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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48 views

How to show $\lim_{n \to \infty}f(x_n)=f(x)?$

consider the Banach space $\ell^1$, and let $e_i$ be the sequence $(0,\dots,1,0\dots)$, with $1$ in the $i$-th position. show that $\{e_i\}$ converges weakly to $0$ in $\ell^1$ but not strongly. My ...
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1answer
69 views

$\mu_n\rightharpoonup\mu$ and $\nu_n\rightharpoonup\nu\implies\mu_n\otimes\nu_n\rightharpoonup\mu\otimes\nu$.

Let $\mu_n,\mu\in P(X)$ and $\nu_n,\nu\in P(Y)$ ($X$ and $Y$ are Polish spaces, or if this doesn't work, maybe $\mathbb{R}^d$). I was trying to reproduce the steps in this answer. Since $\mu_n\...
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2answers
45 views

Narrow convergence and support of the limit measure

Let $(X,d)$ be a Polish metric space and $\{\mu_n\}_{n\in\mathbb{N}}$ a sequence of probability measures such that $\mu_n\rightarrow\mu$ narrowly (i.e. $\int_Xf\,\mathrm{d}\mu_n\rightarrow\int_Xf\,\...
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1answer
69 views

Limiting distribution of $Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$ where $X_i$'s are i.i.d $N(0,1)$

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d standard normal variables. I am looking for the limiting distribution of $$Y_n=\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^3}$$ Since $E(X_1^3)=0$, I don't think I ...
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1answer
41 views

Slutsky's theorem for infinite space

Assume that we have a sequences of random elements $X_{n}$ and $Y_{n}$, taking values, for example in the space $\ell_{2}$. Assume that $$ X_{n}\overset{d}{\to} X $$ and $$ Y_{n}\overset{p}{\to} c \in ...
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1answer
23 views

sum of sequences which converge in distribution

Assume that we have two sequences $X_{n}$ and $Y_{n}$, taking values in some Hilbert space, defined on the same probability space and assume that $$ X_{n}\overset{d}{\to} X' $$ and $$ Y_{n}\overset{d}{...
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1answer
18 views

Restrictions of measure converges to original measure?

Let X be a subset of $\mathbb{R}^d$ and $(X,A,\mu)$ be a measure space. Let $K_n$ be compact sets that exaust $X$, i.e. $X=\bigcup\limits_{n\ge 1}K_n$. I was wondering if the restriction $\mu_n:=\...
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1answer
44 views

Asymptotic Independence of Random Variables

Let $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ be two sequences of random variables such that the sequences $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ converge in law to $...
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1answer
77 views

If $X_n$ converges to $X$ in distribution and $a_n \to 0$ then $a_nX_n$ converges to $0$ in distribution. (Counterexample?)

I am enjoying studying probability theory a lot but sometimes the proofs can be a bit too fiddly for me. Here is the question I am working on now: If $X_n$ converges to $X$ in distribution and $a_n \...
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1answer
28 views

If every bounded sequence has a weakly convergent subsequence, then the inner product space is Hilbert

We can prove that in Hilbert space, bounded space has a weakly convergent subsequence, as we can see here or there. Does the converse hold? That is, in an inner product space $H$, if every bounded ...
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0answers
34 views

Weak* convergence and weak convergence in X*

Suppose $X$ is a Banach space and $\{f_n\}$ is a sequence in $X^*$ such that $f_n$ converges weak* to $f\in X^*$ (meaning $\lim_{n\to\infty}f_n(x) = f(x)$ for all $x\in X$), and $\lim_{n\to\infty}\...
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11 views

What is the relation between weak convergence of measure and convergence of their densities?

