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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2
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1answer
46 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
0
votes
2answers
35 views

Weak convergency vs strong convergency in Hilbert space

Let $\mathcal{H}$ be an Hilbert space and let $(x_n)_n \subset \mathcal{H}$ be a sequence s.t. $$ x_n \rightharpoonup x ~~~,~~~ \| x_n \| \to \|x\| $$ We want to show that $ x_n \to x $. Now, I ...
1
vote
2answers
100 views

Characterization of the weak topology

In our functional analysis lecture we defined the weak topology in a what seems to me like a non canonical way, i.e. not as unions of finite intersections of preimages of open sets in the underlying ...
1
vote
1answer
21 views

Does weak-$\ast$ convergence with an exponential rate imply convergence of measures of sets with the same rate?

Assume that $\mu_n \to \mu$ in the weak-$\ast$ topology with the following rate for any compactly supported continuous function $f$: $$|\mu_n(f) - \mu(f)| \leq C_f e^{-n}.$$ Can we replace $f$ with ...
4
votes
0answers
40 views

Uniform convergence of characteristic functions implies uniform convergence of distribution

Let $F(x)$ and $(F_{n})_{n\geq 1}$ be some distribution functions and let $\varphi(t)$ and $(\varphi_{n})_{n\geq 1}$ be their respective characteristic functions. I am trying to show that if: $\sup_t ...
2
votes
0answers
39 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
2
votes
1answer
29 views

Is the regularity of $u$ necessary to deduce this result? (Evans PDE)

One of the exercises in Evans book on PDEs (at the end of chapter 7) is given as follows: Assume $$u_k\rightharpoonup u\quad\mbox{in}\quad L^2(0,T;H^1_0(U)),$$ $$u_k'\rightharpoonup ...
1
vote
1answer
19 views

multivariate convergence in law

Suppose $X_n \overset{\mathscr{L}}{\longrightarrow} X$, and $Y$ is another random variable which may be depending on $X_n$. Then it seems not true that we have the following joint convergence in law ...
1
vote
1answer
46 views

$(T_n)_{n\in\mathbb{N}}\subseteq L(H)$, $T_n\to T$ weak, why does there exist $C>0$ such that $\|T_n\|\le C$ for all $n\in\mathbb{N}$?

Let $H$ be a Hilbert space, $(T_n)_{n\in\mathbb{N}}\subseteq L(H)$ a sequence such that $T_n^*=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a map $T\in L(H)$ such that $T^*=T$ ...
2
votes
2answers
51 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
1
vote
0answers
13 views

convergence in law for two index

I have a question involving the a sort of central limit theorem which has two index. Suppose that $\{X_{n,m}\}$ is a sequence of random variables which satisfies $$X_{n,m} ...
0
votes
1answer
12 views

If $u_n \rightharpoonup^* u$ in $L^\infty$, does $\int u_n^+ \to \int u^+$?

Let $\Omega$ be bounded. Suppose that $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$ and $u_n \to u$ in $H^{-1/2}(\Omega)$ (that is negative a half, not a typo). Does this somehow imply that ...
3
votes
1answer
37 views

On a problem of weak convergence for a particular convolution of pr. measures

Assume that $\{P_n: n \in N\} $ and $\{Q_n: n \in N\} $ are sequences of probability measures. Assume that $P_n \stackrel{w}{\to} P. $ Also, assume that $Q_n = \delta_{b_n}, $ the Dirac measure and ...
2
votes
0answers
46 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to ...
4
votes
0answers
53 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{Z}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
3
votes
0answers
46 views

The Weak topology on an infinite-dimensional space is not metrizable

Let $X $ be an infinite-dimensional normed space I want to prove that weak topology on $X$ is not metrizable, this is my solution Assume that there is a metric $d$ on $X$ such that induced weak ...
1
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1answer
28 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
2
votes
1answer
29 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
4
votes
1answer
39 views

Weak compactness of a set of translates in $C_0(\mathbb{R})$

Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$. I want to show that $A$ is relatively compact in the ...
2
votes
1answer
58 views

The sequence $f_n=x^n$ is not weakly convergent in $C[0,1]$

Let's consider the sequence $f_n=x^n$ for $n \in \mathbb{N}$ in $C[0,1]$ equipped with the usual supremum norm. How can we show that $f_n$ does not converge weakly in $C[0,1]$ without using an ...
0
votes
1answer
102 views

Weak convergence and convergence almost everywhere

If a bounded sequence $(u_n)$ converge weakly to $u$ in $W^{1,p}(\Omega)$ (Where $\Omega$ is an open bounded from $\mathbb{R}^N$ and $N>p$) Have we that $u_n(x)$ converge to $u(x)$ almost ...
6
votes
1answer
66 views

Do we need completeness for a weak*-convergent sequence to be bounded?

Let $(\phi_n)_n$ be a weak* convergent sequence in the dual of some normed space $X$ with (weak*-)limit $\phi$. If $X$ is Banach then it follows from the uniform boundedness principle that $\sup_n ...
3
votes
2answers
31 views

Absolutely Continuous Weakly Convergent Sequence Need Not Converge Strongly

The following appears as an exercise in Sinai and Koralov's Theory of Probability and Random Processes. Give an example of a family of probability measures $P_{n}$ on ...
1
vote
1answer
41 views

$(X_n)_{n\in\mathbb{N}}$ independent Cauchy-distributed random variables. Convergence of $n^{-\gamma}(X_1+\cdots+X_n)$

I want to solve the following exercise but i am unsure if my ideas are correct or not. Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables with probability density $$ ...
1
vote
1answer
47 views

Continuous mapping theorem - counterexample

The continuous mapping theorem states that Let $g: R^n \rightarrow R^k $ be continuous in every point of a set $C$ such that $\mathbb P\left(X\in C\right)=1$. If $X_n \xrightarrow{d} X $ then ...
3
votes
0answers
39 views

Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq ...
3
votes
1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
1
vote
1answer
26 views

Weak convergence of a double sequence of random variables

Consider two sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Let's assume $X_n\xrightarrow{D} X$, $Y_n \xrightarrow{D} Y$, and $\{X_n\}$ and $\{Y_n\}$ are independent of each other. It is ...
3
votes
0answers
62 views

A sequence of positive finite measures has a positive measure in the weak limit.

