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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1answer
14 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
0
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1answer
70 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
2
votes
1answer
48 views

Missing crucial step in the derivation of the Stirling's Formula via the Poisson Distribution and CLT

I believe that this is a particular neat example that we've done in class. Unfortunately there is one step I do not quite understand and my Professor had to skip due to the lack of time. I think this ...
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vote
1answer
21 views

Showing weak law of large numbers holds

My question: $\{X_n\}$ is a sequence of random variables. Var$(X_n)\le C\ \ \forall \ n$ and $\rho_{ij}=$Cov$(X_i,X_j)\to 0 $ as $|i-j|\to \infty$ . Show WLLN holds. In my book there are 3 ...
0
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1answer
33 views

Is any closed ball compact in the Weak$^*$ topology $\sigma(E^*,E)$ for a Banach Space $E$?

For a Banach Space $E$, the Banach Alaoglu Bourbaki theorem asserts that the closed unit ball in $E^*$: $$B_{E^*}= \{f \in E^* \ | \ ||f|| \leq 1 \} $$ is compact in the weak$^*$ topology $\sigma(E^*...
3
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1answer
73 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
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1answer
32 views

Convergence in probability of a sum of dependent random variables to 0 [closed]

Suppose we know that $A_n + B_n \xrightarrow{p} 0$ as $n \rightarrow \infty$, and we know that $A_n \xrightarrow{d} N(0,1)$. Can we say that $B_n \xrightarrow{d} N(0,1)$? Note that $A_n$ and $B_n$ are ...
3
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1answer
40 views

A problem of nets in topology

Let $G$ be a topological group with neutral element $e$. Let $\pi \colon G \to B(E)$ a (non-continuous) representation of $G$ on a Banach space $E$ by bounded linear operators. Let $T$ an element of ...
2
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1answer
30 views

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Proof of lemma $9.7$ in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations argues as follows: For an element $u \in H_0^1(\Omega)$ we define $D_h u= \frac{u(x+h)-u(x)}...
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0answers
29 views

CLT for continuous functions of random variables

Let $(X_i)$ be a collection of zero mean, unit variance, real valued random variables (I do not assume that they are iid). Let $\mathcal H$ be a separable RKHS with a bounded kernel $k(x,y)$. Suppose ...
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0answers
43 views

Weak convergence of Poisson distributed random variables

I am stucked in the middle of an exercise: Let $$X_n,Y_m$$ independent random variables having the Poisson distribution with parameters n and m respectively. Show that $$\frac{(X_n-n)-(Y_m-m)}{\sqrt{...
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0answers
47 views

Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. (...
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0answers
21 views

Weak-topology separates points on $X$.

Let $X$ be a normed vector space and $X^{\ast}$ be its dual consisting of all continuous functionals on $X$. I must show that weak topology separates points on X, i.e. (?) that for any $x,y \in X$ s....
3
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0answers
24 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures $P^{...
0
votes
1answer
27 views

Equivalence of norm and weak topologies in finite dimensional space.

I haven't solved a lot of problems on functional analysis, so I don't have intuition whether my idea of proof is correct or not. The weakness of weak-topology is obvious from the definition. To show ...
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3answers
38 views

Every bounded sequence of dual space contains a subsequence which is weak* convergent

I were doing this problem in Functional Analysis of Erwin Kreyszig(part 4.9, problem 10, page 269), but got stuck in the last point to come to the conclusion. Can anyone give me some hint to move on? ...
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1answer
23 views

Prove that a set which satisfies every nonempty subset contains a weak Cauchy sequence must be bounded

I got stuck on this problem about weak convergence in normed space. This problem is exercise 9, page 263 in Functional Analysis of Erwin Kreyszig. So the question is basically on the title, let me ...
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2answers
53 views

Student test statistic and self normalizing sum.

I study the asymptotic distribution of self normalizing sums which are defined as $S_n/V_n$ where $S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$ for some i.i.d RV's $X_i$. Motivation ...
7
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1answer
82 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
3
votes
1answer
43 views

Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
2
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1answer
37 views

Q: Exercise 3.10 in Brezis' Functional Analysis

I'm trying to solve exercise 3.10 (page 81) from H. Brezis' "Functional Analysis, Sobolev Spaces and PDE". Let be $E$ and $F$ Banach space such that $T\in \mathcal{L}(E,F)$ and $T^*\in\mathcal{L}(F^*,...
0
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1answer
21 views

Distribution of the radius of a circle uniformly on a square

Suppose U1, U2, . . . , V1, V2, . . . are independent Uniform(1, 1) random variables, so that (U1, V1), (U2, V2), . . . are independent points distributed uniformly over the square $[-1,1]^2$. Let Rn ...
2
votes
1answer
44 views

Are probability measures weak-* closed?

Non-duplicates This is in a different setting, and this only deals with compact spaces which is the easy case. Now for the question. Let $X$ be a locally compact Hausdorff space. $\mathcal{C}_0(X)$ ...
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1answer
23 views

Lower-semicontinuity of quadratic functionals and weak convergence weirdly associated

Each random variable (rv) considered here is absolutely continuous. Let $f_{T}$ be a bounded random variable and $T>0$. Suppose we have: $$f_{T}\rightharpoonup f\tag{1}$$ weakly in probability, ...
3
votes
1answer
28 views

Finding a random variable

Let $X_{1}, X_{2}, \dots$ be i.i.d Uniform[0,1] random variables. Find a random variable $X$ such that: $$(X_{1}\times\cdots\times X_{n})^{\frac{1}{\sqrt{n}}}e^{\sqrt{n}} \overset{d}{\to} X$$ The ...
2
votes
2answers
43 views

