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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3
votes
1answer
38 views

Weak convergence - $f_n$ “goes up the spout”

Fix $1 < p < \infty$. Given $f \in L^p(\mathbb{R})$ define $f_n(x) = n^{1/p}f(nx)$ for $n = 1, 2, \dots$. Prove that $f_n$ converges weakly to $0$ in $L^p$. I'm really confised about this ...
1
vote
0answers
17 views

Which properties are stable under convergence in distribution?

I'm currently looking at convergence in distribution (i.e. weak convergence) of random variables. A question is bothering me since quite a while and I hope I could express it properly: Given weak ...
0
votes
0answers
16 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
0
votes
0answers
14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
0
votes
0answers
22 views

Check Functional Analysis Proof

I seem to have proved something with elementary techniques even though the paper I found it in suggests it requires heavy tools. There could be a mistake but I can't find it if there is one. Theorem: ...
-1
votes
0answers
9 views

bounded weak convergence sequence implies pointwise convegence in a.e. [on hold]

My questios is followings $u_n$ converge weakly $u$ in $H^1(R)$ then it is bounded in $L^{\infty}$. Then $u_n$ converges pointwise a.e. to $u$ is this true or not?
2
votes
1answer
19 views

Slutsky for joint convergence

I am interested whether Slutsky's Theorem also holds in the case of joint convergence. Let $(X_n,Y_n)$ be random variables with $(X_n,Y_n) \rightarrow (X,Y)$ in distribution for $n \to \infty$. ...
1
vote
0answers
20 views

A question about weak convergence in Lp space [duplicate]

Suppose $1 \leq p<\infty$, given $f \in L^p (\mathbb{R})$, define $f_n (x)=n^{1/p} f(nx)$ for n=1,2,3... Prove $f_n$ converges weakly to zero in $L^p$. Now I can just know the that $ \|f_n\|_p$=$ ...
1
vote
0answers
20 views

Convergence in distribution plus convergence of moments.

Suppose that the sequence of r.v $\{X_n\}_{n\geq 1}$ has all the moments, and $X_n\stackrel{D}{\longrightarrow} X\sim N(0,\sigma)$. Assume that $E\{(X_n)^K\} \stackrel{n} {\longrightarrow} E(X^K)$, ...
0
votes
1answer
37 views

weak convergence of $L^2$ implies weak convergence of $W_0^{1,2}$ (up to a subsequence)?

In the paper that I am reading, it says that if $\{u_n\}$ are bounded in $W_0^{1,2} (\Omega)$ (bounded $\Omega\subset \mathbb{R}^N$) and $u_n \rightharpoonup u$ weakly in $L^2 (\Omega)$, then there ...
2
votes
1answer
22 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
1
vote
1answer
35 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
0
votes
1answer
58 views

If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ Let $\phi$ be a smooth, bounded and nondecreasing ...
0
votes
0answers
11 views

weak convergence on C*-algebra

Let A be a $C^*$-algebra and I be a $C^*$-subalgebra such that I** = q A** q (q is a projection in A**). Let $(x_n)$ and $(f_n)$ be weakly null sequences in I and I*. Define $g_n$ in A* by $g_n (a) = ...
1
vote
1answer
36 views

weak* convergence definition in Sobolev space

I have a question which might quite trivial but I would appreciate any assistance. Why does it follow that for Sobolev spaces, say $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$, it follows ...
1
vote
1answer
21 views

Convergence in distribution with exponential limit distribution

Let $X_1,X_2, \ldots$ be independent, identically distributed, positive random variables with probability density function $f$, which is continuous in $(0, \infty)$ and $\lambda :=\lim_{x \searrow 0} ...
0
votes
1answer
41 views

Weak Convergence If and Only If (Pointwise) Convergence of Characteristic Function

This is actually a theorem from lecture notes, with the corresponding proof. Unfortunately, it doesn't prove the last bit, or mention it at all (!), and I have a question about the penultimate bit. ...
1
vote
1answer
32 views

A basic question on weak convergence of probability measures

Let $f_n, f$ be bounded and continuous functions. Given that $$\int f d\mu_n \to \int f d\mu$$ for every bounded and continuous function $f$ under what conition $$\int f_n d\mu_n \to \int f ...
3
votes
0answers
31 views

A basic question on spaces of probability measures

This problem is regarding the space of probability measures. For $N \geq 1$, let $\{e_i^N(.), i\geq 1\}$ denote a complete orthonormal basis for $L_2[0,N]$. Let $\{f_j\}$ be countable dense in the ...
1
vote
0answers
21 views

Existence of nets from weak closure

Hi I am interested in the following question. Given some normed space $X$ with a subset $S \subset X$. If I consider $x \in \text{wcl}(S)$, where 'wcl' denotes the weak closure of $S$, then since the ...
0
votes
1answer
30 views

