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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Weak convergence and convergence of moments

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ such that $X:\Omega\rightarrow \mathbb{R}$. Suppose that $X\sim N(\mu, \sigma^2)$. Consider a random ...
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0answers
12 views

$\pi_n(y)\in O(1)$ for every realisation $y$ of $Y_n$ implies $\pi_n(Y_n)\in O_p(1)$

Consider a sequence of random variables $\{Y_n\}_n$ all defined on the same probability space $(\Omega, \mathcal{F}, P)$ such that $Y_n:\Omega \rightarrow \mathbb{R}$ $\forall n$. Consider a sequence ...
1
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1answer
35 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...
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0answers
18 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
1
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1answer
27 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...
1
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1answer
18 views

Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
2
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1answer
48 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
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0answers
13 views

Asymptotic power of a test

Do you have any insight on the following statement Consider a test with a test statistic weakly convergent to a continuous distribution under any alternative and such that the finite sample power at ...
0
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1answer
33 views

Find the limit of $P_{\theta_n}\Big(\sqrt{n}(T_n-\mu(\theta_n))<z_\alpha \sigma(0)-\sqrt{n}(\mu(\theta_n)-\mu(0))\Big)$

Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow ...
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2answers
118 views

Weak topology and strong topology in a Banach space.

I have a doubt about weak topology in a Banach space. Let $\mathcal{B}$ a infinite dimensional Banach space, I understood that the weak topology in $\mathcal{B}$, is the topology generated by $\Sigma ...
2
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0answers
33 views

Subspace of a weakly sequentially complete is weakly sequentially complete

A Banach space $X$ is called weakly sequentially complete if all weakly Cauchy sequences are weakly convergent. Question: If $Y$ is a subspace of a Banach space $X$, must $Y$ be weakly sequentially ...
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0answers
20 views

Implications of convergence in distribution

I would like to ask you for an help to show $(\star)$ Consider a sequence of real-valued random variables $\{X_n\}_n$ and assume $\sqrt{n}(X_n-\mu)\rightarrow_d (N,\sigma^2)$ as $n \rightarrow ...
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0answers
17 views

Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
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0answers
20 views

Joint distribution of two normal marginal distributions

My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence. Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the ...
2
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0answers
23 views

Moving Average of an Ergodic Markov processes.

Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in ...
3
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1answer
40 views

Question about weak convergence, $\lbrace f(x_{n}) \rbrace$ converges for all $n$, then $x_{n} \rightharpoonup x$

I found the following question in my textbook Let $E$ a reflexive space and $\lbrace x_{n} \rbrace \subset E$ a sequence such that $\lbrace f(x_{n}) \rbrace$ converges for all $ f \in E^{*} $, Show ...
3
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1answer
40 views

Question about weak convergence of random variables

When you start to learn probability theory, for instance the central limit theorem, you learn about convergence in distribution $X_n\to X$ (where, say, both $X_n$ and $X$ are $\mathbb R$-valued random ...
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2answers
27 views

Convergence of product of (weakly) converging sequences in $L^{p}$

Preparing for an exam, I was wondering about general statements about the convergence of products. 1) Let $p, q \in ]1, \infty[$ such that $\frac{1}{p}+\frac{1}{q} = 1$ and $a_n \rightharpoonup a$ ...
4
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1answer
32 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
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0answers
33 views

Pre-compactness in $L \log L$

As far as I know Zygmund class of Orlicz spaces or "$L \log L$" is defined as an Orlicz space with the Young function $Q(t) = t \sqrt{\ln(1+t)}$ (or something similar to this in different ...
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0answers
16 views

Is the space of discrete probabilities measurable in the space of probabilities with weak topology?

