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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Dual of completion and weak$^*$-topology

Let $X$ be a dense subspace of a Banach space $Y$. The restriction map $$r: Y^* \to X^*:\omega \mapsto \omega \vert_X$$ is then an isometric isomorphism that is weak$^*$-continuous. I am wondering if ...
Andromeda's user avatar
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Convergence in Distribution of a Particular Sample Average

Suppose $g_{n}(\cdot)$ defined on $[0,1]$ converges in distribution to a continuous Gaussian process. Let $U_{1},...,U_{n}$ be i.i.d. random variables following $\text{Unif}[0,1]$. Allow $g_{n}$ to ...
Ecthelion's user avatar
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Sequence of functions that converges strongly in $L^3$, weakly in $L^2$ and not strongly in $L^2$

How can I construct a sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f_n \overset{L^3}{\to} 0 \\ f_n \overset{w-L^2}{\to} 0 \\ f_n \overset{L^2}{\not\to} 0 $$ I know ...
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Total variation convergence in context of stochastic processes

Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution $$X_t~\...
Oskar Vavtar's user avatar
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Weak convergence of dependent variables

$X_n \xrightarrow[]{d} X$, $Y_n \xrightarrow[]{d} Y$ where $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$, but $X_n \not\!\perp\!\!\!\perp Y_n$. What do we need to analyze $(X_n, Y_n) \...
Sicco Kooiker's user avatar
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weak convergence in an unit ball.

let $x_n \rightarrow x$ weakly and $x_n \in \overline{B(0,1)}$ for all $n$ and $\|x\|_H = 1$ then $\|x_n-x\|_H \rightarrow 0$. We can write $\|x_n -x \|_H^2 = <x,x>+<x_n,x_n> -2Re<x_n,x&...
voroshilov's user avatar
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Reference for a good multidimensional portmanteau theorem

I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions: The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$; $...
MikeTeX's user avatar
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2 votes
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Is it true that any sequence of random variables that converge in distribution is tight?

Let the sequence of real random variables $\{X_n\} \to X$ in distribution (but not necessarily in probability). Is it true that $\{X_n\}$ form a tight sequence, and if yes, how do we prove it? So we ...
Learning Math's user avatar
1 vote
1 answer
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Definition of a weakly continuous map

I am familiar with the weak topology and a linear functional on a Banach space being weakly continuous, but in PDE I often see a statement along the lines of "$f$ is a weakly continuous solution ...
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$(X_n, Y_n) \to (X, Y)$ in distribution (Le Gall 10.6)

$$ \newcommand{\N}{\mathbb N} $$ I am paraphrasing this textbook question slightly. Question: Let $(X_n)_{n \in \N}$ and $(Y_n)_{n \in \N}$ be two sequences of real random variables, and let $X$ and $...
caitlin's user avatar
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Convergence in distribution in more dimensions

The general definition of convergence in distribution is given by: Let $(X^{(n)})_{n\in\mathbb{N}}$ be a sequence of random variables taking values in a seperable metric space $(S,d)$. We say that $(...
yannik0103's user avatar
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weak convergence imply expectation $\le$ lower limit of sequence expectation

$(X_{n})$ are uniformly integrable stochastic process ,$X_{n}\xrightarrow{d}X$ Proof $E[\left | X \right |]\le \liminf_{n\to\infty}E[\left | X_{n} \right |]$ I see a way that $f_{m}(x)=\begin{cases}\...
Yu GongLian's user avatar
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If whenever $\psi_n\rightharpoonup 0$, $A\psi_n \rightharpoonup 0$ then $A$ is bounded.

This if from a previous Princeton exam on functional analysis Let $A$ be a linear operator $A: X\rightarrow Y$ between normed vector spaces. If $\psi_n\rightharpoonup 0$ implies $A\psi_n \...
Kadmos's user avatar
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How does the weak topology imply the definition of weak convergence?

