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).
2,414
questions
2
votes
1
answer
60
views
On showing that if $\mu_n$ converges to $\mu\in M_1(X)$ in weak* topology, then $\lim_n\int_X fd\mu_n=\int_X fd\mu$ for every Lipschitz function $f$
Let $(X,d)$ be a metric space and $M_1(X)$ the set of Borel probability measures over $X$. Suppose that $\mu_n$ is a sequence in $M_1(X)$ such that $\mu_n$s converge to some $\mu\in M_1(X)$ in weak* ...
- 810
3
votes
2
answers
60
views
Weak convergence of functionals $g_n^*(f) = n\int_0^1 x^nf(x)dx$ [closed]
Show that sequnce of functionals $g_n^*(f) = n\displaystyle{\int_0^1 x^nf(x)dx}, f \in C[0,1]$ converges weakly and find its limit functional. Does it converge in the norm of space $C^*[0,1]$?
I don'...
- 161
1
vote
1
answer
35
views
Weak, strong and uniform convergence of operator on $L^p(\mathbb{R}^d)$ [duplicate]
For every $a \in \mathbb{R}^d$, let $T_a(f)(x) = f(x-a), \forall f \in L^p(\mathbb{R}^d), \forall x \in \mathbb{R}^d$. Prove that $T_a$ is a linear isometry of space $L^p$ on itself. Find $\lim_{a\to ...
- 161
0
votes
1
answer
51
views
Prove $a_nX_n+b_n \Rightarrow aX+b$ by means of characteristic functions
I want to solve the following exercise in Probability and Measure, Billingsly [1994]
According to Example 25.8, if $X_n \Rightarrow X$, $a_n \rightarrow a$ and $b_n \rightarrow b$, then $a_nX_n+b_n \...
- 1,069
1
vote
0
answers
38
views
Proving that convergence of a series implies weak convergence in an inner product space
I'd like to prove the following:
If $\sum_{n=1}^\infty u_n = u$, then $\sum_{n=1}^\infty \langle u_n, x \rangle = \langle u, x \rangle$ for any $x$ in an inner product space.
Is the following proof ...
2
votes
0
answers
24
views
Dual of $L^p_{loc}$ and weak star convergence.
I was reading a paper and stumbled across the terms
$$f^\varepsilon\to f \text{ in } L^{p}_{loc} \Bigl( [0,+\infty) , \left[ L^p_{loc}(\mathbb{R}^2) \right]^* \Bigr)$$ and $$f^\varepsilon\...
- 21
1
vote
1
answer
40
views
Theorem contradicting the necessity of Slutsky's theorem
If $X_n\overset{p}{\rightarrow}X,Y_n\overset{p}{\rightarrow}Y$ then, show that $X_n+Y_n\overset{p}{\rightarrow}X+Y$.
I am not looking for the proof. My issue is, this seems contradictory to Slutsky's ...
- 1,665
1
vote
2
answers
37
views
$X_n\overset{p}{\rightarrow}0\Leftrightarrow E(g(X_n))\rightarrow0$ for $g(x)=\frac{x^2}{1+x^2}$
If $g(x)=\frac{x^2}{1+x^2}$ then, show that $X_n\overset{p}{\rightarrow}0\Leftrightarrow E(g(X_n))\rightarrow0$.
I haven't been able to go too far with the proof. What I know is:
$g(0)=0, |g(x)|\le\...
- 1,665
1
vote
1
answer
39
views
On bounded sets of $L^2$ the weak topology is metriziable
I just read the weak topology is metriziable on bounded sets of $L^2$. Does that mean if $h_n\in B$ and $h_n\to h$ weakly in $L^2$ then $h_n\to h$ w.r.t. $d$? Does this metric have an explicit formula?...
- 63
2
votes
1
answer
39
views
How to prove this characterisation of convergence by distribution?
We solved the following problem in class and I do not understand what happens.
Definition: A sequence of random variables $W_1, W_2, \ldots$ converges in distribution to random variable $W$ if for ...
- 1,572
2
votes
0
answers
38
views
$X^n\not\Rightarrow X$ but finite dimensional distributions converge (Billingsley exercise 12.5)
Chapter 3, section 12 of Billingsley Convergence of Probability Measures contains the following theorem on p. 136:
Theorem 12.6. Suppose that $E \in \mathcal D$ and $T_0$ is a countable, dense
set in $...
