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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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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
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662 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
user66081's user avatar
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Is there a version of Slutsky theorem for stochastic process?

To be more specific, if a stochastic process $X_n(t)$ converges weakly to a tight Gaussian process $G(t)$, and another stochastic process $Y_n(t)$ converges uniformly to a deterministic function $H(t)$...
nahh's user avatar
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A lemma in the application of Concentration compactness principle in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
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Is the topology of weak convergence of probability measures first-countable?

Let $S$ be a separable metric space, and $\mu,\mu_1,\mu_2,\ldots$ be Borel probability measures on $S$. We know that $\mu_n \to \mu$ weakly if and only if $\pi(\mu_n,\mu) \to 0$ where $\pi$ is the ...
Wooyoung Chin's user avatar
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Relation between weak convergence of probability measures and weak-* convergence

I am trying to nail down the relation between probability and functional analysis. In particular, how the notion of weak convergence used in probability theory is related to the weak-* convergence of ...
ttb's user avatar
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$\alpha$-mixing properties and convergence in distribution

I have a stochastic process $\{W_t\}_{t\geq 1}$, of uncorrelated but not indipendent random variables, with $\mathbb{E}(W_t) = 0$ and $Var(W_t)=\frac{t-1}{2}$ $\forall, t\geq 1$ (The $\{W_t\}_{t\geq 1}...
MathRevenge's user avatar
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Application of Banach-Alaoglu theorem to extract convergent subsequence of currents

While reading about currents I came across the following lemma in Lectures on Geometric Measure Theory by Leon Simon on page 135: Lemma. If $\left\{T_j\right\}_{j\in\mathbb{N}}$ is a sequence of ...
Lorago's user avatar
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186 views

Upgrading weak-convergence in $\mathbb{R}$ to strong-convergence in compact sets

Let $s,s'\in\mathbb{R}$ satisfying $s'<s$. Consider $u=u(t,x)\in C(\mathbb{R},H^s(\mathbb{R}))$ and suppose that, as $t$ goes to infinity, we have the following weak convergence: $$ u(t,\cdot)\...
W2S's user avatar
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Exercise 5.22 in Brezis, "Functional Analysis Sobolev Spaces and Partial Differential Equations".

Let $H$ be a Hilbert space, $C\subseteq H$ a nonempty closed convex set and $T:C\to C$ a nonlinear contraction, that is $$ (*)\qquad|Tu - Tv| \leq |u-v|. $$ Let $(u_n)$ be a sequence in $C$ such that $...
Albert's user avatar
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1 answer
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Weak convergence on uniformly convex Banach spaces

I want to prove the following: Let X be a uniformly convex Banach space, $\{x_n\}_{n\in\mathbb{N}} \subset X, x \in X, \|x_n\| \to \|x\|$, $x_n$ converges weakly to $x$. Then $x_n$ also converges ...
Spinnennetz's user avatar
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Joint convergence in distribution of independent random variables

Question: My approach: Fix any $x$ and $y$ in the codomains of $X$ and $Y$ respectively. $\mathbb{P}(X_n \leq x, Y_n \leq y) = \mathbb{P}(X_n \leq x)\mathbb{P}(Y_n \leq y) \rightarrow \mathbb{P}(...
Canine360's user avatar
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1 answer
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Prove that a weak continuous curve in a separable Banach space is a Borel function.

I need some help to approach the following exercise. Let $B$ be a separable Banach space and let $\gamma : [0,1] \rightarrow B$ be a continuous curve with respect to the weak topology of $B$. Prove ...
mrprottolo's user avatar
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Uniform convergence of characteristic functions implies uniform convergence of distribution

Let $F(x)$ and $(F_{n})_{n\geq 1}$ be some distribution functions and let $\varphi(t)$ and $(\varphi_{n})_{n\geq 1}$ be their respective characteristic functions. I am trying to show that if: $\sup_t ...
janesue's user avatar
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Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to L^p(\...
bartgol's user avatar
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359 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ d\mu_{\frac{1}{\sqrt{n}}M_n}\stackrel{P}\rightarrow\int_{\mathbb{...
Integral'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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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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4 votes
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89 views

Central limit theorem on small scales

Say $X_1,\ldots,X_n$ are iid, bounded, symmetric random variables with mean $0$, variance 1 and a smooth density. At what scale does $\frac{X_1+\cdots+X_n}{\sqrt{n}}$ "look like" the ...
euler_pi_i's user avatar
4 votes
1 answer
217 views