Let $X$ be a Banach space. Then the weak topology on $X$ is defined as the coarsest topology such that the linear functionals in $X^*$ are continuous. A sequence $x_n \in X$ weakly converges to $x$ if ...
CBBAM's user avatar
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Prove weak convergence of probability measures

Let $Q \in P(P(\Sigma))$ be given, whereby $\Sigma$ is a Polish space (complete and separable metric space) and $P(\Sigma)$ denotes the space of probability measures on $\Sigma$ and $P(P(\Sigma))$ the ...
user996159's user avatar
2 votes
1 answer
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Weak topology on Baire measures "obviously" Hausdorff

In the book "Weak Convergence of Measure" by Bogachev the Baire $\sigma$-algebra $\mathcal{B}a(X)$ of a (Hausdorff) topological space $X$ is defined as the smallest $\sigma$-algebra ...
Botwinnik's user avatar
1 vote
1 answer
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Weak convergence for infinite sum of random variables

Given two independent sequences of random variables $(X^1_k)_{k=1}^\infty, (X^2_k)_{k=1}^\infty,$ respectively converging to $X^1$, $X^2$ in distribution, we know that $(X^1_k + X^2_k)_{k=1}^\infty$ ...
user808843's user avatar
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1 answer
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Prove that simple functions are not dense in weak $L^{p,\infty}$

Let $p > 0$, and denote by $L^{p,\infty}(\mathbb{R})$ the space of all measurable functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which $$ \|f\|_{p,\infty}:=\sup_{\alpha > 0} \alpha^p |\{x ...
Mr. Proof's user avatar
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2 votes
2 answers
151 views

Showing weak convergence with a trick

I wonder if my reasoning works: Given $1 < p < \infty$ let $(a_n)$ be a sequence of real numbers and define $f_n(x) = a_n$ if $x \in [n,n+1]$ and $0$ otherwise. I claim that if $(a_n)$ is ...
user57's user avatar
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Almost sure convergence of cdf implies uniform convergence for multivariate random vectors

Let $X_1, X_2, \dots \in \mathbb{R}^d$ be random vectors, each with cdf $F_n$. Let $F$ denote the cdf of another random vector $X$. Suppose they are all continuous w.r.t. Lebesgue measure for now. I ...
statstats's user avatar
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1 answer
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Hilbert Space, $f_{n}$ converges weakly to $f$, then $f$ is in the closure of $\text{span}(f_n)$

Let $H$ he a Hilbert space. Let. ${f_{n}}$ be a sequence in $H$. a) Show that if $e_{n}$ is any orthonormal sequence in H, then $e_{n}$ converges weakly to $0$. Sol: For any $g \in H$, $\langle e_{n},...
wolf_pack_32's user avatar
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Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

Assume $(\Omega, \mu)$ is a probability space. Consider a class of functions $S$ all contained in the unit ball of $L^\infty(\Omega, \mu)$. Let $f \in L^\infty(\Omega, \mu)$ be contained in the weak $...
David Gao's user avatar
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Does almost sure convergence of parameters of a parametric probability distribution imply pointwise convergence of the characteristic function?

Consider a family of distributions $\mathcal{P}=\{P_{\theta}: \theta \in \mathbb{R}^d\}$, with identifiable parameterization. If a sequence of estimators $\hat{\theta}_n \overset{a.s.}{\to} \theta_0$, ...
statstats's user avatar
4 votes
2 answers
220 views

Definition of weak convergence

Weak convergence came up in my PDE class and I'm trying to understand it even though I lack background in topology and functional analysis. Please check if my understanding is correct. Definition: Let ...
Len's user avatar
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Is strict positivity preserved by weak convergence?

If $\{u_n:\Omega\to \mathbb{R}\}$ is a continuous function sequence such that $u_n>0$ for all $n\in \mathbb{N}$ and suppose that $u_n$ converges weak to $u \in L^2(\Omega)$. Can I obtain $u>0$?
Luiza Camile's user avatar
1 vote
0 answers
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Metric for weak convergence on space of non-negative measures

Let $X$ be a Polish space, let $\mathcal{M}_+(X)$ be the space of non-negative measures on $X$. What is the metric that metrizes topology of weak convergence (i.e. where test functions are bounded ...
lulli_'s user avatar
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2 votes
1 answer
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Sequence with bounded $L^p$ norm which converges in measure also converges weakly

Let $\{f_n \}_{\mathbb{N}}$ be a bounded sequence in $L^p(X, \Sigma, \mu)$ (i.e, there exists $M > 0$ such that $\|f_n \|_{L^p} \leq M$ for all $n \in \mathbb{N}$), where $(X, \Sigma, \mu)$ is a ...
Matheus Andrade's user avatar
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1 answer
106 views

