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).

learn more… | top users | synonyms

1
vote
1answer
12 views

FDD convergence of subsequences

First, sorry for this probably very stupid question. Let $(X_n)_{n\geq1}$ be a sequence of random variables (e.g. in $\mathbb{R})$ s.t. $X_n\stackrel{d}{\to} X$ in distribution. Now look at the ...
1
vote
0answers
25 views

Weak convergence of a sequence of elements of a compact set implies strong convergence

Let E be a Banach space and $K \subset E$ a compact subset in the strong topology. Let $(x_n)_{n \geq 1} \subset K$ such that $x_n \rightharpoonup x$ in $\sigma (E, E^*)$. K is compact, so $(x_n)_n$ ...
2
votes
1answer
24 views

Strong convergence of convex combinations of a weakly convergent sequence

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
1
vote
1answer
32 views

How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
2
votes
1answer
22 views

Continuity sets are neccesary for weak convergence.

Portmanteau theorem In particular if $\mu_n \to \mu$ weakly then $\mu_n(A)\to \mu(A)$ for each continuity set. I want an example to show that the hypothesis of $\mu(\partial(A))=0$ is neccesary. ...
2
votes
1answer
23 views

Weak and vague convergence of normal distribution

Let $\mu_n = \mathcal{N}(0,n)$ be the normal distribution with mean $0$ and variance $n$ on $\mathbb{R}$, $\nu$ the zero-measure (which is defined by $\nu(A) = 0$ for any ...
0
votes
0answers
27 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
1
vote
0answers
18 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
0
votes
1answer
30 views

sequence in $L^1$ converging pointwise a.e., but not weakly.

Find a $(X,\Sigma,\mu)$, a $\sigma$-finite measure space and a norm-bounded sequence $\{f_n\}$ in $L^1(X)$ that converges almost everywhere to $f$ but does not converge weakly to $f$. Can you help me ...
1
vote
1answer
29 views

Weak convergence and lim inf and lim sup of the sequence of norms

Assume $x_n$ is a sequence in a Banach space that converges weakly to $x$. Then we know that $\|x\| \leq \lim \inf \|x_n\|$. 1)But can we say that $\lim \inf \|x_n\| < \infty$ or is this in ...
2
votes
1answer
40 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
3
votes
2answers
100 views

Weakly convergent in different spaces

Given $\Omega \subset \mathbb{R}^n$ open connected and $k\geq 0$. Let $f_n, f$ be distributions such that $$f_n \rightharpoonup f \in \mathcal{D}'(\Omega)$$ as $n\to \infty$. Assume that $f_n \in ...
2
votes
0answers
54 views

weakly converges distribution sense implies uniform convergence

Given a set of equicontinuous and uniformly bounded functions $\{f_n, f\}$ defined over an open connected $\Omega \in \mathbb{R}^n$. Suppose $(f_n)_n$ weakly converges to some $f$ in distribution ...
0
votes
1answer
35 views

Convergence in $L^2$ norm given inner product converges

Suppose $(f_n)$ and $f$ are in $L^2$ and $\int_E f_ng \rightarrow \int_E fg,\,$ for $g \in L^2(E)$. If $\|\,f_n\|_2 \rightarrow \|\,f\|_2$, then show that $f_n \rightarrow f$ in $L^2$. (All functions ...
1
vote
0answers
62 views

Galerkin method for Poisson's equation

This is problem 3 from chapter 7 of Evans book: Suppose $f\in L^2(U)$ and assume that $u_m=\sum_{k=1}^md_m^kw_k$ solves $\int_UDu_m\cdot Dw_k=\int_Uf\cdot w_kdx$ for k=1,...,m. Show that a subsequence ...
3
votes
0answers
141 views

Complicated convergence of nonlinear term

Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in ...
1
vote
0answers
20 views

Weak Convergence in Metric Space proof

I have been reading Billingsleys book where I came across this theorem and proof. I am having difficulty understanding the theorem/proof. I feel there is a better, more complete way to prove it. Does ...
1
vote
1answer
38 views

If the sequence of distribution functions weakly converge, the sequence of corresponding subprobability measures converges weakly, too

Let $\mu,\mu_n$ be subprobability measures on $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$ $F,F_n$ be the distribution functions of $\mu,\mu_n$ with ...
6
votes
1answer
160 views

Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and strongly converge to $u$ in $L^p(\Omega).$ We define a function $f:\Omega\times ...
1
vote
0answers
40 views

