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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Weak convergence against upper invariant measure

Setting I am studying invariant measures and their weak limits. In a book about probability on graphs the following setting is presented in chapter 6.3 (this is a short form of the actual presentation)...
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2 votes
2 answers
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Setwise convergence of measures implies weak convergence under special hypothesis

I'm struggling with producing a proof of the following result: Let $X = \overline{\mathbb{C}}$ be the Riemann sphere, and consider $M(X)$ the space of finite Borel measures on $X$ with norm given by ...
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$\lim_{\delta\to 0} \delta^{-1}\int \phi\, dL^n\vert_{B(0, 1+\delta)\setminus B(0,1)} = \int \phi \, d\sigma^{n-1}$ for every $\phi\in C_0(\Bbb R^n)$

The surface measure $\sigma^{n-1}$ on the sphere $S^{n-1}$ is defined in Folland text, in the following way: Consider the homeomorphism $\Phi:\Bbb R^n\setminus\{0\} \to (0,\infty)\times S^{n-1}$ ...
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Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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Does $\max{\{ f_{1}(X_{1,t}),f_{2}(X_{2,t})\}} \overset{d}{\to} \max {\{ f_{1}(X_{1,\infty}),f_{2}(X_{2,\infty})\}}$ when $f_{i}$ is continuous?

Let $X_{i,t}$ be a random variable. Support $X_{i,t}$ converges to $X_{i,\infty}$ in distribution, whose the distribution is $f_{i,\infty}$. Let $Y_{t} = \max {\{f_{i,\infty} (X_{i,t}) : \forall i=1,\...
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1 answer
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Exercise 4.19 (1) of Brezis

I am trying to solve the following exercise of Brezis' book on Functional Analysis. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $L^p(\Omega)$ with $1 < p < \infty$ and let $f \in L^p(\Omega)$...
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If $\mu_i \to \mu$ weakly, then do we have $\operatorname{supp}(\mu_i) \to \operatorname{supp}(\mu)$ in Hausdorff distance?

Suppose $\mu_i$ are Radon measures with support in $\mathbb{R}^{n}$. Is it true that if $\mu_i \to \mu$ weakly then their supports must converge in Hausdorff distance?
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Existence of sequence of linear combinations of weakly convergent sequence such that this sequence converges strongly [closed]

If $\left(x_{n}\right)$ is a weakly convergent sequence in a normed space $X$, say, $x_{n} \stackrel{w}{\longrightarrow} x_{0}$, show that there is a sequence $\left(y_{m}\right)$ of linear ...
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There exists such result for any convergence of sequence definiton

Assume that $L(x_n) $ stands for the limit of the sequence $(x_n) $ in some sense, not necessarily the usual limit. For example, could be the almost convergence limit or the statistical limit. I ...
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Weak convergence in Hilbert space, convergence in distribution and pointwise convergence

I am learning about weak convergence in Hilbert spaces of functions and I am wondering if it is the same as pointwise convergence or convergence in distribution. I do not really see a difference ...
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4 votes
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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 $\mathcal{Q}$ -- $\...
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5 votes
1 answer
177 views

How to show $\frac{1}{\delta} \mathbb{P}[ \sup_{t\leq s \leq t+\delta} |X(s)-X(t)|\geq \epsilon] \leq \eta$?

If a random element $X$ of $D[0,1]$ has the property that $$\lim_{\delta\to 0} \sup_{0\leq t\leq 1-\delta} \frac{1}{\delta}\mathbb{P}(|X(t+\delta)-X(t)|\geq \epsilon) = 0.$$ for every $\epsilon >0$,...
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2 votes
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Question about proof that every probability measure on a metric space is regular

I have a question from the first proof in Billingsley's Convergence of Probability measures. In the following proof, why does the result follow from showing that $\mathcal{G}$ is a $\sigma$-field? ...
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1 answer
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Doubts about $D[0,1]$ set