A relatively compact family of probability measures that is not tight

We know, thanks to Prokhorov's theorem, that a family of probability measures $\mathcal{H}$ in a metric space $X$ is tight if, and only if, it's relatively compact whenever $X$ is polish. The ...
Abito's user avatar
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2 votes
0 answers
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showing almost convergence of product of random variables

Let $\left(X_j\right)_{j \geq 1}$ be i.i.d. positive with $\log X_j \in \mathrm{L}^4$. Prove that $\left(\prod_{j=1}^n X_j\right)^{1 / n}$ converges almost surely as $n \rightarrow \infty$. Does it ...
noidea's user avatar
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0 answers
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Possibility that Sum of Random Variable Converges Weakly to Uniform Distribution

Suppose $\{\xi_n\}$ are independent random variables such that $|\xi_n|\leq C$, $\forall n$. They also have densities $p_n(x)$, and $|p_n(x)|\leq C$, $\forall n,x$. Is it possible that $\sum\xi_n$ ...
喵喵露's user avatar
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1 vote
0 answers
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Domain of attraction of $\alpha$-stable distributions

I have been trying to understand why in Remark 2.4.1 c) of these lecture notes, $a_n$ can be chosen as $0$ for $\alpha\in(0,1)$. So I have a distribution function $F$ given which satisfies $1-F(x)+F(-...
yannik0103's user avatar
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1 answer
38 views

Sequence of expectation doesn't converge despite weak convergence

Can someone give example where sequence of random variable $X_{n}$ converge in distribution to random variable $X$. $\sup E(X_{n})< \infty$ over all $n$, $\lim_{ n \to \infty} E(X_{n})\neq E(X)$?
Document123's user avatar
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Summation of random variables converging to exponential

I want to know for what iid random variables $X_i$, the sum, $\sum_{i=1}^n\frac{X_i}{n^\alpha}$ for some $\alpha$ converges in distribution to an exponential distribution as $n \to \infty$. From this ...
stochs's user avatar
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4 votes
0 answers
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Central limit theorem on small scales

Say $X_1,\ldots,X_n$ are iid, bounded, symmetric random variables with mean $0$, variance 1 and a smooth density. At what scale does $\frac{X_1+\cdots+X_n}{\sqrt{n}}$ "look like" the ...
euler_pi_i's user avatar
1 vote
1 answer
49 views

Weak convergence argue

If we take a sequence $\{u_n\}$ that converges weak to u in $L^2(\Omega)$, where $\Omega$ is bounded and $g_n\to g$ weak-* in $L^{\infty}(\Omega)$ then how can I obtain this limit, for all $\varphi \...
Luiza Camile's user avatar
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0 answers
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Can $f(Ax_k)-f(Ax^*)\le o(1/k)$ derives $f(x_k)-f(x^*)\le o(1/k)$?

$\{x_k\}$ is a sequence which converges to $x^*\ (when\ k\to \infty)$, and $f$ is a convex and L-lipschitz smooth function, that means, for $\forall x,y\in dom(f),|\nabla f(x)-\nabla f(y)|\le L|x-y|$. ...
QX_Sen's user avatar
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3 votes
0 answers
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Normal approximation for sum of dependent indicators from triangular array

Suppose there is a triangular array of random variables $X_{k,n}$ with the following properties: $$ \mathbb{P}(X_{k,n} = 1) = p_n, \quad \mathbb{P}(X_{k,n} = 0) = 1 - p_n, \quad \lim_{n\to\infty}p_n =...
Yalikesifulei's user avatar
5 votes
1 answer
98 views

Weak convergence implies convergence in probability?