Let $S$ be a polish space and $\mathcal{P}(S)$ be the space of all probability measures with weak topology. I am wondering whether the set of all discrete probability measures is measurable. Thanks ...
3
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1answer
25 views

Finding a random variable $X$ such that $X_n$ (given) converges in distribution to $X$

For every $n\in\Bbb{N}$, let $X_n$ be a random variable which gets the values $\{-1, -\frac{n-1}n,...,-\frac 1 n, 0, \frac 1 n,...,\frac{n-1}n, 1\}$ with equal probability. Find a random ...
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16 views

Compute the limit and show that uN converges weakly

full question I already know that the norm is 1, and that you can use the definition of weak convergence but that's where I get lost. Somebody told me I can use the Riesz representation theorem since ...
2
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1answer
43 views

about a product of random variables that converges weakly

Let $(\Omega,\mathcal{F},P)$ be a probability space. Suppose $f_n,g_n, n\in \mathbb{N}$ are sequences of functions on this space such that their product $f_ng_n$ converges weakly in $L^2$ to $h$, say. ...
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1answer
19 views

Connection between weak topology in probability and weak* topology in functional analysis

In functional analysis, Definition A: for any normed linear space $(X, \| \cdot \| )$, the weak star topology $\sigma (X^*, X)$ on $X^*$ is generated by the collection of seminorms $\{ p_x ...
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1answer
27 views

Simplifying expressions with Big O notation

Consider a real-valued sequence $\{h_n\}_n$ with $\lim_{n \rightarrow \infty}h_n=0$. Could you help me to simplify $(O(h_n^2))^2+3O(h_n^2)$? My attempt: (1) ...
3
votes
1answer
37 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
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0answers
13 views

Relation between stochastic and deterministic big O

Could you help me to clarify the relation between stochastic and non-stochastic Big O notation? Suppose I have a sequence of real-valued random variables $\{X_n\}_n$ defined on the probability space ...
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0answers
10 views

Inequalities with stochastic Big O

I have a very general question related to inequalities containing stochastic big o notation. Introduction: consider two sequences of real-valued random variables $\{X_n\}_n$, $\{Y_n\}_n$ ...
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0answers
18 views

Properties of stochastic little o notation

I have a question related to stochastic little o properties. Let $\{X_n\}_n$ be a sequence of real-valued random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such ...
2
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0answers
42 views

Weak Convergence in $\ell^p$

First, my definition of weak convergence in $X$ is that $x_n \rightharpoonup x$ if $\phi(x_n) \to \phi(x)$ for all $\phi \in X^*$. I recently read the statement that $e_n \rightharpoonup 0$ in ...
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0answers
15 views

From deterministic little o to stochastic little o notation

I have a question on Lemma 2.12 in van der Vaart which can be found here. Consider statement (i): if $R(h)\in o(||h||^p)$ as $h\rightarrow 0$ $\Rightarrow $ $R(X_n)\in o_p(||X_n||^p)$. Among the ...
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1answer
52 views

What is Vitali theorem [closed]

What is the definition of weak and strong convergent sequence in L^p? I can not seem to find anywhere the proper definition and to see difference between the two. Also What is Vitalis theorem really ...
2
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0answers
17 views

Stochastic Convergence

I need help figuring out if a series of "apparently random" digits are the result of the same (possibly non-polynomial) function, ergo, not-random but deterministic. The highest level math I know is ...
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0answers
26 views

Pointwise and weakly convergent limits are the same [duplicate]

Consider $L^1(\mu)$ for an arbitrary measure space, and let $f_n$, $f$, be in $L^1(\mu)$. Suppose $f_n$ converges to $f$ almost everywhere, and $f_n$ converges to $g$ weakly. Is it true that $f=g$ ...
0
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1answer
26 views

Asymptotic Normality of one-sample U-statistic in van der Vaart

I have a question related to the proof of Theorem 12.3 (asymptotic normality of the one-sample U-statistic) in Van der Vaart "Asymptotic Normality" here. My question is related to the point in ...
4
votes
1answer
170 views

Properties of Weak Convergence of Probability Measures on Product Spaces

EDIT: For the Bounty, I made a substantial edit revision concerning the structure of the question, to make it more readable (hopefully). Moreover I added a question on problem 2.7 of Billingsley’s ...
4
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0answers
53 views