The weak convergence between $L^2$ and $H^1$

If the sequence $u_n\in H^1(R^n)$, $n\geq 3$ and $u_n$ weak converges to $u$ in $H^1(R^n)$, if we can get $u_n$ weak converges to $u$ in $L^2(R^n)$? Furthermore, if we can have $\nabla u_n$ weak ...
1
vote
1answer
37 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
1
vote
0answers
19 views

Improving weak convergence of random variables

Assume you know $X_n\Rightarrow X$ in a (nice, e.g. Polish) space $(E,d)$, that is $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for all $f\in\mathcal{C}(E,\mathbb{R})$. Now the topology on $E$ determines ...
4
votes
2answers
47 views

Show that there is a probability such that $P_n$ converges weakly/in distribution as $n \to \infty$.

Suppose that $P_n$ $n \ge 1$ is a sequence of probabilities concentrated on $[a,b]$. Suppose that one may show for each positive integer $r$ that $\int_{[a,b]}x^rP_n(dx) \to m_r \in R$ as $n \to ...
1
vote
0answers
25 views

Seeking help understanding steps in a proof, in “Convergence of probability measures by P.Billingsley”.

To not waste anyones time: this question is directed at people who either have the mentioned book in their possesion, or have at some point read it. I am reading through Patrick Billingsley's book: ...
1
vote
2answers
41 views

Convergence in distribution (weak convergence) of sum of real-valued random variables

Suppose that $\{(X_n,Y_n)\}^\infty_{n=1}$ is a sequence of pairs of real-valued random variables that converge in distribution to $(X,Y)$. Show that $X_n + Y_n$ converges in distribution to X+Y. ...
2
votes
0answers
18 views

Convergence of time-slice measures to convergence in law

Question: Are there some conditions that allow one to go from finite dimensional distribution convergence to convergence in path space? Motivation: Consider a sequence of measures for e.g. solutions ...
1
vote
1answer
34 views

Weak and strong convergence

I have the sequence $(v_n)\subset H^1_0(0,1)$ such that $v_n\rightharpoonup v $ (weakly) in $H^1_0(0,1)$ and $v_n\rightarrow v$ in $L^2(0,1)$ and $v_n\rightarrow v$ in $C^0(0,1)$ My question is why ...
4
votes
1answer
132 views

Comparison between weak convergences in Banach spaces

Let $X$ be a Banach space and let $Y=BC(\mathbb{R},X)$ be the Banach space of all bounded continuous functions from $\mathbb{R}$ to $X$ equipped with the supremum norm. Let $(f_n)_n$ be a sequence of ...
0
votes
0answers
26 views

weak limit of $u_n(t,x) = \sin(nt)g(x)$ in $L^2(0,T;L^2)$

Let $u_n(t,x) = \sin(nt)g(x)$ for a function $g \in L^2(\Omega)$. We know that $\lVert u_n \rVert_{L^\infty(0,2\pi;L^2(\Omega))} < C$ because $\sin(nt) \leq 1$ and $C$ does not depend on $n$. So ...
0
votes
0answers
15 views

Weak limit of $u_n(t,x) = sin(nt)v(x)$ in $L^2(0,2\pi;H^1(\Omega))$ where $v \in H^1(\Omega)$.

Let $u_n(t)(x) = sin(nt)v(x)$ where $v \in H^1(\Omega)$ and $\Omega$ is bounded. Does $u_n$ have a weak limit in the space $L^2(0,2\pi; V)$? If we had no spatial part, then the sequence would ...
3
votes
0answers
40 views

If $u_n \rightharpoonup u$ in $L^2(0,T;L^2)$ and $u_n$ bounded in $L^\infty(0,T;L^2)$, does $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ a.e. $t$?

Let $u_n$ converge weakly to $u$ in $L^2(0,T;L^2(\Omega))$ and let $u_n$ be bounded in $L^\infty(0,T;L^2(\Omega))$. Is it true that $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ (weakly) for a.a. ...
1
vote
1answer
50 views

Boundness of funcions in $L^2(0,T;H)$

Let $H$ be a Hilbert space and $u_{k} \rightharpoonup u$ in $L^2(0,T;H)$ (the $\rightharpoonup$ means "weakly convergent to") Assume one has the uniform bounds $$\mathrm{essential~sup}_{0\leq t\leq ...
0
votes
0answers
30 views

Proving that if a sequence converges weakly, then their set of norms is bounded.