Weak and strong convergence in $L^p$

Another practice qual question: Let $X = [-\pi,\pi]$ and consider the Lebesgue measure. Let $p$ be a real number with $1 \leq p < \infty$. Define for each integer $k \geq 1$ that $f_k(x) = \sin(...
1
vote
1answer
16 views

A weaker characterization of convergence in distribution

$X_n$ and $X$ are real-valued random variables. If $X_n \to X$ in distribution, then we know that $$P\{X_n \leq a\} \to P\{X \leq a\}$$ for every $a$ at which $x\mapsto P\{X \leq x\}$ is continuous. ...
0
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1answer
52 views

Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x $ in the ...
0
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1answer
43 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of $X'...
2
votes
1answer
57 views

Does weak convergence ($X_n\Rightarrow X$) imply weak convergence of the difference to zero ($X_n-X\Rightarrow 0$)?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am wondering if this implies that $X_n-X\Rightarrow 0$ as $n\rightarrow ...
1
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1answer
49 views

Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty $ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
3
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1answer
59 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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0answers
29 views

On the completeness of Weak Operator Topology

Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...
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0answers
35 views

Weak convergence in probability and functional analysis

Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have $$\int\...
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0answers
38 views

Continuity in dual space with weak$^*$- topology

Let $X$,$Y$ be locally convex topological vector spaces. Assume now I have an operator $T:Y'\rightarrow X'$ where $Y'$ and $X'$ are equipped with the weak$^*$-topology. Does this imply that $T$ is ...
1
vote
1answer
43 views

Does this joint weak convergence result hold?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am trying to determine the most general conditions on a function $f$ ...
0
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1answer
28 views

(Uniform) continuity of push-forward operator

I am wondering about extension of the the answer given here. Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: \...
2
votes
1answer
26 views

Variance converging to zero implies weak convergence to delta measure?

I have the following question: Suppose that I have a sequence of random variables $X_n$ such that all moments exists and are finite. I have that $E[X_n]\to a$, where $a$ is a finite number and ...
0
votes
1answer
46 views

An example of a sequence of functions that is not pointwise convergent

I've been searching for an example of a sequence $f_n(x)$ of functions that is not pointwise convergent, i.e.: $$\lim_{n\rightarrow \infty}\left | f_n(x) - f(x) \right | = 0$$ but I cannot find one.
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0answers
46 views

Notion of conditional weak convergence

I am looking for references or lecture notes which define the notion of conditional weak convergence of a sequence of random variables. In the case of (usual) weak convergence, we say that a sequence $...
3
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1answer
73 views

Are the weak* and the sequential weak* closures the same?

I have a question that might be easy for the experts. Let $E$ be a Banach space (non-separable) and let $E'$ be its dual space. Suppose that $X\subset E'$ and assume that $X$ is separable with ...
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2answers
64 views

Characterization of weak convergence in Lp

Weak convergence in $X=L^p(0,1)$ for $1<p<\infty$ can be characterized as following: $f_n\rightharpoonup f$ if and only if $f_n$ is bounded in $X$ and $\int_{(0,t)}\;f_n\;\rightarrow \int_{(0,t)}...
3
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1answer
65 views

Weak convergence in $W^{1,p}(\mathbb{R})$

Let $\varphi\in\mathcal{C}^\infty_0(\mathbb{R})$ a map with compact support. For all $n\in\mathbb{N}$, I define $u_n$ by $u_n(x)=\varphi (x+n)$. I would like to prove that $u_n$ converges weakly ...
1
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1answer
38 views

Weak convergence preserver pointwise inequality

The proof of boundedness of Hardy-Littlewood maximal function in Sobolev spaces in Kinnunen's paper has the following argument: "... Hence $(v_k)$ is a bounded sequence in $W^{1,p}(\mathbb{R}^n)$ ...
3
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0answers
75 views

Weak-* bounded, closed convex set is compact?

Suppose $E$ is a Banach space, and $K\subseteq E^*$ is convex, and is closed and bounded with respect to weak-* topology. Is it true that $K$ is compact? If $E$ is reflexive, then this is the case, ...
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1answer
50 views

If $u_n \rightharpoonup u$ in $H_0^1(\Omega)$, what can we say about $\{\nabla u_n\}$?

I have a bounded sequence $\{u_n\}$ in $H_0^1(\Omega)$. Then, a general result about Hilbert spaces and weak convergence, implies that there exists a subsequence $\{u_{\sigma(n)}\}$ in $H_0^1(\Omega)$ ...
2
votes
1answer
139 views

How to prove convergence of process and stopping time

We consider the random function $X^n=(X^n_t)_{t\geq 0}$ with values in the Skorokhod space $\mathcal{D}$ of càdlàg paths, and suppose that it weakly converges (i.e in distribution) to $X=(X_t)_{t\geq0}...
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2answers
29 views

Problem on moment convergence for sufficiently tight measures.

Let $\mu_n$ and $\mu$ Borel probability measures on $\mathbb{R}_+$ such that $\mu_n \Rightarrow \mu$ (converges weakly). Show that if $$ \sup\limits_{n \geq 1} \int x^a d \mu_n(x) <\infty $$ for ...
2
votes
1answer
36 views

Characteristic functions of infinite dimensional random elements

I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that $\{X_n:n\ge1\...
0
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1answer
38 views

Does there exist an unbounded function $f$ such that for all $\mu_n$ converging to some probability measure $\sup_n \int f(z) \mu_n(dz) <\infty$

Does there exist an unbounded positive function $f$ such that for all sequence of probability measures $\mu_n$ converging to some probability measure $$\sup_n \int f(z) \mu_n(dz) <\infty$$