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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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1answer
19 views

Convergence in Probability - Random function evaluated in random argument.

Suppose that $g_n: \mathbb{R}^k \to \mathbb{R}^h$ are random linear mappings such that $$ g_n \stackrel{P}{\to} g, \quad \quad \text{as } n\to\infty, $$ where the non-random limit $g$ is injective. ...
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0answers
41 views

Merging with respect to bounded uniformly continuous functions in terms of characteristic functions

I would like to know if there are any results, where merging of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic ...
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1answer
26 views

Weakly convergence

If $g \in L^p(\mathbb{R})$ be a given non-trivial function, show that following sequences converge weakly in $L^p$ but not strongly in $L^p$. (a) $g_k(x)=k^{1/p}g(kx)$. (b) $h_k(x)=g(x+k)$. I need ...
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1answer
26 views

Weak convergence to 0 iff bounded and pointwise convergence to 0

I am working on a problem from functional analysis that has me stumped. Let $B$ be a reflexive Banach space on some subset of $\mathbb{R}^n$ s.t. point evaluations are continuous. Show that if $f_n\...
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1answer
44 views

Convergence of Indicator function in weak*-topology

Let $\omega_n$ and $\omega$ be measurable subsets of $[0,1]$. Also let indicator function $\chi_{\omega_n}$ converge to $\chi_{\omega}$ in weak*-topology on $L^\infty(0,1)$, that is $$\int_0^1f(x)(\...
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1answer
16 views

Can you have a weakly convergent sequence of unbounded linear operators? (Example)

Is it at all feasible to have a weakly convergent sequence of unbounded linear operators? If so, what is a concrete example of a sequence of not necessarily bounded linear operators who converge in ...
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2answers
41 views

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$ My trial I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded....
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1answer
29 views

Is every complex Banach space with Schur's property hereditarily $l^1$?

An infinite-dimensional Banach space $X$ is hereditarily $l^1$ if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l^1$. And $X$ has Schur's property if every weakly ...
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0answers
29 views

What is the largest class of functions $f$ such that the map $\pi \mapsto \int_{\mathbb{R}^n} f \ d\pi$ is continuous?

What is the largest class of functions $f$ such that the map $\pi \mapsto \int_{\mathbb{R}^n} f \ d\pi$ is continuous, where $\pi$ is a probability measure? Further, what if $\pi$ is a probability ...
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0answers
18 views

How to prove that the marginals of the limit distribution are the limits of the marginals of a sequence of measures?

Suppose $(\pi_n)_{n \in \mathbb{N}}$ is a sequence of measures on two Polish spaces $X$ and $Y$, each with marginals $\mu$ and $\nu$. I want to prove that if it exists, the limit of the sequence, ...
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2answers
60 views

Schur’s property in $L^1( \mathbb X )$ where $\Bbb X$ is a mearsure space

We know $\ell^1( \mathbb N )$ enjoys the Schur property, I want to know if $L^1( \mathbb X )$ also enjoys the Schur property. A Banach space is said to have Schur’s property if weak convergence ...
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1answer
23 views

Simple example on uniformly convex spaces

In the lectures we showed the following result: Theorem: Let $(E,\|\cdot\|_E)$ be a uniformly convex space. Consider a sequence $\{x_n \}\rvert_{n\in\mathbb{N}} \subset E$ and $x \in E$ such that it ...
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0answers
18 views

Construction of oscillating sequence in $\mathcal{L}^{\infty}(\Omega, \{z_1, …, z_N\})$

Let $N \in \mathbb{N}$, $\lambda_n \in (0,1)$ , $z_n \in \mathbb{R}^N$ for $n \leq N$ such that \begin{equation} \sum_{n=1}^N \lambda_n = 1,\quad \sum_{n=1}^N \lambda_n z_n = z \in \mathbb{R}^N. \...
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0answers
33 views

About weak convergence and convergence of norms

Today in an exam on functional analysis the following question was posed: Let $H$ be a Hilbert space and $(x_n)_{n\in \Bbb{N}} \subseteq H$ be a sequence that converges weakly to $x\in H$ satisfying $...
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0answers
26 views

Weak convergence of discrete probability measures in separable metric space

Suppose $P$ is a Borel probability measure on the separable metric space $(S,\rho)$. Suppose $\{x_1,x_2,\cdots\}$ is a countable dense subset of $S$. Since $\bigcup_{i=1}^{\infty} B\left(x_i,\dfrac{1}{...
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1answer
23 views

If $S$ is compact then the separating classes are same as convergence determining classes

Suppose $(S,\rho)$ is a metric space with Borel $\sigma-$algebra $\mathcal S$. A class of subsets $\mathcal A\subset\mathcal S$ is called a separating class if whenever two Borel probability ...
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1answer
39 views

How to show the following weak convergence using characteristic functions

Suppose $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: $\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{'}(\mu)^{2} ...
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1answer
40 views