We are given that if $\{\Lambda_n\}\subset X^*$ (where $X^*$ is the dual space of some Banach space $X$) converges weakly to some $\Lambda\in X^*$. That is $\Lambda_n x\to\Lambda x$ as ...
0
votes
1answer
16 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
1
vote
1answer
46 views

Weak convergence of Dirac measures converges to a Dirac measure?

Let $X$ be a metrizable space and $\{x_n\}$ be a sequence in $X$. Suppose the sequence of Dirac measures $\delta_{x_n} \xrightarrow{w} P$ where $P$ is some probability measure. Prove that $P = ...
1
vote
0answers
19 views

What is the norm of an element in $L^\infty(\mathbb{D})$ using the weak-star topology?

Under the convention that $L^1(\mathbb{D})$ has normalized Lebesgue measure for the unit disc in $\mathbb{C}$, its dual space can be regarded as $L^\infty(\mathbb{D})$. Hence we can equip ...
1
vote
0answers
31 views

Convergence in distribution of normal random variables

Let $X_n \sim \mathcal{N}(\mu_n,\sigma_n^2)$. Prove that if $X_n \rightarrow X$ in distribution, then either $X$ is normally distributed or there exists a constant $c$ such that $X = c$ almost surely. ...
0
votes
0answers
22 views

passing to a liminf from weak convergence. [duplicate]

Say i have a sequence $(||x_n||)$ which is bounded above, and say $x_n$ converges weakly to $x$. Then, how can I show that $||x|| \le \liminf_{n \to \infty}||x_n||$. Well, clearly, the $\liminf$ ...
3
votes
0answers
44 views

Weak vs strong convergence for unitary operators

Suppose $H$ is a separable complex Hilbert space with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$, where $\|u\|^2 = (u,u)$. Suppose $u, u_1, u_2, \dots \in H$. Then $\lim_{n \to \infty} u_n = ...
6
votes
1answer
171 views

Moment method for weak convergence

Say, we have a sequence of probability distributions $\mu_n$ on $\mathbb R$, that are uniformly subgaussian in the sense that $$\mu_n(\mathbb R\setminus[-R,R])\leq Ce^{-CR^2}$$ for some positive ...
2
votes
1answer
24 views

Showing Stronger result of Weak Law of Large Numbers

So, Khintchine's form of the Weak Law of Large Numbers asserts that $i) E(X_1)=0 \Rightarrow (S_n/n) \rightarrow 0$ The stronger result is: $ii) E(X_1)=0 \Rightarrow E(\|S_n\|)=o(n)$ Now ii) is ...
0
votes
2answers
48 views

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

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$ \int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
1
vote
0answers
35 views

Find $p$ such that $f$ defined by $f(x)=(|x|^\frac{d}{2}+|x|^d)^{-1}$ is in $L^p(R^d)$

The function f is definited in $\Bbb R^d$ by $$f(x)=\frac{1}{(|x|^\frac{d}{2}+|x|^d)}$$ How do you find $p\in[0.+\infty]$ such that $f(x)\in L^p$ ? Any help is appreciated. My attempt: $r=|x| $, ...
1
vote
1answer
46 views

weak* convergence in Sobolev space

Consider $\Omega \subset \mathbb{R}^{N}$, open and bounded. If $u_{n} \rightharpoonup^{*} u$ in $W^{1,\infty}_{0}(\Omega)$, then does it follow that $u_{n} \rightharpoonup^{*} u$ in ...
1
vote
0answers
28 views

A interesting question about moments.

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
4
votes
2answers
120 views

Convergence in distribution to derive the expectation convergence

If $X_n\longrightarrow X$ in distribution, $\mathbb{E}(X)\lt\infty$, Do we have the following conclusion: $\mathbb{E}(X_n)\longrightarrow\mathbb{E}(X)$?
1
vote
0answers
30 views

A question about weak convergence of random variables

I am reading my lecture notes and our definition of weak convergence or random variables is: First another definition: A sequence $\mu_n$ of probability measures on $\mathbb R$ converges weakly to a ...
3
votes
1answer
49 views

$X_n \stackrel{d}{\to}X$, $Y_n \stackrel{d}{\to} c \implies X_n+Y_n \stackrel{d}{\to} X+c$

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show that $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
3
votes
1answer
30 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
3
votes
1answer
48 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 ...