I'm striving to understand a proof from a paper. Denote the symbol $\rightsquigarrow$ as a weak convergence, and $l^{\infty}(\Omega)$ as the space of bounded functions on a metric space $(\Omega, d)$. ...
d8g3n1v9's user avatar
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necessary and sufficient condition about weak convergence on N (the set of natural numbers)

I would like to show that if all natural number $k$ satisfy that $\displaystyle\lim_{n\to\infty}\mu_{n}(\{k\})=\mu(\{k\})$, then $\{\mu_n\}_{n=1}^\infty$ converges on $\mu$ weakly, where $\{\mu_n\}_{...
lymd's user avatar
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2 votes
1 answer
63 views

Proof of weak convergence when domain of rv is N

I need to show: If $X_i, i\geq1, X$ are random variables with domain $\mathbb{N}$, then $ X_n \rightarrow X$ weakly iff $\forall i \in \mathbb{N}: P(X_n = i) \rightarrow P(X=i)$. The direction $\...
num2333's user avatar
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0 answers
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Exercise 3.1.23 in Salamon's Functional Analysis

I'm reading the book Functional Analysis written by Dietmar A. Salamon.The following problem is the exercise 3.1.23 in the book. Let X be a Banach space and suppose the dual space of X is separable....
L Zhang's user avatar
1 vote
1 answer
31 views

Convergence of a sequence in $L^2(\mathbb{R}^N)$ that is also bounded in $H^1(\mathbb{R}^N)$

Let $(u_n)_n\subset H^1(\mathbb{R}^N)$ be a sequence and $u\in L^2(\mathbb{R}^N)$ verifying: $(u_n)_n$ is bounded in $H^1(\mathbb{R}^N)$ $(u_n)_n$ converges to $u$ in $L^2(\mathbb{R}^N)$ Then, does $...
UCL's user avatar
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1 vote
0 answers
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The equivalence definitions of convergence in distribution

There is a statement about the equivalence properties of weak convergence in wikipedia: In the case $S$ ≡ $\mathbf{R}$ with its usual topology, if $F_n$ and $F$ denote the cumulative distribution ...
jcm22's user avatar
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0 votes
1 answer
101 views

Does $\|f_n -f\|_{L^2} \to 0$ imply $f_n \to f$ in the weak topology of $H^1 (I)$?

Let $I$ be the open interval $(0, 1)$. Let $f, f_n \in H^1 (I)$. We consider (S1) $f_n \to f$ in the weak topology of $H^1 (I)$. (S2) $\|f_n -f\|_{L^2} \to 0$. Is it true that (S1) $\implies$ (S2) ...
Akira's user avatar
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2 votes
0 answers
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Exercise 3.19 in Brezis' Functional Analysis

Exercise 3.19: Let $E = \ell^p$ and $F = \ell^q$ with $1 < p < \infty$ and $1 < q < \infty.$ Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $$|a(t)| \leq C|t|^{\frac{p}...
D4c's user avatar
  • 59
3 votes
1 answer
85 views

Convergence in distribution of a trig sequence of random variables

Let $\theta_i\sim\mathcal{U}(0,2\pi)$, $i = 1,\dots,n$ be $n$ i.i.d. uniformly distributed random variables. Let: $$ \frac{D_n^2}{n} = 1 + \frac{2}{n}\sum_{i<j}^{n}\cos(\theta_i-\theta_j) $$ I have ...
MathRevenge's user avatar
5 votes
0 answers
46 views

$\alpha$-mixing properties and convergence in distribution

I have a stochastic process $\{W_t\}_{t\geq 1}$, of uncorrelated but not indipendent random variables, with $\mathbb{E}(W_t) = 0$ and $Var(W_t)=\frac{t-1}{2}$ $\forall, t\geq 1$ (The $\{W_t\}_{t\geq 1}...
MathRevenge's user avatar
0 votes
1 answer
53 views

When does convergence in distribution imply convergence of integrals?

Suppose $f_n \rightarrow f$ in distribution, where $(f_n)_{n \geq 1}$ is a sequence of integrable functions. Also suppose $\sup_{n \geq 1} |f_n|$ is well-defined and measurable. Does this imply that $$...
banana_free's user avatar
3 votes
1 answer
57 views

Relationship between strong and weak operator topologies

I am looking for citations in a book or article for the following fact: Let $T_n$ be a sequence of operators in Hilbert space and $T$ also be an operator in Hilbert space. If $T_n$ is convergent in a ...
james stan's user avatar
-2 votes
2 answers
62 views

Sequence in C[0,1] for which the weak limit is 0 but the infinity norm is 1.

I want to find a sequence $(x_n)\in C[0,1]$ for which the weak limit is 0 ($\sigma-\lim_{n\to\infty} x_n = 0$), but $||x_n||_\infty=1$ for all $n$. Here is our definition of the weak limit: Let $(x_n)\...
Sven's user avatar
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