Let $D[0,1]$ be the space of real functions $f$ on $[0,1]$ that are right continuous and have left limit. That is space of cadlag functions. Consider $D_n = \{x \in D[0,1] | x \ \mbox{has at most $n$ ...
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1 answer
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Proving the existence of a weak limit given the convergence of $\lim_{n\to\infty}(x_n,v) \forall v\in H$

I've got a problem I'm a little stuck on and was curious as to a way to prove this. I've always sort of just assumed that when defining weak convergence, the weak limit automatically exists but now I ...
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2 votes
0 answers
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Why we need $\min_{1\leq i \leq v} (t_i - t_{i-1}) > \delta$?

Let $D[0,1]$ be ta space of real functions $f$ on $[0,1]$ that are right continuous and have left limit. That is space of cadlag functions. Now for $f\in D$ and $T\subset [0,1]$, define $$w_f(T) = \...
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2 votes
1 answer
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How to prove that weakly convergence in $L^{p_2}$ implies weakly convergence in $L^{p_1}$ where $1\leq p_1\leq p_2<\infty$?

As the title suggested, suppose that $1\leq p_1\leq p_2<\infty$. Let $\{f_n\}$ be a sequence of functions in $L^{p_2}([0,1])$ and $f\in L^{p_2}([0,1])$. Show that $f_n \rightharpoonup f$ in $L^{p_2}...
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6 votes
1 answer
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Does the limit superior of every subsequence equals to the same measurable function imply convergence of the original sequence?

In an abstract measure space $(X, A, \mu)$, we consider a sequence of measurable functions $(f_n)_n$ with $f_n : X \to \mathbb{R}$ such that there exists some measurable function $f : X \to \mathbb{R}$...
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0 answers
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Using the monotone class argument to prove that the set of transport plans is closed in the weak topology

i'm having some problem understanding the first answer given to the following question: Proof that the set of transference plans is closed in the weak topology. In particular i can't prove the second ...
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1 answer
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Portmanteau theorem with lower semi-continuous and bounded from below functions

I'm trying to prove this version of Portmanteau theorem. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. ...
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0 answers
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Euler discretization scheme

I am trying to understand the euler discretization scheme, but I am super confused about the distribution of the error term which is a random variable. The error term is defined by where, ST,k denotes ...
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Exchange convergence in distribution and integration

I would like to know if it is possible to apply dominated convergence theorem to show that if a sequence of integrable measurable functions (random variable depending on some real number) $f_n(x) \...
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1 answer
39 views

Example of a sequence of Borel measures that converges weakly, but does not converge in Lévy–Prokhorov metric

Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. We endow $\mathcal P(X)$ with the Lévy–Prokhorov metric $d_P$. Let $\mathcal C_b(X)$ be the space ...
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1 answer
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The space of Borel probability measures is closed in that of finite Borel measures w.r.t. Lévy–Prokhorov metric

I'm trying to prove this intuitive result. Could you have a check on my attempt? Let $(X, d)$ be a metric space. Let $\mathcal{M} :=\mathcal{M}(X)$ and $\mathcal{P} :=\mathcal{P}(X)$ be the sets all ...
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1 vote
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Demonstrate a certain inequality related to the accompanying law theorem.

Suppose I have a triangular array of r.v. $(X_{nj})_{1\leq j \leq k_n}$- $X_{nj} \sim \mu_{nj}$, $k_n \uparrow\infty$ - which are independent in each row and satisfies the uniformly asymptotically ...
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Tightness vs Equi-integrability, Prokorov

I am a bit confused by this Theorem in the book 'Optimal Transport for Applied Mathematicians' of Santambrogio. This question concerns absolutely continuous (w.r.t Lebesgue) probability measures on $\...
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3 votes
2 answers
419 views

The set $C[0,1]$ is nowhere dense in $D[0,1]$.

$D[0,1]$ be the space of real functions $x$ on $[0,1]$ that are right continuous and have left hand limit. On the other hand $C[0,1]$ is set of all continuous real functions on that interval. ...
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3 votes
1 answer
58 views

Is Probability Measure always tight?

