Questions tagged [strong-convergence]

A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x\in X$ such that $\lim_{n\rightarrow \infty } \| x_n - x\| =0 $

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strong and weak* limits

I have the sequence $\Phi_n(x)=\sin(nx) \ \forall x\in(0,\pi),n\in \mathbb{N}$ and I know that it weakly* converges to $0$ in $L^{\infty}((0,\pi))$. Now I have to show that it doesn't converge ...
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42 views

Strong convergence in $L^p(\mathbb{R}^N)$

I would like to ask your help about this: Let $q>1$ fixed. Consider $\{m_n\}$ a sequence of non-negative functions in $L^s(\mathbb{R}^N)$ for every $s\in[1,q)$ such that $\int_{\mathbb{R}^N}m_n=M$ ...
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34 views

Equivalent definitions for Strong Operator Topology in Banach Spaces

Strong Operator Topology (definition-$1$): We called a sequence $\{T_n\}$ in $\beta(X,Y),$ for $X,~Y$ are Banach spaces and $\beta(X,Y)$ denotes the family of bounded linear operators from $X$ to $Y$, ...
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Will $L^1\log L^1$ bound gives strong $L^1$ convergence?

I am trying to prove this statement: given a sequence $\{f_n | f_n > 0, c_1 \leq \int_{\Omega} f_n \log{f_n} \leq c_2 \}$, here $\Omega$ is a bounded domain; can we prove the $L^1$ strong ...
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Proof of a.s. convergence in Continuous Mapping Theorem

Suppose $g$ is a.e. continuous on the support of $X$. Then the continuous mapping theorem (CMT) asserts (among other things) that \begin{align*} X_n\overset{a.s.}{\rightarrow}X\implies g(X_n)\overset{...
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25 views

Does Skorohod Dudley Wichura theorem imply Continuous Mapping Theorem?

The continuous mapping theorem (CMT) asserts that for $g$ a.e. continuous on the support of $X$, $X_n\overset{a.s.}{\rightarrow}X\implies g(X_n)\overset{a.s.}{\rightarrow}g(X).$ $X_n\overset{p}{\...
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Showing different definitions of almost sure convergence are equivalent.

There are a couple different equivalent definitions of almost sure (a.s.) convergence: $\forall \varepsilon>0\quad P(\liminf_{ n\uparrow \infty}\{\|X_n-X\|\leq\varepsilon\})=1$ $\mathbb{P}(\{\...
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169 views

Brezis Exercise 3.13 item 3.

I work in the following question: Let $E$ be a space Banach, $\{x_n\}_{n=1}^{\infty} \subset E$, $x \in E$ and \begin{equation*} K_{n}= \overline{con(\bigcup_{i=n}^{\infty}\{xi\})}. \end{...
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Doubt about weak and strong covergence in $L^p$

The problem is the next: $Let\ 1<p<\infty\ and \ let \ f_n=n^{1/p}f(nx)\ where\ f \ is\ a \ L^p\ function, \ examine\ the\ weak \ and\ strong\ convergence\ in \ L^p $ i could prove that $\...
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Strong Law of Large Number for a weighted sum of two sequences.

Let's say we have two sequences of independent random numbers $X_1, X_2, X_3, ...$ with $X_i \sim F_i(x_i)$ and $E[X_i] = \mu_i$, and $Y_1, Y_2, Y_3, ...$ with $Y_i \sim G_i(y_i)$ and $Y_i>0$ and $...
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Is there a relation between weak-* convergence in $L^\infty((0,T); L^2(\Omega))$ and strong convergence in $L^2(\Omega)$ uniformly in $t$?

I have problems in understanding a proof. So my question is: Let $u_n$ be a sequence that converges weak-* in $L^\infty((0,T); L^2(\Omega))$ to $u$. Can I imply that $u_n$ converges strongly in $L^2(\...
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Using SLLN To Estimate A Probability

Suppose we flip a coin to figure out the probability p to get head. We flip $n$ times and get the frequency $\hat{p}$. By strong law of large number, $P(\lim_{n \rightarrow \infty} \hat{p}=p)=1$, so I ...
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Generalization of Strong Law of Large Numbers - dividing by a sequence rather than $n$

Suppose $X_1,X_2,\ldots$ are i.i.d. with mean $\mu$. The SLLN says that $$ \frac{X_1 + \ldots + X_n}{n} \rightarrow \mu \text{ almost surely}. $$ Now suppose $a_n$ is a sequence of integers such that $...
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97 views

Prove that if $T_n \rightarrow T$ and $x_n \rightarrow x$ then $T_n x_n \rightarrow T x$

$X$ - Banach space and $T_n$ bounded linear operators that is strongly convergent to $T$ as well as $x_n$ is strongly convergent to $x$. How to prove that $T_n x_n$ is strongly convergent to $Tx$? I ...
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77 views

Show strong operator convergence

I have a problem in which I need to prove that a sequence of operators is strongly convergent and that it isn't uniformly convergent. The operators are defined like so: $ T_j: L^1 \to L^∞ $ And the ...
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114 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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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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248 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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38 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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Rate of convergence - Stochastic Euler Method

The absolute error criterion of the pathwise approximation of an Itô process $X$ by an Euler approximation $Y$ is: $$ \epsilon=E\left(\left|X_{T}-Y(T)\right|\right) $$ We shall say that a time-...
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55 views

Show that $\{P_n\}_{n \geq 1}$ does not converge in norm topology.