- 5,890
1
vote
0
answers
39
views
Is asymptotic unbiasedness is a necessary condition for consistency: counter-example [duplicate]
Suppose $T_n$ is a sequence of estimators of $\theta$ and suppose $ET_n\nrightarrow\theta$. Is is true that $T_n$ is NOT consistent for $\theta$?
I don't think unbiasedness is a necessary condition ...
- 1,665
0
votes
1
answer
28
views
Weak convergence on separable and complete product space
I read a paper in which the authors seem to have a simplified definition of convergence in distribution of random variables in a product space. The paper itself is very specific, so I can link it but ...
- 1
1
vote
0
answers
78
views
Show that $\int_{\mathbb R^p} x d\nu = 0$ in a problem related to weak convergence of measures
This question is related to another question that was answered correctly, but not in the way I wanted, as I forgot to state a hypothesis.
Let $(X_{jn})_{1\leq j \leq n}$, $X_{jn} \sim \mu_{jn}$, be a ...
- 616
0
votes
0
answers
22
views
Distribution of limit of empirical measures with noise
Suppose that we have a sequence of i.i.d. random variables $X_1,X_2,...$ on $\mathbb{R}^2$ with probability distribution $\mu$ and define random empirical measures as
$$\mu_n=\frac{1}{n}\sum_{k=1}^n \...
- 81
1
vote
0
answers
37
views
Proof of Helly Bray theorem: $X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for bounded continuous $g$
Helly Bray Theorem
$X_n\overset{d}{\rightarrow}X\Leftrightarrow Eg(X_n)\rightarrow Eg(X)$ for all bounded, continuous functions, $g$.
For the only if part of the proof, I am a little stuck. I have ...
- 1,665
4
votes
1
answer
44
views
Weak convergence in $W^{1,2}$ implies strong convergence under extra condition?
Suppose $U \subset \mathbb{R}^d$ is a smooth bounded domain. Let $(u_n)_{n\geq 1}$ be a sequence in the Sobolev space $W^{1,2}(U)$ such that it weakly converges to zero: $u_n \rightharpoonup 0$ as $n \...
- 65
0
votes
0
answers
16
views
Is a bounded non linear functions of weakly convergent sequence also a weakly convergent sequence?
Let $A$ be a compact subset of $\mathbb{R}$ and $B$ a compact subset of $\mathbb{R}^n$. Assume that the sequence $(x_n)_n\in L^2(A;B)$ weakly converges to some $\bar{x}\in L^2(A;B)$. Now, let $f:L^2(A;...
- 21
2
votes
0
answers
35
views
Weak convergence and strong convergence of solutions of ODEs
I would like to check if the proof of the following result is true.
Proposition: Let us note $y_u:[0,T]\mapsto \mathbb{R}^n$ the solution of the following ODE
$$y_u'(t) := f(y_u(t), u(t))$$
where $y_u(...
- 21
1
vote
0
answers
34
views
Joint distribution of $\left(\xi_1, \xi_2, \xi_3\right)$ converges to a 3D Gaussian distribution
I'm trying to teach my self the first 10 chapters of the book out of interest: "Leonid Koralov, Yakov G. Sinai - Theory of probability and random processes".
In chapter 8 (Weak Convergence ...
- 45
0
votes
1
answer
25
views
$\sigma$-compact and weak* convergence
It is from Stein's Functional Analysis Chapter 1, Problem 4.
Suppose $X$ is a $\sigma$-compact measurable metric space, and $C_b(X)$ is separable, where $C_b(X)$ denotes the Banach space of bounded ...
- 762
0
votes
0
answers
19
views
Trace 0 and weak convergence in $\mathbb{R}^+ \times \Omega$
Suppose I have a sequence of $\{u_j\} \in H^1(\Omega \times \mathbb{R}^+)$ with uniform bounds such that $u_j$ is converging strongly in $L^2(\Omega \times \mathbb{R}^+)$ and $\{\nabla u_j\}$ are ...
- 751
1
vote
2
answers
88
views
Two independent weak convergence sequences have dependent limit?
Let $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ as $n\to\infty$, where all these $X$ and $Y$ are well defined random variables and $\Rightarrow$ is the convergence in distribution.
Could you give me an ...
- 61
0
votes
1
answer
36
views
Continuity set and his preimage in $f$
Let $\pi$ is probability measures on $(X,\mathscr{B}\left(X\right))$, where $\mathscr{B}\left(X\right)$ is borel $\sigma\text{-algebra}$ on $X$. Consider the set of all continuity sets in $\mathscr{B}\...
