawyer's user avatar
1 vote
1 answer
67 views

There exists the converse of this corollary from Brezis?

In Brezis's Functional Analysis, there is a corollary Corollary 3.30. Let $E$ be a separable Banach space and let $\left(f_{n}\right)$ be a bounded sequence in $E^*$. Then there exists a subsequence $...
Francesca's user avatar
  • 103
4 votes
1 answer
60 views

Dual of completion and weak$^*$-topology

Let $X$ be a dense subspace of a Banach space $Y$. The restriction map $$r: Y^* \to X^*:\omega \mapsto \omega \vert_X$$ is then an isometric isomorphism that is weak$^*$-continuous. I am wondering if ...
Andromeda's user avatar
  • 790
2 votes
0 answers
26 views

Convergence in Distribution of a Particular Sample Average

Suppose $g_{n}(\cdot)$ defined on $[0,1]$ converges in distribution to a continuous Gaussian process. Let $U_{1},...,U_{n}$ be i.i.d. random variables following $\text{Unif}[0,1]$. Allow $g_{n}$ to ...
Ecthelion's user avatar
  • 135
1 vote
1 answer
43 views

Sequence of functions that converges strongly in $L^3$, weakly in $L^2$ and not strongly in $L^2$

How can I construct a sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f_n \overset{L^3}{\to} 0 \\ f_n \overset{w-L^2}{\to} 0 \\ f_n \overset{L^2}{\not\to} 0 $$ I know ...
Niklas's user avatar
  • 105
1 vote
0 answers
21 views

Total variation convergence in context of stochastic processes

Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution $$X_t~\...
Oskar Vavtar's user avatar
0 votes
1 answer
45 views

Weak convergence of dependent variables

$X_n \xrightarrow[]{d} X$, $Y_n \xrightarrow[]{d} Y$ where $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$, but $X_n \not\!\perp\!\!\!\perp Y_n$. What do we need to analyze $(X_n, Y_n) \...
Sicco Kooiker's user avatar
0 votes
0 answers
37 views

weak convergence in an unit ball.

let $x_n \rightarrow x$ weakly and $x_n \in \overline{B(0,1)}$ for all $n$ and $\|x\|_H = 1$ then $\|x_n-x\|_H \rightarrow 0$. We can write $\|x_n -x \|_H^2 = <x,x>+<x_n,x_n> -2Re<x_n,x&...
voroshilov's user avatar
0 votes
0 answers
46 views

Reference for a good multidimensional portmanteau theorem

I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions: The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$; $...
MikeTeX's user avatar
  • 2,088
2 votes
1 answer
97 views

Is it true that any sequence of random variables that converge in distribution is tight?

Let the sequence of real random variables $\{X_n\} \to X$ in distribution (but not necessarily in probability). Is it true that $\{X_n\}$ form a tight sequence, and if yes, how do we prove it? So we ...
Learning Math's user avatar
1 vote
1 answer
77 views

Definition of a weakly continuous map

I am familiar with the weak topology and a linear functional on a Banach space being weakly continuous, but in PDE I often see a statement along the lines of "$f$ is a weakly continuous solution ...
CBBAM's user avatar
  • 6,063
0 votes
2 answers
62 views

$(X_n, Y_n) \to (X, Y)$ in distribution (Le Gall 10.6)

$$ \newcommand{\N}{\mathbb N} $$ I am paraphrasing this textbook question slightly. Question: Let $(X_n)_{n \in \N}$ and $(Y_n)_{n \in \N}$ be two sequences of real random variables, and let $X$ and $...
caitlin's user avatar
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1 vote
0 answers
34 views

Convergence in distribution in more dimensions

The general definition of convergence in distribution is given by: Let $(X^{(n)})_{n\in\mathbb{N}}$ be a sequence of random variables taking values in a seperable metric space $(S,d)$. We say that $(...
yannik0103's user avatar
0 votes
1 answer
20 views

weak convergence imply expectation $\le$ lower limit of sequence expectation

$(X_{n})$ are uniformly integrable stochastic process ,$X_{n}\xrightarrow{d}X$ Proof $E[\left | X \right |]\le \liminf_{n\to\infty}E[\left | X_{n} \right |]$ I see a way that $f_{m}(x)=\begin{cases}\...
Yu GongLian's user avatar
3 votes
1 answer
91 views

If whenever $\psi_n\rightharpoonup 0$, $A\psi_n \rightharpoonup 0$ then $A$ is bounded.

This if from a previous Princeton exam on functional analysis Let $A$ be a linear operator $A: X\rightarrow Y$ between normed vector spaces. If $\psi_n\rightharpoonup 0$ implies $A\psi_n \...
Kadmos's user avatar
  • 2,147
2 votes
1 answer
58 views

How does the weak topology imply the definition of weak convergence?

Let $X$ be a Banach space. Then the weak topology on $X$ is defined as the coarsest topology such that the linear functionals in $X^*$ are continuous. A sequence $x_n \in X$ weakly converges to $x$ if ...
CBBAM's user avatar
  • 6,063
1 vote
0 answers
39 views

Prove weak convergence of probability measures

Let $Q \in P(P(\Sigma))$ be given, whereby $\Sigma$ is a Polish space (complete and separable metric space) and $P(\Sigma)$ denotes the space of probability measures on $\Sigma$ and $P(P(\Sigma))$ the ...
user996159's user avatar

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