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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uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
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20 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
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1answer
36 views

Weak convergence of random variables implies $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$

Proof that, if $X_n \rightarrow X$ weakly, then $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$. I know, that I should use Fatou's lemma but I don't know what can I do first.
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1answer
21 views

Show a bounded linear operator is weakly sequentially continuous

Let $X$ and $Y$ be normed linear spaces, $T \in B(X,Y)$, and $\{x_{n}\}_{n=1}^{\infty}\subset X$. If $x_n \rightharpoonup x$, I need to prove that $T x_{n} \rightharpoonup T x$ in $Y$, where ...
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24 views

Portmanteaus Theorem and Lipschitz functions

I was looking at some basics in the theory of convergence of distributions, e.g. Portmanteaus Thm. One part of it asserts that it suffices to check $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for bounded ...
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44 views

If a sequence converges weakly in a closed subspace $M$ of a Banach space, then the (strong) limit point is in $M$.

Let $M$ be a closed subspace of the Banach space $X$ and let $x_{n}\in M$ converge weakly to $x$. Show that $x \in M$. We use the following definition $x_n\rightharpoonup x$ in $X$, $x_n$ ...
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27 views

Neighborhood base of weak topology

If $X$ is a Banach space, the weak topology on $X$ is the weakest topology in which each functional $f$ in $X^\ast$ is continuous. I have some difficulties in understanding its neighborhood basis in ...
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1answer
30 views

Weak star convergence leads to weak convergence of derivative?

Let $u_\epsilon \in W^{1,p}(\Omega)$ be a sequence with $\Omega \subset \mathbb{R}^n$ bounded. Let $u_\epsilon \rightharpoonup^* u$ weakly* in $L^\infty (\Omega)$ (for a subsequence) with $u \in ...
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1answer
22 views

$f \in L^1$ differentiable almost everywhere with $f' \in L^1$ implies $f$ has weak derivative $f'$?

I got a function $f:[a,b] \to \mathbb{R}$ where $f \in L^1(a,b)$. It is differentiable a.e. with derivative $f' \in L^1(a,b)$. How can I show that $f$ has a weak derivative? I know that $\int ...
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37 views

Weak Convergence of piecewise constant function (Evans PDE, Section 8.7, Q1b)

I'm having problems with the following question from Evans' PDE book (From Section 8.7, Question 1b); Fix $a,b \in \mathbb{R}$ and $0<\lambda <1$. Define $$u_k(x)= \begin{cases} a ...
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1answer
17 views

Proof of convergence in distribution

I want to prove that a sequence of random variables $X_n$ converges in distribution to $N(0,1)$, if we have the following condition for an arbitrary $\epsilon>0$: $$(1-\epsilon)Y_n \le X_n \le ...
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27 views

Showing mutual contiguity

The problem: Let $P_n$ and $Q_n$ be the distribution of the mean of a sample of size $n$ from the $N(0, 1)$ and the $N(\theta_n, 1)$ distribution, respectively. Show that $P_n$ and $Q_n$ are ...
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1answer
44 views

Weak convergence exercise.

Let $(f_n)$ be a sequence in $L^2(\mathbb R)$ and let $f\in L^2(\mathbb R)$ and $g\in L^1(\mathbb R)$. Suppose that \begin{eqnarray*} f_n\rightharpoonup f \hbox{ weakly in }L^2(\mathbb R)\,, \\ ...
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2answers
68 views

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$.

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$. My attempt: $$\|\phi\| = \sup_{\|x\| = 1} |\phi(x)| \leq \sup_{\|x\| = 1} \lim_n |\phi_n(x)|$$ Using an ...
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0answers
20 views

Weak convergence in L^2

Suppose that $\{f_n\}_{n\in\mathbb{N}}$ converge weakly to $0$ in the $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2\subset\mathbb{R}^2$ denote the torus. If $u_n=R^{\perp}f_n=(-R_2f_n,R_1f_n)$ where $R_1$ ...
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35 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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1answer
36 views

What is the Conditional law?

I am sorry to ask such a question, but I always cant understand what is meant by the notation I am currently reading. In a mathematical paper I witness a formula as such $\mathcal{L}\left( ...
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0answers
43 views

weak convergent sequence in $L^p(\mathbb{R})$ with $(1\leq p < \infty)$ implies norm is bounded

$f_n \rightharpoonup f$ in $L^p(\mathbb{R})$ with $(1\leq p <\infty)$ implies $||f_n||_p$ are bounded. And for $p = \infty$, if $f_n \xrightarrow{w^*} f$, then $||f_n||_\infty$ are ...
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1answer
32 views

Understanding the scalar product in the duality

I was trying to solve an exercise for my class, but then I have found somewhere a solution. I need to understand the meaning of a certain step. The exercise and the solution read as follow. Exercise ...
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1answer
23 views

Does negative part of a standardized random variable converge to negative part of a $\mathcal{N}(0,1)$?

