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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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1answer
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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 ...
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1answer
27 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 ...
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2answers
39 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 ...
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1answer
28 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 ...
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2answers
57 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 ...
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0answers
26 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 $Y_n-X_n\xrightarrow{P}0$?
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2answers
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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 ...
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1answer
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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 ...
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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 ...
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1answer
70 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 ...
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1answer
63 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 < ...
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1answer
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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 $W^{1,1}(B)$? ...
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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| ...
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1answer
29 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 ...
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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 ...
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1answer
44 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. ...
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35 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 ...
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1answer
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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 $L^q$ ...
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A question involving weak and strong convergence

Let E be a Banach space, $K \subset E$ a compact subset in the strong topology and $(x_n)_{n \geq 1} \subset K$, $x_n \rightharpoonup x$ weakly in $\sigma (E, E^*)$. If there is a subseqence ...
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1answer
55 views

Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
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1answer
36 views

Dose weak convergence imply tight?

$X$ is a separable metric space, $\{P_n\}_{n=1}^{\infty}$, $P$ are probability measures on $X$, and $P_n$ converges weakly to $P$, can we conclude that $\{P_n\}_{n=1}^{\infty}$ is tight? I know if ...
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Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
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1answer
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Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
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weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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1answer
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A multiple of a characteristic function is the weak limit of a sequence of characteristic functions

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...
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1answer
25 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) ...
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1answer
20 views

Weak convergence equivalent to convergence of distribution functions

I'm trying to understand the following proof: I don't see where the Lipschitz property of the $f$ is needed. Why isn't bounded and continuous enough? Can somebody explain it to me? Thanks.
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1answer
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Convergence in Distribution and Exponential Function

There's a well known fact that if a sequence of real numbers, $\{x_{n}\}$ converges to $x$, then: \begin{equation*} \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x_{n}}{n}\right)^{n} ...
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Understanding Prohorov's theorem

I've seen Prohorov's theorem in the following formulation (I'll cite only one direction). $\mathcal{M}_{\leq 1}(E)$ is the set of subprobability measures on $(E, \mathcal{B}(E))$: Let $(E, d)$ be ...
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Proving the uniqueness of the weak limit

In "A First Look at Rigorous Probability Theory" by J. S. Rosenthal there is the following exercise: Prove that weak limits, if they exist are unique. That is, if $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
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Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
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1answer
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Evaluating limit of a characteristic function (Fourier Transform) in $R^k$

I am trying to evaluate this limit: $$\lim_{n \to \infty} \left[(\text{det}\ \Gamma_n)^{-\frac{1}{2}}\exp \left\{ {-\frac{1}{2}(x-m_n)\cdot (\Gamma_n)^{-1}(x-m_n)} \right\} \right]$$ where ...
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1answer
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$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
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1answer
41 views

About the weak compactness of a certain set.

Why is the following set weakly compact in $L^1(d\mu)$? $$\left\{-\frac{|x|^2}{2}+O(l)\right\}$$ where $\mu$ is a probability measure in $\mathbb{R}^n$ with finite second order moment: ...
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A question involving weak convergence in $\sigma(L^1, L^\infty)$

Exercise 4.15 from H. Brezis - "Functional analysis..." Let I = (0, 1) and $f_n = n e^{-n x}$ a sequence of functions. Show that $$ f_n \;\; \text{does not converge weakly to} \;\; 0 \;\; \text{in} ...
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1answer
99 views

weak convergence lim inf sequence example

I have been stumped by the following problem , I was able to answer the first part of the problem which involved the inequality by using weak convergence. But when it came to looking for such an ...
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1answer
18 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 ...
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42 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 $(x_n)_n$ ...
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1answer
44 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 ...
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1answer
64 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 ...
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1answer
27 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
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1answer
39 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 ...