Articles of weak convergence

Weak and strong convergence

I have the sequence $(v_n)\subset H^1_0(0,1)$ such that $v_n\rightharpoonup v $ (weakly) in $H^1_0(0,1)$ and $v_n\rightarrow v$ in $L^2(0,1)$ and $v_n\rightarrow v$ in $C^0(0,1)$ My question is why $$\int_0^1 v_n(x) (v_n(x)-v(x)) dx\rightarrow 0$$ I say that $\int_0^1 v_n (v_n-v) dx=\int_0^1 (v_n-v+v)(v_n-v) dx= \int_0^1 (v_n-v)^2 dx +\int_0^1 v(v_n-v) dx=$ $ ||v_n-v||^2_{L^2(0,1)} +\int_0^1 v(v_n-v) dx$ By the […]

Weak convergence and weak convergence of time derivatives

I am working in $H^1(S^1)$, the space of absolutely continuous $2\pi$-periodic functions $\mathbb R\to\mathbb R^{2n}$ wih square integrable derivwtives. I have a sequence $z_j$ (for the record, it comes from minimizing a functional, in the middle of Hofer-Zehnder’s proof that a Hamiltonian field has a periodic orbit on a strictly convex compact regular energy surface) […]

Composition of a weakly convergent sequence with a nonlinear function

Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain. Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ such that $u_m$ is uniformly bounded i.e. $\|u_m\|_{H^2}\leq M$ and given the function $f(u)=u^3-u$. If I know that $u_m\rightharpoonup u(u\in H)$ in $L^2$ sense i.e. $\int_{\Omega}u_m v\to \int_{\Omega}u […]

Tight sequence of processes

Let $X_{n} \in \mathbb{R}^{\infty}$ be a tight sequence of processes in metric space $(\mathbb{R}^{\infty}, l_{2})$ and for each $x\in\mathbb{Z}_{+}$ we have that $X_{n,x}\stackrel{d}{\to} Y_{x}$. Does it follow the weak convergence of the process $X_{n}$ to $Y$?

Compactness of Sobolev Space in L infinity

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to […]

Central limit theorem and convergence in probability from Durrett

I saw following exercise from Durrett’s probability theory book and I managed to solve the 1st part, but couldn’t get the 2nd part. Let $X_1, X_2, \dots$ be i.i.d samples with mean $0$, and finite non-zero variance. Denote $S_n = X_1 + X_2+\dots + X_n$. Use central limit theorem and Kolmogorov $0-1$ law to show […]

Prove weak convergence of a sequence of discrete random variables

People, I have no idea how to start proving weak converge. I know the definition that limit of distribution of $X_n$ should be equal to distribution of $X$. BUT no idea how to prove it. I think that there should be some hint with summation. So we have: $P(X_n=i/n)=1/n$ for $i=1,\ldots,n$, and I need to […]

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of $F_{X(x)}(y) = \mathbb{P}(X(x) \leq y) $, then $\mathbb{E}[f(X_n(x)]$ converges to $\mathbb{E}[f(X(x)]$ for any $f$ which is continuous and bounded. Suppose I have uniform convergence in distribution […]

definition of “weak convergence in $L^1$”

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set $A$. 2) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n f)\rightarrow \mathrm{E}(X\mathrm{1}f)$ for every (essentially) bounded measurable function $f$. my question: are 1) and 2) equivalent? I see that 2) implies 1) (indicators […]

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 $E$ be a reflexive space, and let $\lbrace x_{n} \rbrace \subset E$ be a sequence such that $\lbrace f(x_{n}) \rbrace$ converges for all $ f \in E^{*} $. Show that exists a $x \in E$ such that $x_{n} \rightharpoonup x$. I can show that $x_{n}$ bounded […]