Finding a subsequence converging in distribution

Let $\{X_n\}$ be a sequence of real random variables, such that $$\displaystyle\liminf_{n\to \infty}E({X_n}^2)<\infty$$ Then show there exists an integrable random variable $X$ and a subsequence $\{...
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1answer
31 views

Examples reflexive spaces

Let $(E, \| \cdot \|_E)$ be a normed vector space over a field $\mathbb{K}$ and $(E', \| \cdot \|_{\mathrm{op}})$ its dual. Theorem 1). If $(E, \| \cdot \|_E)$ is reflexive, then each bounded ...
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1answer
19 views

Example of weak limit not in the set

Let $(E, \| \cdot \| _E)$ be a normed vector space over a field $\mathbb{K}$ and $E'$ be its dual space. Theorem: Let $C\subset E$ be a closed (respect to the strong topology $\| \cdot \| _E$) and ...
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1answer
36 views

Bounded sequence in dual with any weak*convergent subsequence

Let $(E,\|\cdot\|_E)$ be a separable normed vector space over a field $\mathbb{K}$ and $E'$ its dual. Theorem: Each bounded sequence in $E'$ has a weak-* convergent subsequence. Question: With $(E,\...
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1answer
38 views

Strong and weak limits in a dense set for a sequence of bounded operators

I am working through Lax's "Functional Analysis", and I'm trying to prove Theorem 6 of section 15.2, dealing with weak and strong convergence in the operator space. We denote by $s-\lim$ the fact that ...
3
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1answer
39 views

Counterexample of weak convergence

Let $E$ be a vector space over a field $K$, $x\in E$ and a sequence $\{x_n \}_{n \in \mathbb{N}} \subset E$. Question: I need to find a counter example of two different linear functionals $$\psi,\...
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1answer
49 views

On a sequence $f_k$ in $L^{2+\frac{1}{k}}$

Suppose that $f_k\in L^{2+\frac{1}{k}}(\Omega)$ with the property that $\|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1$ for all $k\ge 1$. $\Omega$ is a bounded domain in $\mathbb{R}^n$. Can such a sequence ...
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2answers
44 views

Why this sequence converges to $0$ and not to $1$ in probability?

I didn't understand why this sequence bellow converges to $0$ in probability? Shouldn't this sequence converge to $1$ in probability? This is the definition of convergence in probability:
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1answer
30 views

Proof of $O_P(1) o_P(1) = o_P(1)$

Consider $U_n=O_P(1)$ and $X_n=o_P(1)$. Show that $U_nX_n=o_P(1)$. Since $o_P(1)$ is equivalent to convergence in probability, it suffices to show that $P(|U_nX_n|>\varepsilon) \to 0$. However, I ...
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1answer
48 views

Mazur's lemma without Hahn-Banach theorem/axiom of choice?

In the development of gradient-flow theory (in Hilbert-space $H$), we soon stumble on the question whether the function $u \mapsto \varphi[u]:=\frac{1}{2}\|u\|^2+I[u]$ -where $I:H \to \mathbb{C}$ is ...
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1answer
48 views

weak convergence of weak derivative in Sobolev Spaces

Short question cornerning some lectures notes in my current calculus of variation class: Let $\Omega \subset \mathbb{R^n}$ be open and bounded. It is now stated that if $(\phi_j)_{j \in \mathbb{N}} \...
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0answers
31 views

Central Limit Theorem- Lapunow, Linderberg

I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg ...
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1answer
39 views

Does an empirical distribution converge to the underlying distribution?

Let $\mu_n$ be an empirical distribution of $n$-iid points from the underlying distribution $\mu$. In 1D, it is well-known by Kolmogorov's theorem, Glivenko–Cantelli theorem that for any $x$, let $...
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1answer
78 views

Why does weak-$L^2$ convergence not imply pointwise convergence for continuous functions?

This question shows that $L^2$ convergence does not show pointwise convergence, even when the functions involved are continuous. This strongly contradicts my intuition, because I thought that weak-$L^...
2
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1answer
64 views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$ Another way to ...
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0answers
22 views

Convergence of conditional distributions given convergence of conditioning set

Suppose that $X_n \stackrel{p}\rightarrow X$. Is it true that $\mathbb{P}(Y|X_n)$ converges in distribution to $\mathbb{P}(Y|X)$? I.e. $\mathbb{P}(Y\leq y |X_n) \rightarrow \mathbb{P}(Y\leq y |X)$ ...
3
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1answer
28 views

Showing weakly continuous operators are continuous without using weak topology

Let $X$ and $Y$ be Banach spaces, and let $T:X\rightarrow Y$ be a linear map such that $f\circ T$ is continuous for all $f\in Y'$. Show that $T$ is continuous. Now I think this problem is trivial ...
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0answers
39 views