We know that probability measures are tight if the metric space is separable and complete. Here tight means there exists a compact set in that metric space say $K$ such that $P(K) > 1- \epsilon$. I ...
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3 votes
1 answer
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A sequence in $L^p$ that converges in measure but not weakly and vice versa

I am able to cook up an example A sequence in $L^p$ that converges in measure but not weakly. Namely, let $f_n = n \chi_{[0, 1/n]}$ for $n \in \mathbb{N}.$ Then we have that $\{f_n \}$ converges in ...
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1 answer
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If the underlying metric space is separable, then weak convergence is equivalent to convergence in Prokhorov metric

I'm rewriting the proof that weak convergence is equivalent to convergence in Prokhorov metric in separable metric space. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\...
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0 votes
0 answers
21 views

Convergence in L^p for function multiplied for another function

Problem: Let $\Omega \subset R^n$ and let $f_{\epsilon}, f \in W^{1,p}(\Omega)$ be functions such that $f_{\epsilon} \rightarrow f$ in $L^p(\Omega)$. Which condition should have another function $g$ ...
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1 answer
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If $\mu_i(A) \to \mu(A)$ for all $A$ with $\mu(\partial A) = 0$, then $\int_E g \mathrm d \mu_i \to \int_E g \mathrm d \mu$

I'm trying to prove below equivalence of weak convergence of finite Borel measures. Let $(E, d)$ be a metric space and $\mu, \mu_1, \mu_2,\ldots$ finite Borel measures on $E$. Let $g:E \to \mathbb R$ ...
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  • 1,153
0 votes
1 answer
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Convergence in probability of the difference between the inverse of two sequences

Say $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ are two sequences of random variables defined on the same space such that $$W_n:= X_n - Y_n$$ converges in probability to zero. Does it hold that $$Z_n:= \...
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Uniform bounded sequence in $L^1\cap L^{\infty}$ is converging weakly (weak$^{*}$) along a subsequence to some $f\in L^1\cap L^{\infty}$?

Let $(f_n)_{n\in \mathbb{N}}$ be some uniform bounded sequence in $L^1\cap L^{\infty}(\mathbb{R}^n)$. Let $p_1>1$ and $\infty>p_2>p_1$. Then there exists some subsequence and $f_1\in L^{p_1}$ ...
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2 votes
1 answer
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Example in $\ell^\infty$ of a sequence that is bounded and componentwise convergent that is not weakly convergent

After resolving this related question, I thought about this case but couldn't think of anything. I have seen that, for $1 < p < \infty$, a sequence in $\ell^p$ is weakly convergent if and only ...
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0 votes
1 answer
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Example in $\ell^1$ of a bounded and componentwise convergent sequence that is not weakly convergent

I have seen that, for $1 < p < \infty$, a sequence in $\ell^p$ is weakly convergent if and only if it is componentwise convergent and bounded. Is there a counterexample for $p = 1$? That is, ...
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2 votes
0 answers
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Weak- L^1 convergence and lower semicontinuity of an integral functional

Let $f,f_n: \mathbb{R^3} \to \mathbb{R}$ be positive functions. Suppose $f_n \rightharpoonup f$ in $L^1(\mathbb{R}^3)$ as well as $f_n|v|^k \rightharpoonup f|v|^k$ in $L^1(\mathbb{R}^3)$ for all $0\le ...
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3 votes
1 answer
43 views

Is $\{a_n X_n +c\}$ this sequence bounded in probability?

Suppose $\{X_n\}$ converge weakly to standard normal distribution. Now we define a sequence $\{a_nX_n +c\}$ like that where $c>0$ is any constant and $a_n\rightarrow\infty$. Can we conclude that ...
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2 votes
3 answers
150 views

Brownian motion has unbounded variation.