I think the statement in the header is true, but I haven't been able to find a proof for it. Consider the measurable space $(\mathbb{R},\mathcal{B})$, i.e. the real number line with the Borel ...
0
votes
1answer
19 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
0
votes
1answer
25 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
1
vote
0answers
41 views

Prove that $\sin(n\pi x)$ weakly converges to $0$ in $L^2(0,1)$ [duplicate]

Let $$f_n(x):=\sin(n\pi x)\;\;\;\text{for }x\in (0,1)$$ and $$\langle f,g\rangle:=\int_{(0,1)}fg\;d\lambda^1\;\;\;\text{for }f,g\in L^2(0,1)$$ I want to show, that $(f_n)_{n\in\mathbb{N}}$ weakly ...
2
votes
1answer
27 views

Convergence in distribution with finite mean

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with another one of the exercises our professor gave us. Let $X_n, n=1,2,\ldots,$ and $X$ be nonnegative ...
0
votes
1answer
32 views

Convergence in distribution and weak convergence implies convergence of expectations

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with one of the exercises our professor gave us. Let $x_n, n=1,2,\cdots$, and $X$ be random variables ...
1
vote
2answers
46 views

Strong convergence from weak convergence

I am trying to show that a sequence $(x_n)_n \subseteq \mathcal{H}$ converges strongly to $x$ if it converges weakly to $x \in \mathcal{H}$ and $\|x_n\| \to \|x\|$ as $n \to \infty$ $\mathcal{H}$ is ...
3
votes
1answer
31 views

Convergence in probability, in the sense of weak convergence of measures

I am reading a paper where the author has a family $(\rho_t : t \geq 0)$ of random probability measures (on the real line with Borel sigma-algebra), and a measure $\rho$. One of his theorems says that ...
3
votes
2answers
75 views

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of ...
2
votes
0answers
28 views

Does almost sure ordering, same distributional convergence imply convergence in probability?

Suppose $X_n$, $Y_n$ are real valued random variables such that $X_n \leq Y_n$ a.s. If $X_n \xrightarrow{d} X$, and $Y_n \xrightarrow{d} X$ can I say that $Y_n-X_n\xrightarrow{P}0$?
2
votes
2answers
28 views

Prove that $S_n/\sqrt{n}-S_{4n}/\sqrt{4n} \overset w \longrightarrow \mathcal{N}(0,\sigma^2)$

For $X_1, X_2,\ldots$ i.i.d., $\mathbb{E}[X_1]=0$ and $\mathbb{E}[X_1^2]=\sigma^2$ it should be possible to prove $$\frac{S_n}{\sqrt{n}}-\frac{S_{4n}}{\sqrt{4n}} \overset w \longrightarrow ...
1
vote
1answer
64 views

Why is the strong law of large number stronger than weak law? [closed]

The weak law is easy to prove, but the strong law (which of course implies the weak law, by Egoroff’s theorem) is more subtle. I'd like to know for which mathematical reason is the strong law ...
2
votes
0answers
33 views

Why is convergence of measures tested against functions?

This question is to help my intuition. Why do we test the convergence of measures against different classes of functions and not use definitions like: If $(B,\mathcal{B})$ is a measurable space then ...
1
vote
1answer
75 views

Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$ [duplicate]

I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$. We know trivially that $0 \in L^2(0,1)$. I need to show that $\langle u^*,\sin(kx) \rangle \to \langle ...
1
vote
1answer
65 views

Prove that the sequence is a weakly convergent sequence

I am trying to solve the following exercise: Fix $a,b \in \mathbb{R}^n, 0 < \lambda < 1.$ Define $$ u_k(x)= \left\{\begin{array}{ccc} a & \text{if } & \frac jk \leq x < ...
0
votes
1answer
24 views

Question about a bounded sequence in $W^{1,1}$ admitting a weakly convergent subsequence

Let $B$ be the unit ball in $\mathbb{R}^{12}$ and $f_n\in L^7(B)$ such that $\lVert f_n \rVert_{W^{1,4}(B)}$ is bounded. Is it true that there exists a subsequence weakly convergent in $W^{1,1}(B)$? ...
1
vote
2answers
27 views

Does $a_n \in H^1(\mathbb R^n)$ and $b_n \rightharpoonup 0$ in $H^1(\mathbb R^n)$ imply $\langle a_n, b_n \rangle \to 0$?

I have a question mainly in functional analysis. Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows: Does $a_n \in H^1(\mathbb R^n)$, $|a_n| ...
4
votes
1answer
32 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...
1
vote
0answers
16 views

multivariate convergence in distribution for Skorohod topology

If $X_n$ is a $R^d$-valued sequence of stochastic processes and $X$ is a $R^d$-valued the limiting process, all having Cadlag paths. I know what it means by $X_n \Rightarrow X$, i.e. $X_n$ converges ...
2
votes
1answer
53 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
2
votes
1answer
50 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
1
vote
1answer
73 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...