Pointwise Convergence of Convolutions

While reading chapter 4 of "Deep Learning Architectures, A Mathematical Approach" by Ovidiu Calin, I stumbled upon the following statement: Using the fact that the Gaussian tends to Dirac ...
I H's user avatar
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4 votes
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65 views

Brezis' exercise 5.19

I'm trying to solve below exercise in Brezis' Functional Analysis Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $u_n, u \in H$ such that $u_n \to ...
Akira's user avatar
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4 votes
0 answers
117 views

Convergence of Hilbert transform of a converging sequence

Fix $n$, and consider random variables $x_1, \dots, x_n$ whose joint p.d.f. is $p_n$. Assume that the empirical distribution of $x_1, \dots,x_n$ converges weakly almost surely to the probability ...
Rostam22's user avatar
  • 504
4 votes
1 answer
136 views

Convergence of non iid observations on the empirical distribution

Let $f$ be a function on domain $X$ with binary output $f: X\to \{0,1\}$. We define an arbiatry distribution $\mathcal{Q}$ over $X$ and the empircal distribution of $n$ samples from $\mathcal{Q}$ -- $\...
user2757771's user avatar
4 votes
0 answers
326 views

weak convergence of joint distribution and conditional distribution

It is known that the weak convergence of joint distribution does not imply the weak convergence of conditional distribution (for example, see this post). What happens if we assume that the density ...
causalMeasurable's user avatar
4 votes
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140 views

A proof without using net in Brezis's Ex 3.14

I'm doing Ex 3.14 in Brezis's book of Functional Analysis. Let $E$ be a reflexive Banach space and $I$ a set of indices. Consider a collection $(f_{i})_{i \in I}$ in the dual space $E'$ and a ...
Analyst's user avatar
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When does there exist a subsequence which converges to the Cesaro mean

I have a sequence of continuous (even analytic) functions on $[0,1]$. I know that the Cesaro mean of the sequence of functions converges uniformly on $[0,1]$: $$\frac{1}{N}\sum_{n=1}^Nf_n\rightarrow f$...
GSofer's user avatar
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4 votes
0 answers
107 views

Joint asymptotic convergence/normality of partitioned sums

Define $S = \{1,\dots, |S|\}$, and for each $s \in S$ let $\{X_{i,s}\}_{i=1}^\infty \overset{iid}{\sim} P_s$ and assume that $E[X_{i,s}] = 0$ for each $s$. Now, for each $n$, define $n_s(n) \leq n$ ...
jackson5's user avatar
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251 views

Why is this $L^1$-sequence relatively weakly sequentially compact?

Let $(E,\mathcal E,m)$ be a probability space, $\theta$ be a measurable map on $(E,\mathcal E)$ with $m\circ\theta^{-1}=m$, $s_n$ be a real-valued nonpositive integrable random variable on $(E,\...
0xbadf00d's user avatar
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4 votes
0 answers
796 views

About closed sets and sequences in the weak topology

Let $E$ be a normed vector space. In an exercise for a homework in my Functional Analysis class, my professor defined weak convergence for sequences as $(x_n)$ is weakly convergent to $x \in E$ iff ...
user480840's user avatar
4 votes
0 answers
547 views

Prokhorov Metric and Weak Topology of Measure

Let $X$ be separable metric space and $\mathscr{M}\left(X\right)$ be the space of all probability measures on $X$ and $d_{P}$ are Prohorov metric on $\mathscr{M}\left(X\right)$. We denote $\mu_{n}\...
Leoalan.Huang's user avatar
4 votes
0 answers
40 views

Showing that $C=\{h \in L^1(\Omega) : u_1(z) \leq h(z) \leq u_2(z)\}$ is weakly compact.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $u_1, u_2 \in L^1(\Omega)$ with $u_1(z) \leq u_2(z)$ almost everywhere in $\Omega$. We let $C$ be the set $C=\{h \in L^1(\Omega) : ...
Rebellos's user avatar
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4 votes
0 answers
103 views

In Wasserstein, what is the relationship between $W_2 (\widehat{\mathbb{P}}_{N},\mathbb{P})$ and $W_2 (\widehat{\mathbb{P}}_{N}^{x},\mathbb{P}^{x})$?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem: Definition: The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ ...
Diego Fonseca's user avatar
4 votes
0 answers
278 views