Relation between weak convergence of probability measures and weak-* convergence

I am trying to nail down the relation between probability and functional analysis. In particular, how the notion of weak convergence used in probability theory is related to the weak-* convergence of ...
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2answers
40 views

Proof involving weak and weak-* convergence

"Prove that if $(f_n)_{n=1}^\infty\subset X'$ converges strongly in $X'$ and a sequence, $(x_n)_{n=1}^\infty\subset X$ converges weakly in $X$, then, $f_n(x_n)\to f(x),\,n\to\infty$." My attempt: ...
2
votes
1answer
34 views

Proving $\|u\|_{L^\infty(0,T;H)}\leq C$ from a given hint.

My question concerns to the problem 6, chapter 7, of Evans PDE book (2nd edition). In the book a hint is given but I couldn't get a solution from it. On the other hand, I got a solution without ...
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0answers
41 views

Density and Pointwise convergence imply strong convergence for Bi-orthogonal system?

Let $H=\ell^2$ and we denote $x = (x(k))_k$ for $x\in H$. Let $(e_n,f_n)_{n=1}^\infty$ be a bi-orthogonal system in $H$ , that is, $\langle e_m, f_n\rangle=\delta_{mn}$. For $x\in H$ we define, $$x_n ...
3
votes
2answers
40 views

Weakly convergence but not strongly - properties of limsup and liminf

Let $X$ be a Banach space and suppose we have a sequence $\{x_n\}$ which is convergent weakly but not strongly. Define $y_n:=\sum\limits_{k=1}^{n}x_k$. What we can say about ...
4
votes
1answer
32 views

Tightness, relative compactness and convergence of stochastic processes

For proving the convergence of a certain sequence of stochastic processes (which take values on a compact set), I am taking the following approach (as taken in previous papers I am looking at): 1) ...
2
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0answers
36 views

Weak convergence of measures time functions

Let us consider a sequence of measures $(\mu_n)_n$ which converges weakly to the measure $\mu$ on a metric space $X$. To simplify let us take $X=\mathbb{R}^n$ and $L$ be the Lebesgue measure. Now on ...
2
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1answer
20 views

Weakly Convergent Subsequence: Iff-Condition?

Helly's theorem states that we can extract a convergent subsequence from any sequence of distribution functions. Moreover, one can show that the limit of this subsequence is a distribution function ...
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1answer
21 views

Convergence in probability to a sequence converging in distribution

Consider two sequences of real-valued random variables, $\{X_n\}_n$ and $\{Y_n\}_n$, and a real-valued random variable $Y$. Suppose that $X_n\overset{p}{\rightarrow}Y_n$ and ...
2
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1answer
20 views

Marginal convergence in distribution implies joint convergence of a subsequence?

Consider two sequences of real-valued random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_n:\Omega\rightarrow \mathbb{R}$ and $Y_n:\Omega\rightarrow \mathbb{R}$. ...
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1answer
19 views

How to understand the demonstration that that if $f_n(x)$ uniformly converges to $f(x)$ then $(f_n(x))_{n\in \mathbb{N}}$ weakly-converges to f(x)

I don't understand the implication in the following demonstration which shows that if $f_n(x)$ uniformly converges to $f(x)$ then $(f_n(x))_{n\in \mathbb{N}}$ weakly-converges to f(x) $f_n$ ...
4
votes
2answers
48 views

Limsup of a probability

I've encountered the following questions. Suppose $X_n \rightarrow X$ in distribution, and $a<b$ Prove that $$ P( a \le X\le b) \ge \limsup_{n\to \infty}P(a\le X_n\le b) $$ I know how to find ...
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1answer
39 views

Generalised version of le Cam's Third Lemma

I'm confused with the generalised version of Le Cam's Third lemma presented in Theorem 6.6 of van der Vaart asynptotics Statistics here: ...