Problem on sequence of probability measures

Let $\mu_n$ and $\mu$ be probability measures such that $$\lim_{n\to \infty} \mu_n(A) = \mu(A)$$ for Borel $A \subset S$ satisfying $\mu(\partial A) =0$ (we call it a $\mu$-continuity set). I have ...
0
votes
1answer
70 views

Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in ...
1
vote
0answers
55 views

Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
1
vote
1answer
42 views

weak* convergence for sequence in $ L^\infty$

Let $ \Omega \subset \mathbb{R}^d $ be a bounded and open set. Suppose $ \{f_n\} \subset L^{\infty} (\Omega) , f \in L^{\infty} (\Omega) $. Prove that $ f_n \rightharpoonup^* f \ \ \text{in} \ \ ...
1
vote
1answer
20 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
1
vote
1answer
37 views

Exercise on functional analysis

I have a problem with this exercise of functional analysis about weak convergence: Let $U:=\prod_{i}^{d}(a_{i},b_{i})$ a subset of $\mathbb{R}^d$ with $a_{i}<b_{i}$ for each i and let ...
1
vote
1answer
73 views

Central limit theorem extends to absolutely continuous measures

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ that is absolutely continuous w.r.t. $\mathbb{P}$. Let the ...
3
votes
2answers
51 views

Limit law of real-valued independent random variables

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp. i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$. Then, are $X$ ...
2
votes
1answer
36 views

Weak Convergence for a specific example

Let $X=(0,1)$ and $u_v(x) = v^{1/p}1_{(0,1/v)}$, where $1 < p < \infty$. It needs to be shown that $\lim_{v \to \infty} \int_X u_v \phi = 0$ for all $\phi \in \mathcal{L}^{p^\prime}$ (The ...
0
votes
0answers
14 views

limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
7
votes
3answers
93 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
0
votes
1answer
35 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
1
vote
1answer
45 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
3
votes
0answers
35 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
2
votes
2answers
42 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
1
vote
1answer
35 views

$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$. Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty. Can anyone help me with this problem? ...
1
vote
0answers
23 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
2
votes
1answer
57 views

Product of weak/strong converging sequences

Let's consider two sequences $u_n$ and $v_n$ such that $$u_n\to u\,,\,\,\,\rm{in}\,\,\,L^\infty(\mathbb{R}^n)$$ and $$v_n\rightharpoonup v\,,\,\,\,\rm{in}\,\,\,L^2(\mathbb{R}^n)$$ What can I say of ...
0
votes
0answers
40 views

Sequential compactness in weak topology

When the Banach space $V^*$ is reflexive, we have the unit ball in $V^*$ is weak$^*$ sequentially compact. For a Banach space $V^*$ that might not be reflexive, we have to assume that $V$ ...
1
vote
0answers
29 views

Projections as a dense subset of $\ell_p^*$

Okay, it seems I wrote a confusing formulation of my question: $\textbf{(1)}$ I'm dealing with the following problem: prove that if $ \left(\{a_n\}_{n=1}^{\infty}\right)_k \in \ell_p$, then ...
1
vote
1answer
41 views

Weak convergence in Hilbert space implies strong convergence of averages for some subsequence

Let $H$ Hilbert Space. Show that if $x_n\rightharpoonup x$ then there exists a subsequence $\{x_{nk}\}$ of $\{x_{n}\}$ such that the sequence $\lim_{m\rightarrow \infty } ...
2
votes
2answers
43 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
1
vote
1answer
46 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...
2
votes
1answer
46 views

Show that a bounded linear transformation is continuous

I am not sure what this question is asking. A linear operator $T$ between normed spaces X and Y is bounded if and only if it is a continuous linear operator. But weak topology is not metrizable. I ...
1
vote
1answer
45 views

Isn't this just the definition of weakly convergence?

Hi I come across this question when reviewing my notes. But isn't this just the definition of weak convergence? Am I failing to understand the problem? The definition given on my notes is A ...
0
votes
1answer
35 views

Find a sequence in the unit ball of L^1([0,1]) that has no convergent subsequence in the weak topology.

This is a question I come across when reading my notes. Find a sequence in the unit ball of $L^1$([0,1]) that has no convergent subsequence in the weak topology. By a corollary of Banach-Alaoglu, ...
1
vote
0answers
21 views

A basic question on convergence in Skorohod metric

Consider a sequence of piecewise constant (positive) function in the space $D$ with Skorohod topology such that each of the functions is zero at the origin and any two discontinuities are at least ...
5
votes
2answers
87 views

$e_n \to 0$ weakly in $l^\infty$

Given the the sequence $(e_n)_n$ in $l^\infty$, I want to show that that $e_n$ converges weakly to $0$ in $l^\infty$, i.e. $$e_n\rightharpoonup 0 \text{ as } n\to \infty.$$ By $e_n\in l^\infty$, ...
1
vote
0answers
24 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
1answer
67 views

Weak Convergence and Weak Topology

In discussing weak topology of a normed space $X$, a lemma is given as follows. If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded. I understand the proof of this ...