Computing the Scaling Limit of a Nonnegative Markov Chain

Fix $\alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves': From $x^t$, move to $x^{t+1/2} = x^t + y^t$, where $y^t \sim \text{Gamma}(\...
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2answers
61 views

If the sample mean converges in distribution to F, the mean of the odd and even observations converge as well

could somebody please help with this question: Consider a random sample of $X_1, X_2, \ldots ,X_n$, such that $E[X_i] = \mu$ and $n $ is even. Define $P_n = \frac{2}{n}\sum_{i}X_{2i}$ and $I_n = \...
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0answers
32 views

Convergent subsequence in $L^1$

If $\{f_n\}_{n \in \mathbb{N}}$ is a sequence in $L^1(\Omega)$, we know that if the BV($f_n$) is uniformly bounded then there exists a subsequence which converges in $L^1$ norm(Because BV is embedding ...
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0answers
22 views

Question on Bochner spaces: Convergence of the weak time derivative

Let $U$ be a $C^2-$compact manifold and consider two non negative sequences $f_n,g_n$ such that $f_n \overset * \to f$ weakly in $L^{\infty}(0,T;\mathcal M_{*}(U))$ $g_n \overset * \to g$ ...
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1answer
50 views

$Weak^*$ convergence in $L^1$

Suppose we have a sequence $\{f_n\}$ of $L^1$ functions such that $||f_n||_1 \leq K_1$, then viewing $L^1(\mathbb{R}) \subset \mathcal{M}(\mathbb{R})$ where $\mathcal{M}(\mathbb{R})$ is the space of ...
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1answer
39 views

Weak Convergence Lemma - Is Banach needed?

Lemma. Let $X$ be a normed space. If $x_n \rightharpoonup x$ in $X$ and $x_n^* \to x^*$ in $X^*$, then $\lim_{n \to \infty} x_n^*(x_n) = x^*(x)$. If $X$ is even Banach, then $x_n \to x$ in $X$ and $...
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2answers
31 views

Marginal convergence with independence implies jointly convergence

If $X_n$ and $Y_n$ are independent random vectors for every $n$, then $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to}Y$ imply that $(X_n,Y_n) \overset{d}{\to} (X,Y)$ where $X$ and $Y$ are ...
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0answers
24 views

What does it mean for a map taking a measure to an integral to be continuous?

Suppose $\mathcal{M}$ is a set of probability measures. What does it mean to say that a mapping $\mu \rightarrow \int_{\mathbb{R}}f \ d\mu$ is continuous on $\mathcal{M}$? How could one prove that ...
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1answer
12 views

Counterexample of convergence in distribution at points of discontinuity

We define convergence in distribution where $F$ is continuous. To illustrate the importance of continuity in the definition, an example was given in the class which I did not understand. Consider a ...
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2answers
36 views

Unit ball of $X^{**}$ is weakly compact!

Is it true that the closed unit ball in $X^{**}$ is compact with respect to the weak topology on $X^{**}$, where $X$ is a Banach space? If so, how can we prove it?
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1answer
41 views

Does $X_n \stackrel{d}{\to X}$ and $Y_n \to 0$ almost surely imply that $X_n Y_n \to 0$ almost surely?

I'm stuck with the following problem: I have a sequence $X_n$ of random variables, which converge in distribution to some random variable, which is finite almost surely. The other sequence $Y_n$ ...
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1answer
34 views

Verification of convergence of random variables

Let $(X_n)_{n \in \mathbb{N}}$ a series of random variables with: $$P(X_n = 2^n) = \frac{1}{2^n} \hspace{15pt}\text{and}\hspace{15pt} P(X_n = 0) = 1-\frac{1}{2^n}$$ for all $n \in \mathbb{N}$. ...
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1answer
46 views

Do weak convergence and convergence of norms imply convergence in $L^1$?

I know this does hold in $L^2$, since it's a Hilbert space. I suspect that this is not true, but I cannot think of a counterexample. Specifically, I want to know if $f_n \xrightarrow{w} f$ and $\...
3
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1answer
24 views

$f_n \in L^{\infty}(\mathbb R)$ such that $f_n \to f$ and ${f_n}^2 \to g$ weakly$^*$ with $f^2 \neq g$

The exercise metioned in the title was a part of my exams in Functional Analysis, which took place the previous week. Unfortunately I wasn't able to think of a sequence $f_n\in L^{\infty}(\mathbb R)$ ...
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2answers
26 views

Literature on relationship between strong and weak* (weak star) convergence

I am trying to follow a proof in the paper Wasserstein Generative Adversarial Networks by Arjovsky et al. (proof A in supplementary material). They show that the convergenc of the total variation ...
6
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2answers
191 views

Norm and Weak Topologies agree?

so my professor mentioned that when a normed space is finite dimensional the norm and weak topologies agree. To show the topologies agree it should be enough to show that they have the same convergent ...