I tried to solve this problem in the following way. Suppose $\{B_t | t\in [0,1]\}$ is our Brownian motion. Define, $$f_n(w) = \sum_{k=1}^{2^n} \bigg|B_{\frac{k}{2^n}}(w)-B_{\frac{k-1}{2^n}}(w)\bigg|.$$...
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1 vote
0 answers
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Does weak convergence imply almost sure continuity of mapping

Consider standard Borel spaces $\mathcal{X}, \mathcal{Y}$ and $[0,1]$ with random variables $X, Y, U$ and joint distribution $\mathbb{P}(X, Y, U)$, where the marginal $\mathbb{P}(U) = \lambda$, i.e., $...
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3 votes
1 answer
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A sequence of functions in $L^1$ that does not converge weakly

I have attempted the following part from exercise 5.49 in https://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf : For every $n \in \mathbb{N}$, let $f_n \colon \mathbb{R} \to \mathbb{R}$ be defined by: $...
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0 votes
0 answers
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Weak convergence and continuity [check of proof]

I wanted to prove the following statement: Let $X,Y$ be Banach, $T:X\rightarrow Y$ be linear. Further for every sequence $(x_n)_{n \in \mathbb{N}}$ in X weakly convergent to $x\in X$, also $(Tx_n)_{n \...
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1 vote
0 answers
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Why $w'_x(\delta) \leq w_x(2\delta)$?

In the book by Billingsley Convergence of Probability Measures at page number 123. Author stated that $$w'_x(\delta) \leq w_x(2\delta), \ \delta < 1/2.$$ and $$w_x(\delta) \leq 2w_x(\delta) + j(x)$$...
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2 votes
1 answer
65 views

Number of discontinuity points in [0,1]

In the book Convergence of Probability Measures by Billingsley. Chapter 3, there is a lemma (lemma 1) as follow, For each $x$ in $D[0,1]$ and each positive $\epsilon$, there exist points $t_0,...,t_v$...
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2 votes
0 answers
40 views

Semicontinuity w.r.t. weak convergence of probability measures

Let $(S, \mathcal{S})$ be a Polish space and consider the space $\mathcal{P}(S)$ of all Borel probability measures on $S$ endowed with the topology of weak convergence of measures. We know that if $f:...
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2 votes
1 answer
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Show there exists a subsequence that converges weakly

In PDE by Evans, Chapter 8.4.2. we want to minimize the energy functional $$ I[w]: = \int \frac{1}{2} |Dw|^2 - fw dx $$ among all functions in $$ A: = \{ w \in H^1_0(U): w \geq h \; \; a.e. \}$$ with ...
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1 vote
0 answers
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Convergence of expectation and its implications

I understand that $T_n\xrightarrow{\mathbb P}t$ does not imply that $\mathbb E[T_n]$ converges (including to $t$). At the same time, we know that convergence in probability, implies convergence in ...
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2 votes
1 answer
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Weak convergence and convergence of functionals in dual space

Let $X$ be a normed space, $(x_n)_{n \in \mathbb{N}} \subset X$ and $(x_n')_{n \in \mathbb{N}} \subset X'$ such that $(x_n)_{n \in \mathbb{N}}$ weakly converges to $x \in X$ and $(x_n')_{n \in \mathbb{...
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1 vote
1 answer
49 views

Is this sequence $\{\mu_n\}$ of probability measures weakly convergent?

Given $X=[a,b]$. Assume $\{\mu_n\}$ is a sequence of probability measures on $X$ such that for each polynomial $p\in\mathbf{R}[x]$, $$\lim_{n\rightarrow\infty}\int_{a}^{b}p\,\mathrm{d}\mu_n$$ exists. ...
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1 vote
1 answer
24 views

Continuous from Strong to Weak Star Topology Can be Characterized as Sequence Continuity?

This is a problem from Brezis Exercise 3.11: In the solution section, Brezis gave the following solution: I wonder why does it suffices to argue on sequences and why does there exists some $y \in E$ ...
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