I have the following question. Let $(\mu_n)_n$ and $\mu$ be measures on $\mathbb{R}^d$ that are absolutely continuous with respect to the Lebesgue measure $\lambda$ with derivative $g_n=\frac{d\mu_n}{...
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30 views

Probability Convergency / Stochastically bounded [duplicate]

We have a random variable $Y_n$ stochastically bounded, so for each $\epsilon \geq 0$ , $\exists$ a constant K and a value $n_0 = n_0 (\epsilon)$ such that $P(\left | Y_n \right |\leq K) \geq 1- \...
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62 views

Taking limits in a variational PDE with respect to a parameter

Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain and $B \subset \Omega$ be an open ball, for a given $\epsilon>0$ consider $$a_\epsilon(x) = \begin{cases} 1, \qquad x \in \Omega \...
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1answer
37 views

Almost sure convergency Uniform distribution

We have $f: \left [ 0,1 \right ] \rightarrow \left [ 0,1 \right ]$ continuous. $\left \{ \xi \right \}_{n\geq1}$ random variables; $\xi_1\sim Uniform \left [ 0,1 \right ]$ And we consider, $X_n:= f(\...
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1answer
48 views

sum of two random variable converges in distribution

It's simple question. $\{X_n\}, X, Y$ are random variables such that $X_n + cY$ converges to $X+cY$ in distribution for each $c >0$. How can I show that $X_n$ converges to $X$ in distribution?? I ...
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0answers
27 views

Convergence of second moment implies 2-uniform integrability

$(X,d):Polish\ space$ $\mathcal P_2(X):= \{ μ \in \mathcal P (X)|\int_X{d^2(x,x_0)dμ\ \text{for some }x_0\in X} \}$ Definition $\mathcal S \subset \mathcal P_2(X)$ is said 2-uniformly integrable if ...
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29 views

Weak* and Strong type product convergence in the space of Radon measures

Let $L^1(\mathbb{R})$ denote the space of all Lebesgue integrale functions and $\mathcal{M}(\mathbb{R})$ be the space of Radon measures which is the dual of continuous function with compact support on ...
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26 views

Prove that $\ell^1(\mathcal{P}(\mathbb{N}))$ is not separable with weak star topology

Let $X = \mathcal{P}(\mathbb{N})$, first consider $g_n(E)= 1$ if $n\in E$ $g_n(E)=0$ otherwise, for $E\in \mathcal{P}(\mathbb{N})$. We can easily see that $(g_n)$ is bounded in $\ell^\infty(X)$, but ...
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1answer
60 views

Application of Radon Nikodym Theorem

Let $f_n,g_n \in L^1(\mathbb{R})$ be two sequences of non negative functions such that $h_n:=\frac{g_n}{f_n} \leq K$. Suppose, $f_n \buildrel\ast\over\rightharpoonup \mu_f$ and $g_n \buildrel\ast\over\...
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24 views

Convergence in distribution of normally distributed random variables

Suppose $X_1, X_2, \dots$ are independent random variables, with $X_k \sim N(0,2^{k-1})$ for each $k \in \mathbb{N}$, i.e., each $X_k$ is normally distributed with mean $0$ and variance $2^{k-1}$. For ...
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1answer
68 views

Convergence in Distribution of a random variable with an upper bound implies convergence of higher moments.

Let $k\in \mathbb{N}$ and $\{X_n\}_{n \in \mathbb{N}}$ be random variables such that they converge in distribution to some $X$ and $|{X_n}|\leq Y$, where $Y$ is a random variable with $\mathbb{E}\left[...
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0answers
24 views

$ \mu_n \overset{w}{\longrightarrow} \delta_0 $ $\iff$ $\lim_{n \to \infty} \mu_n((-\epsilon,\epsilon)^c) = 0$

Let $(\mu_n)_{n=1}^{\infty}$ be a sequence of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. I want to show that $$ \mu_n \overset{w}{\longrightarrow} \delta_0 \quad \text{ for } \ n \...
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22 views

Topological Properties of Measures with a Common Barycenter

Let $A$ be a convex subset of a locally convex vector space $X$ and consider $M(A)$ the set of regular Borel probability measures on $A$ with the weak* topology. For a fixed point $x_0 \in \overline{A}...
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1answer
47 views

Convergence for a Random Normal Process defined by Recursion

A question from my Random Processes exams: Let $ W_0,W_1,W_2,...$ be a sequence of independent Gaussian random variables with Mean 0 and Variance $ \sigma ^ 2 > 0 $. Define the sequence $ (X_n : n \...
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1answer
46 views

Closed convex hull of the tail sequence is exactly the weak limit [Haim Brezis Exercise 3.13]

This question is from Haim Brezis' functional analysis exercise 3.13, but generalizes it a little bit. Two related questions are posted here: closed convex hull of weak convergent sequence and here: ...
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0answers
16 views

Weak convergence (in the context of Hilbert spaces) of sqaure integrable random variables.