Let $\mathcal H = \ell^2 (\Bbb N)$ and $P_n$ be the projection onto $\text {span}\{e_0,e_1, e_2, \cdots, e_n \}.$ Show that the sequence $\{P_n\}_{n \geq 1}$ does not converge in norm topology. ...
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38 views

Prove that the unit ball in $L^{p}([0,1]),$ where $1 \leq p \leq \infty,$ is not strongly compact.

Prove that the unit ball in $L^{p}([0,1]),$ where $1 \leq p \leq \infty,$ is not strongly compact. Give an example of a bounded sequence in $L^{1}([0,1])$ that does not have a weakly convergent ...
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52 views

Proving strong continuity of semigroup in $L^2(\mu)$ ($\mu$ invariant measure) from non-cadlag Markov process

I am trying to derive the definition of a Markov semigroup from Section 1.1 Bakry, Gentil, Ledoux Analysis and Geometry of Markov Diffusion Operators from a Markov process. I am unsure about my proof ...
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77 views

Weak convergence equivalent to weak convergence at $0$

Let $(X, \lVert \cdot \rVert)$ be a Banach space and $(T(t))_{t\geq0}$ a semigroup of linear operators , i.e for all $t, s \geq 0$ we have $T(t):X \to X$ is a linear operator, $T(t+s)=T(t)T(s)$ and $T(...
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60 views

weak and strong $L^p$ convergence

for my class I need to find two sequences of functions such that each satisfies one of the conditions below: $f_n \in L^p(\Bbb{R})$ such that it converges weakly toward $0$ in $L^p(\Bbb{R})$ but ...
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44 views

Examples of $L^p \cap L^1$ convergence

I'm struggling to find examples of a sequence of functions in two different cases: let $1<p<2$ functions $(f)_n \in L^p(\Bbb{R},dx)\cap L^1(\Bbb{R},dx)$ such that $(f)_n$ converges strongly in $...
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47 views

Liminf of Pointwise Norms of a Weakly Convergent Sequence

Let $X_1, X_2, \cdots$ be a sequence of $p$-integrable $\mathbb{R^d}$ valued random variables. Assume that $X_n$ converges $0$ weakly, then can we say that $r(\omega) = liminf\{ |X_1 (\omega)|, |X_2 (\...
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221 views

Are strong convergence of measure and almost sure convergence of a random variable related?

I am studying measure theory and probability I am wondering if and how these two concepts are related. A sequence of random variables $X_n$ defined on some probability space $(\Omega,\mathscr{A},\mu)$ ...
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34 views

Does weak convergence in $L^\infty(\mathbb{R},L^p(\mathbb{R}^n))$ imply strong convergence in $L^p_{loc}(\mathbb{R}\times\mathbb{R}^n)$?

Does weak convergence in $L^\infty\left(\mathbb{R},L^p\left(\mathbb{R}^n\right)\right)$ imply strong convergence in $L^p_{loc}\left(\mathbb{R}\times\mathbb{R}^n\right)$ up to a subsequence ?
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Does left-multiplication by compact operators turn strong-convergence into norm-convergence?

If $\{T_i\}_{i\in I}$ is a bounded net of operators on a Hilbert space $\mathscr H$, converging strongly to some operator $T$, and if $K$ is a compact operator on $\mathscr H$, then the net $\{T_iK\}...
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41 views

A question about Sobolev space and the relationship between the weak convergence on it and the strong convergence on L^p space.

Suppose that $\Omega$ is a bounded open set in $\mathbb{R}^n$, $H_0^1(\Omega)$ is the Sobolev space $W_0^{1,2}(\Omega)$. If $u_n$ convergent to $u$ weakly in $H_0^1(\Omega)$, can I get the conclusion ...
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Lim sup nets proof, is it valid?