I know how to prove that any standardized random variable converge in distribution to a $\mathcal{N}(0,1)$, I was wondering if even $f((S_n-n)/ \sqrt{n}))$ converge to $f(\mathcal{N}(0,1))$, in ...
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1answer
68 views

Weak convergence of scaled elements implies norm convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
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1answer
36 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
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1answer
28 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
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38 views

Asymptotic Equivalence implies same asymptotic distribution?

A book I'm reading stated that if we have nonnegative random variables, and if $X_n\to X > 0$ in distribution and $\frac{Y_n}{X_n} \to 1$ in probability then $Y_n \to X$ in distribution. However, ...
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1answer
32 views

Convergence in $L^p(0,T;L^q(\Omega))$

If $\Omega\subset\mathbb{R}^3$ is bounded, $$f_n\to f\mbox{ in }L^q(0,T;L^p(\Omega)),\,1\leq q<\infty,\,1\leq p<2 $$ and $$f_n\to g\mbox{ weak-star in } L^\infty(0,T;L^2(\Omega)),$$ then $f=g$ ...
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161 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
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61 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
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2answers
59 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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1answer
28 views

Weakly sequentially continuous operators in Hilbert space are norm continuous.

Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous. I feel that this statement ...
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45 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
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1answer
56 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
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1answer
40 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
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66 views

Weak convergence $f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$ implies $f^2\leq g$ a.e.

$f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$, then $f^2\leq g$ a.e. Could you guys help me check the proof please, thanks! Proof: to show $f^2 ...
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0answers
48 views

Krein-Milman theorem

Let $E$ be a Riesz space with order unit $u$. With $$\left\|f\right\|_u: = \inf\{\lambda \in [0,\infty): -\lambda u\leq f\leq \lambda u\} $$ $E$ becomes a normed space. The following sets are subsets ...
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Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
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41 views

Questions about $L^p$ spaces and convergences

I would like to sort out the relations for strong/weak convergences for $L^p(X)$ mainly between $[p=1; p>1]$ and $[\mu(X) <\infty ; \mu(X) = \infty]$ For the purpose of strong/weak ...
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1answer
41 views

Weak convergence plus strong convergence

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to ...
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1answer
17 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
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1answer
32 views

Weak convergence and infinite sum

Suppose that $\psi_n$ converges weakly to $\psi$ in a Hilbert space $H$. Assume further $\{\phi_k\}$ is an orthonormal sequence in $H$. Is it plausible that ...
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80 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
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1answer
53 views

Weak convergence with respect to the uniform topology on cadlag functions

Suppose I have a random sequence $X_n$ of cadlag functions on $[0,1]$ that converge weakly to $X$. In general this is meant with respect to the Skorkhod metric but suppose here I have that $X$ is ...
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14 views

weak convergence, tail

in my situation i have a weak convergence of distribution functions like: $\lim_{s\ \to \infty}P(X_{s} <=x)=G(x)\ \forall\ x \in\ C(G)$ (C(G) is the set of all continuity points of G) with G a non ...
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1answer
28 views

Weak convergence in some space.

I have the sequence $\{u_{k}\}_{k}$ weak convergent in the space $L^{2}(0,T; W^{1,2}(\Omega))$. What exactly does it mean? Does it imply weak convergence $\{u_{k}\}_{k}$ in $L^{2}(0,T;\Omega)$ or $\{ ...
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1answer
59 views

Weak convergence of distribution family

I know the convergence in distribution and the weak convergence. but I have two questions: First one: does weak convergence implies pointwise convergence or it is the same? And second one: I have ...
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20 views

Cauchy property of weak convergence

I found in the measure and integration theory book from Bauer (Remark 6, §20) that if $\mu$ is a finite measure, then the weak convergence of a sequence $(f_n)$ is equivalent to $$ \lim_{n,m \to ...
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96 views

Fast Convergence of marginal distribtution

Let $(q_n)$ be sequence of probability density functions of the couple $(x,y)\in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. $p_n(x):=\int q_n(x,y)dy$. Another sequence of functions ...
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1answer
131 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
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0answers
38 views

Weak star convergent sequence and $L^\infty(0,T;L^\infty(\Omega))$

Suppose $u_n$ is uniformly bounded in $L^\infty(0,T;L^\infty(\Omega))$ where $\Omega$ is a bounded domain. From this answer, we know that the dual space of $(L^1(0,T;L^1))^* = ...
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1answer
44 views

Weak-star lower semicontinuity in $L^\infty$

Let $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$. Do we get something like $$\lVert u \rVert_{L^\infty} \leq \liminf_{n \to \infty} \lVert u_n \rVert_{L^\infty}$$ i.e. a weak-star lower ...
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1answer
36 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...