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and consider the Hilbert space $$\mathcal{X}:=\{X:\Omega\to \mathbb{R}~\mbox{measurable}: \mathbb{E}[X^2]<\infty\}$$ with scalar product ...
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1answer
22 views

Compactness in $W^{1,p}_0(B_R)$

I have a sequence of functions $\{u_k\}_k \in u + W^{1,p}_0(B_R, R^N)$, where $B_R \Subset \Omega $ is the ball of radius R and $\Omega \subset R^n$ open and bounded, $u \in W^{1,p}(B_R, R^N)$. I want ...
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1answer
34 views

How does the Glivenko-Cantelli theorem improve the stochastic convergence of the empirical distribution $F_n(x)$?

Let $X_i$ be iid random variables with empirical cumulative distribution function $F_n(x)$ and CDF $F(x)$. From the central limit theorem and the strong law of large numbers, we know that $F_n\...
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1answer
33 views

How to prove that $\sqrt{n}(\ln S_n-\mu)$ converges in law to $\mathcal N(0,\sigma^2)$ where$S_n$is the logarithm of geometric mean of some iid $X_i$?

Let $X_n$ be some positive random variables, independent and identically distributed with $(\ln{X_n})^2$ integrable, $\textbf{E}[\ln X_n]=\mu$ and $\textbf{Var}[\ln{X_n}]=\sigma^2$. Define the ...
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0answers
24 views

Convergence of normal quantile approximations (or convergence of QQplot)

Suppose we have standard normal random variables: $X_1, X_2, ..., X_n \sim N(0,1)$ and we denote by $X_{(i)}$ the corresponding ordered statistics, i.e. $$X_{(1)} \leq X_{(2)} \leq ... \leq X_{(n)}.$$ ...
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1answer
24 views

without weak convergence, can strong convergence be true? [closed]

I need help on part (b) for the linked question (please click on this link to see the question). I understand that the the omitted condition implies a weak convergence. Weak convergent alone does not ...
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2answers
59 views

Bound in reflexive Banach space + monotonicity of sequence implies weak convergence of sequence?

Let $V$ be a reflexive Banach space with a partial order relation $\leq.$ Furthermore, suppose it is a lattice. What further conditions does one need to have this property: if $v_n$ is a bounded ...
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2answers
37 views

Weak convergence in $L^2$ implies strong convergence?

Consider a sequence of (real-valued) functions $f_n$ in $L^2(\Omega)$ where $\Omega$ is a smooth, bounded domain in $\mathbb{R}^d$. If $\lVert f_n \rVert_{L^2} < M$ (uniformly bounded in $n$), then ...
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0answers
57 views

About the ‘exist for each’ ‘exist for every’

I have a problem in understanding the description ‘exist for each’ and ‘exist for every’. This is a question from weak convergence. Prove: Let $X$ be a normed linear space and let $B$ be a dense ...
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1answer
68 views

Why is the hydrodynamic limit a convergence in probability?

The hydrodynamic limit is a matter that involves several notation and definitions and so I will will assume that whoever reads this question has basic knowledge about the notation and understands it (...
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0answers
20 views

Eberlein-Šmulian theorem and “Whitley's construction”

The Eberlein-Šmulian theorem states that if $X$ is a Banach space, $\sigma(X,X')$ denotes the weak topology on $X$ and $A\subseteq X$, then $A$ is (relatively) $\sigma(X,X')$-compact if and only if $A$...
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16 views

the two scale convergence of $(u^{out} - u^{int})$ having different trace on opposite face of $Y$.