I just get really nervous around lim sup and nets for some reason... Please help! Problem I was given: Suppose $(x_i)_{i\in I} \subset B(\mathcal{H})$ converges to $x \in B(\mathcal{H})$ in the SOT (...
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72 views

Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
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91 views

Some doubts about proof of Strong Law of Large Numbers

I quote Jacod-Protter. Theorem: Let $\left(X_n\right)_{n\geq1}$ be independent and identically distributed and defined on the same space. Let$$\mu=\mathbb{E}\{X_j\}$$ $$\sigma^2=\sigma_{X_j}^2<\...
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92 views

Weak and strong convergence of a sequence of operators in $L^2(\mathbb{R})$

Let $M_\alpha = sin(\frac{x}{\alpha})f(x)$ a sequence of operators in $L^2(\mathbb{R})$. Prove that $M_\alpha$ does not converge strongly to $0$, but converges weakly to $0$, for $\alpha \to 0$. ...
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41 views

Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
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168 views

When does strong convergence imply convergence in operator norm?

I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show $\lim_{t \to \infty} T_t =0 $ with respect to the operator norm. After some effort, I was able to prove $\lim_{t \to \infty} T_t =0 $ ...
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Let $B_nx=x(t-\frac{1}{n})\in L^2(\mathbb{R})$. Show that $B_n\overset{s}{\to}Id$ but $B_n\not\rightrightarrows Id$

Let $B_nx=x(t-\frac{1}{n})$ be a sequence of operators in $L^2(\mathbb{R})$. Show that $B_n\overset{s}{\to}Id$ but $B_n\not\rightrightarrows Id$ where $Id$ is the identity operator. So for the strong ...
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53 views

Weak and Pointwise Convergence question

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
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1answer
31 views

Range of strong limit of a semigroup belongs to the fixed space?

Let $\left(T(t)\right)_{t\geq0}$ be a $C_0$-semigroup on a Banach lattice $E$ such that $T(t)$ converges strongly to a positive operator $S$ as $t \to \infty.$ Then $$T(t)S=S \text{ for all }t\geq0.$$ ...
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32 views

If $H$ is Hilbert and $T\in B(H)$ is left shift w.r.t. the OB $(e_{k})_{k\in\mathbb{N}}$, then $T^{n}x\to0$ for all $x\in H$

Let $H$ be Hilbert and $T\in B(H)$ the left shift operator w.r.t. the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$. That is, $Te_{1}=0$ and $Te_{k}=e_{k-1}$ for $k>1$. Then how do I show that $T^{n}...
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83 views

Does weak convergence imply strong convergence for parameterized measure?

Define for some density $q_{\kappa_n}$ (with respect to the Lebesgue measure) on $\Theta$, parameterized by $\kappa_n \in \mathbb{R}^d$. Next, define the sequence of measures $\mu_n$ as $$\mu_n(A) = \...
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46 views

Laws of large numbers problem: Convergence of a symmetric sequence [closed]

I am looking for good solutions to this problem. Could you please help me with this? Any solution would be appreciated. Let $\mathit{X_n}$ be independent and identically distributed. Assume $\...
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1answer
53 views

Weak, Strong, a.e. convergence parameters in $L^2(R^3)$ excercise

I'm doing this exercise, and I wish for corrections on my solution: a) Find the index set $I:=\{ \alpha \in R\} : \{f_n\} \in L^2(R^3), > f_n(x) = n^\alpha X_{B_{1/n}}$ converge weakly in $L^2(R^...
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41 views

Operator satisfying certain properties

Give an example of a Hilbert space $H$ and a sequence of compact operators $(S_n)_{n=1}^{\infty}$ on H such that: (i) $||S_n||\leq 1$ (ii) The operators $V_N=\sum_{n=1}^N\dfrac{1}{n}S_n$ converge ...
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97 views

Poke holes in this proof of the SLLN

I have a proof (sketch) of the Strong Law of Large Numbers, at least the "sufficiency" half of it, that seems a little too easy. This is the version where you only assume i.i.d. random variables, and ...
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114 views

About weak convergence and convergence of norms

Today in an exam on functional analysis the following question was posed: Let $H$ be a Hilbert space and $(x_n)_{n\in \Bbb{N}} \subseteq H$ be a sequence that converges weakly to $x\in H$ satisfying $...
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81 views

On a sequence $f_k$ in $L^{2+\frac{1}{k}}$

Suppose that $f_k\in L^{2+\frac{1}{k}}(\Omega)$ with the property that $\|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1$ for all $k\ge 1$. $\Omega$ is a bounded domain in $\mathbb{R}^n$. Can such a sequence ...
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190 views

Example that strong convergence does not imply convergence in norm

I have found an example that strong convergence does not imply convergence in norm. Let us take $T_{n} = P_{span\{e_{1},...,e_{n}\}}$ be a projection on $span\{e_{1},...,e_{n}\}$ in Hilbert's space ...
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50 views

Weak and strong convergence of operator sequence

Let $|e|\leq1$ such that $e\neq0$, we define the lineal operator $F_e: C^1([-1,1],R)\rightarrow R $ as $$ F_e x(t) =\frac{x(e) - x(-e)} {2 e} . $$ I need to know if the sequence of operators ${\{F_e\}...