- 604
4
votes
1
answer
51
views
Folland: Convergence in weak topology iff convergence in dual
I am self-learning real analysis from Folland and got stuck on weak topology convergence. He defines weak topology as follows:
If $X$ is a normed vector space, the weak topology on $X$ is the weak ...
- 559
1
vote
0
answers
43
views
Metrization of topologies of weak convergence of measures and other notions of convergence
Let $(X,d)$ be a metric space. Denote $\mathcal P(X)$ the space of probability measures on $X$. Let, $(\mu_n) \subset \mathcal P(X)$, the usual notion of weak convergence of probability measures reads:...
- 96
3
votes
0
answers
43
views
Convergence in distribution and in probability
I am trying to show that if $X_n \rightarrow X$ in distribution there exist $Y_n \sim X_n$ and $Y \sim X$ such that $Y_n \rightarrow Y$ in probability.
I take $U \sim \mathcal{U}([0,1])$ and define $...
- 175
0
votes
0
answers
18
views
Can I verify the weak order of convergence of a numerical scheme for SDEs by checking convergence in distribution?
let $X_{t=0}^T$ be a continuous-time stochastic process defined by a certain stochastic differential equation
$$
X_t=a(t,X_t)dt+b(y,X_t)dW_t
$$
where W_t is a Wiener process.
Let $\tilde X_{i=0}^{N}$ ...
- 237
0
votes
1
answer
21
views
On the conceptual difference between almost sure convergence and mean convergence
Let me start by saying I'm not a mathematician, but I really need some clarification on this topic.
There are three notions (actually more than three but anyways) of convergence:
convergence in ...
- 51
0
votes
1
answer
29
views
Convergence in distribution doesn't imply convergence in probability
I know that convergence in distribution does not imply convergence in probability. However, I saw these two lemmas;
Lemma 1: $X_n \xrightarrow{p} X \implies X_n \xrightarrow{d} X$.
Lemma 2: $X_n \...
- 669
0
votes
0
answers
21
views
Carleman's theorem on a compact interval
Let $\mu_n$ be Borel probability measures on $[0, 1]$. I'm trying to prove a special case of Carleman's theorem, i.e.,
Theorem If the sequence $(\int_0^1 x^k \mathrm d \mu_n (x), n\in \mathbb N)$ ...
- 3,902
1
vote
2
answers
37
views
If $\lim_n \mu_n ([0, r]) = \mu ([0, r])$ for all $r \in D$, then $\mu_n \to \mu$ in distribution
Let $T>0$ and $\mu, \mu_n$ be Borel probability measures on the closed interval $[0, T]$. Let $D$ be a dense subset of $[0, T]$. I would like to prove that
Theorem If $\lim_n \mu_n ([0, r]) = \mu (...
- 3,902
0
votes
1
answer
22
views
Is the set of discrete probability measures with exactly $n$ atoms Borel measurable?
Let $(E, d)$ be a compact metric space and $\mathcal P(E)$ the space of Borel probability measures on $E$. We endow $\mathcal P(E)$ with topology of weak convergence. Let
$\mathcal G_n (E)$ be the ...
- 3,902
2
votes
1
answer
45
views
Relation between a convex lower semi-continuous function and its integral functional.
I'm looking for a reference for the following theorem.
In some multidimensional calculus of variations script I found the following theorem.
Theorem: Let $\Omega \subset \mathbb{R}^n$ be open and ...
- 41
2
votes
1
answer
31
views
How can I prove that this set is weakly closed in $L^1((0,1))$
Let $g:(0,1)\rightarrow \Bbb{R}$ and define $$Q_g:=\left\{f\in L^1((0,1)): |f(x)|\leq |g(x)|~~\text{for almost every}~~x\in (0,1)\right\}\subset L^1\left((0,1)\right)$$ I want to show that this set is ...
- 2,457
1
vote
1
answer
26
views
Let $X$ be locally compact and separable. Can we replace $C_c(X)$ with the space of Lipschitz functions with compact supports in weak convergence?
Let $\mu_n, \mu$ be Borel probability measures on a metric space $X$. We have the following equivalence (known as Portmanteau's theorem):
$\int f d \mu_n \to \int f d \mu$ for all $f$ bounded ...
- 3,902
2
votes
1
answer
51
views
Sequence of projections on a sequence in Hilbert space
I need help with the following related to my research. Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$. Let $V_n:= \text{span}\{e_1, e_2, \ldots , e_n\} \subset H$, and let $P_n: H \to ...