I have this geometry (given below) and want to find the two-scale convergence of $[u_{\varepsilon}]$ and $\{u_{\varepsilon}\}$. If anyone has the idea or a reference it would be very helpful. Let $\...
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0answers
23 views

Strong convergence vs Weak convergence _ compactness of integral varifolds

I am reading the proof of the compactness of Integral varifolds on L.Simon's book " Lecture on geometric measure theory", there is a part of the proof concerning the conclusion that the ...
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0answers
13 views

Convex Functionals Weak Inferior Semicontinuity

I don't understand the step $\underline{\lim}f(x_n)=\lim f(x_m)$ in the prove. the subsequence in the prove is from Banach-Sacks below, which is the subsequence of a weakly convergent sequence, whose ...
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0answers
40 views

Bounding a double sequence of integrals

I have a double sequence of probability density functions $\rho_{n,m}\in C^\infty(\mathbb R^d$). Suppose that for every test function $\varphi\in C_c^{\infty}(\mathbb R^d)$ $\lim_{m\to\infty}\lim_{n\...
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1answer
53 views

Show that $(X_1 + \cdots + X_n )/n \to 0$ in distribution by characteristic function

There's a random variable $X$ whose characteristic function is $\phi(t) = E(e^ {itX})$ is given by $$\phi(t) = e^{-|t|^{3/2}}$$ Let $X_1, X_2, \ldots$ the independent random variable with the same ...
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1answer
26 views

Are the pointwise and the weak* limit in $L^1$ the same?

Suppose $\Omega\in \mathbb{R}^n$ is open and bounded and $f_n\in L^1(\Omega)$ converge weakly* (as signed measures) to some $f\in L^1$ and pointwise almost everywhere to some $g\in L^1$. Is $f=g$ ...
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2answers
47 views

Does weak convergence imply $L^2$-norm convergence?

Let $\{f_n\}_{n \geq 1}$ be a sequence of functions in $L^2[0,1]$ converging to $f$ weakly. Does it imply that $f_n \to f$ in $L^2$-norm? This question appeared in an exam paper on functional ...
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1answer
37 views

Unitary conserving Schrödinger equation

If we work in the spaces of $N\times N$ matrices we can write the propagator Schrödinger equation as: $$ \dot{U}(t)=-iH(t)U(t) $$ For some time-dependent Hermitian Hamiltonian $H(t)$. If we assume ...
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3answers
267 views

Conditions for weak convergence and generalized distribution functions

I am having some trouble proving Corollary 6.3.2 in Borovkov's Probability Theory (for reference, this material is on pages 147 to 149 in the book). For convenience, I provide some definitions and ...
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1answer
38 views

Hilbert And Banach dual spaces

Are, and in general, all Hilbert spaces identified to their dual ?, i.e $H'\equiv H$ What about Banach spaces? And can someone please state the weak convergence in each?
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1answer
32 views

Weak convergence in $L^2$ of powers of a sequence which is weakly convergent in $W^{1,2}$

Let $\Omega\subset \mathbb{R}^3$ be a bounded domain. Assume that $(f_n)_{n\in \mathbb{N}}\subset W^{1,2}(\Omega)$ is weakly convergent to $f\in W^{1,2}(\Omega)$. Consider now $(f_n^3)_{n\in \mathbb{N}...
2
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2answers
67 views

Corollary of convergence in distribution

Can I say that if $\xi_n \xrightarrow {d} \xi$ and $\xi_n, \xi$ are non-negative random variables with finite expected value then $$\mathbb{E}\xi \le \lim{\inf{}_{n\to\infty}\mathbb{E}\xi_n}?$$ I have ...
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0answers
24 views

Uniform convergence of functions of a integral

Suppose $\left\{\mu_n\right\}_{n=1}^\infty$ is a sequence of measures in $\mathbb R$ that converges weakly to $\mu$. The function $x\mapsto h(y,x)$ is bounded and continuous for every $y\in \mathbb R$....

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