Convergence of densities vs. other modes of convergence for random variables

While I was summarizing the relations between different modes of convergence for random variables, I got stuck at the convergence of the densities... My question: $X_n \to X$ a.s. (or in $L_1$/ in ...
julbern's user avatar
  • 402
4 votes
0 answers
255 views

Convergence in Law implies a.s. convergence (Donsker-like statement)

I am currently working on a Donsker-like convergence result and I am not quite sure, whether my conclusions are correct (I am dropping the technicalities here): Let $\hat{N}_n(t)$ be a an estimator ...
user190080's user avatar
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4 votes
0 answers
165 views

The Cramér–Wold theorem for complex random vectors

The Cramér–Wold theorem states that $k$-dimensional real random vectors $X_n$ converge in distribution to a $k$-dimensonal real random vector $X$ if and only if $a^TX_n\xrightarrow{d}a^T X$ for all ...
Cm7F7Bb's user avatar
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4 votes
0 answers
323 views

In Calculus of Variation: Problem applying variational principle theorem

Let $f:\mathbb R^m \rightarrow [0,+\infty)\;$ be a smooth function that vanishes on a finite set $A\;$ where $\vert A \vert\; \ge 2$ and the maps $v:(l^{-},l^{+}) \rightarrow \mathbb R^m\;$ ...
kaithkolesidou's user avatar
4 votes
0 answers
112 views

Properties of the space of random variables for different types of convergence

Suppose we have a probability space $(\Omega,\mathcal{A},\mathbb{P})$, a sequence of random variable $(X_n)_n$ and some extra random variable $X$ on $(\Omega,\mathcal{A},\mathbb{P})$. There are a lot ...
HolyMonk's user avatar
  • 1,135
4 votes
0 answers
202 views

Weak convergence counter-example

Given probability measures $\mu,\mu_n: \mathcal{B}(X)\to R$. Construct an example such that $$\langle f,\mu_n\rangle \rightarrow \langle f,\mu\rangle$$ for all convex function $f:X\to R$ but $\mu_n$ ...
Sung-En Chiu's user avatar
4 votes
0 answers
1k 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, ...
Yai0Phah's user avatar
  • 9,803
4 votes
0 answers
102 views

weak convergence of probability measures on a topological but non-metrizable space

Let $X$ be a topological space and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $\Delta$ be the space of all (countably-additive) probability measures on $(X,\mathcal{B})$. Can I define on $\...
Jason Troi's user avatar
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
0 answers
270 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{R}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
Sudheer's user avatar
  • 353
4 votes
0 answers
510 views

A sequence of positive finite measures has a positive measure in the weak limit.

I think the statement in the header is true, but I haven't been able to find a proof for it. Consider the measurable space $(\mathbb{R},\mathcal{B})$, i.e. the real number line with the Borel $\sigma$-...
northwiz's user avatar
  • 325
4 votes
0 answers
502 views

Asymptotic uniform integrability and moments of Student's $t$

I am working on an exercise where I am trying to show that the moments of a $t$ distribution converge in probability to the moments of a standard normal. I'm in need of help with what I think is the ...
hejseb's user avatar
  • 4,745
4 votes
0 answers
1k views

On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...
impartialmale's user avatar
4 votes
0 answers
144 views

A basic problem on weak convergence

Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that ...
aaaaaa's user avatar
  • 2,676
4 votes
0 answers
235 views

Convergence in distribution of bernoulli rv over square root of uniform rv

This is a question from an old comprehensive exam: Let $U$ be a $\operatorname{Uniform}[0,1]$ random variable and let $X$ be a $\operatorname{Bernoulli}(1/2)$ random variable independent of $U$. ...
statgrad's user avatar
  • 227
4 votes
0 answers
2k views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
David's user avatar
  • 527
3 votes
1 answer
42 views

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
  • 409
3 votes
0 answers
65 views

Normal approximation for sum of dependent indicators from triangular array

Suppose there is a triangular array of random variables $X_{k,n}$ with the following properties: $$ \mathbb{P}(X_{k,n} = 1) = p_n, \quad \mathbb{P}(X_{k,n} = 0) = 1 - p_n, \quad \lim_{n\to\infty}p_n =...
Yalikesi's user avatar
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