- 1,544
0
votes
1
answer
56
views
Some problems in the application of Arzelà–Ascoli theorem
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ with a smooth boundary. Consider the sequence $\{u_n(\cdot,s)\} \subset L^2(0,T;L^2(\Omega))$ such that $\lVert u_n(\cdot,s) \rVert_{ L^2(\Omega)} $ ...
- 419
2
votes
1
answer
47
views
Weak Convergence in Hilbert Spaces & Inner Product
I have a question about weak convergence in Hilbert spaces. Suppose we have a Hilbert space $H$ equipped with a scalar product $(\cdot / \cdot)$, and that in this space we have $x_n \rightharpoonup x$ ...
0
votes
0
answers
22
views
Clarifying when a weak limit can be assumed to be a strong limit
In the second answer to this post this it is stated:
Since you've already proved that there is a strongly convergent subsequence, let's say $ Tu_{n_k} \to u^* $ for $ k \to \infty $. Then by the weak ...
- 3,295
0
votes
2
answers
40
views
Linear operators in $L^\infty$ and convergence in weak*-topology
I tried to solve the following problem and want to know if my approach is correct and some help in order to finish. Thanks.
Let $V = L^\infty(\mathbb{R})$ equipped with the norm $\left\lVert \cdot\...
1
vote
1
answer
68
views
$ f_n(x) := n^{\frac{1}{p}} \chi_{[0,1]}(n x) $ weakly converges to $0$ for $p>1$, but not for $p=1$
As the title says, I am trying to show the weak convergence of the function sequence $ n^{\frac{1}{p}} \chi_{[0,1]}(n x) $, which is supposed to converge to $0$ for $p>1$ but not for $p = 1$. I ...
- 67
1
vote
0
answers
41
views
showing weak convergence for scaling limit of basic Markov jump process
Let $X_t$ be a continuous-time, finite state, real-valued Markov jump process with generator matrix $G$. We can just assume the process takes values in $\{0,1,2,\ldots,m\}$. I think this works more ...
- 5,890
2
votes
0
answers
30
views
Notion of convergence of random variables of growing dimensionality
When $Y_1, Y_2, \dots Y_n$ be a sequence of $d$-variate random variables we have the notion of convergence in distribution where we say $Y_n \implies Y_{\infty}$ if the corresponding distribution ...
- 21
2
votes
2
answers
70
views
Failure of Banach Alaoglu in $L^1$
We know that dual of $L^{\infty}(\mathbb{R})$ is not $L^{1}(\mathbb{R})$ and hence Banach Alaoglu theorem is not applicable for the $(L^1,L^{\infty})$ pairing.
In other words if $\{f_n\}$ is a ...
- 364
0
votes
0
answers
43
views
Is there a version of Levy's Continuity Theorem for stochastic processes?
We know that a stochastic process $X = (X_{t})_{t \in \mathbb Z}$ defined in $(\Omega, \mathcal F, \mathbb P)$ with real values can be expressed as a measurable function with values in some space of ...
- 616
0
votes
0
answers
40
views
Application of Continuous Mapping theorem to logarithm of random variables
In the Wikipedia page of Continuous Mapping Theorem, it reads if $g$ is any function between two metric spaces and $X_n$ converges in law to $X$ which almost surely will not take value in the ...
- 1,011
1
vote
1
answer
40
views
Why is the operatornorm not a weak* continuous map?
Let $X$ be a normed space and consider $$||\cdot ||:X^*\rightarrow \Bbb{R};~~f\mapsto ||f||$$ Then in a lecture side note I read that $||\cdot||$ is not weak* continuous.
As I understood it, we say $|...
- 2,457
0
votes
0
answers
42
views
A Characterization of Weak Closure in a Real Hilbert Space
$\textbf{Question}$
Let $\left(\mathcal{H}, \| \cdot \| \right)$ be a real Hilbert space and let $C$ be a subset of $\mathcal{H}$ such that $( \forall n \in \mathbb{N} ) ~ C \cap B(0;n)$ is weakly ...
- 453
2
votes
1
answer
53
views
Weak convergence of symmetries measures
Consider a sequence of probability measure $\mu_1, \cdots, \mu_N$ with support on non-negative real numbers, $\mathbb{R}_{\geq 0}$. Now, let for each $1 \leq i \leq N$, $\hat{\mu}_i$ be the ...
- 462