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Weak continuity vector-valued functions
Let, for each $n\geq 1$, $x_{n}:[0,1]\longrightarrow X$ continuous, where $X$ stands for a banach space, with a norm $\|\cdot\|$. Assume that $X$ is reflexive and $x_{n}([0,1])\subset B(0,r)$ (the ... functional-analysis weak-convergence weak-topology- 107
Every contractive self-adjoint operator is the weak limit of projectors
I try to show that every self-adjoint operator $A$ on Hilbert space $H$ with $\|A\| \leq 1$ is the weak limit of the projectors. My teacher told me that the most important part is to show that for one-... functional-analysis operator-theory weak-convergence self-adjoint-operators- 492
Why is "weak convergence of a sequence to $0$" equivalent to "accumulation points of the sequence are all $0$" in the proof of mean ergodic theorem?
For some context, I am referring to a proof of the Mean Ergodic Theorem from Ergodic Theory and Dynamical Systems by Yves Coudene. Suppose $H$ is a Hilbert space, and let $U \colon H \to H$ be a ... functional-analysis hilbert-spaces weak-convergence ergodic-theory- 31
How to prove $x_n\rightarrow x^*$ if $\{x_n\}$ has a subsequence that converges weakly to $x^*$
Let $X$ be a Banach space and $T\in B(X)$ is a bounded linear operator satisfying that $$ \sup_n\left\|\frac{1}{n}\sum_{k=0}^{n-1}T^k\right\|<\infty,\quad\lim_{n\rightarrow\infty}\frac{1}{n}\|T^n\|=... functional-analysis banach-spaces weak-convergence- 309
Convergence in probability implies weak convergence to a Dirac Delta
I am trying to show that, for $c \in \mathbb{R}^d$ constant, if $X_n \rightarrow^{\mathbb{P}} c$, then $\mathbb{P}^{X_n} \Rightarrow \delta_c$, where $X_n:\Omega \rightarrow \mathbb{R}^d$ is a random ... probability-theory measure-theory weak-convergence- 308
weak convergence + bounded second moment implies convergence of the moment?
Let $\{\mu_N\}$ be a sequence of random measures which converges almost surely in the weak sense to a deterministic measure $\mu$ with impact support. The weak convergence does not necessarily imply ... probability-theory measure-theory weak-convergence- 421
Weak convergence of law of scaled biased random walk
Let $(X_n:n\in\mathbb{N})$ be a sequence of independent, identically distributed random variables of finite mean $m$ and finite variance $\sigma^2$. Set $S_0=0$ and $S_n=X_1+\dots+X_n$ for $n\in\... probability-theory stochastic-processes weak-convergence wiener-measure- 1,011
Uniform integrability of conditional quantile functions
Let $Z^n$ be $\mathbb{R}$-valued random variables which are uniformly integrable, i.e. $$ \lim_{a \to \infty} \sup_{n} E[1_{\{|Z^n| \geq a\}} |Z^n|] = 0. $$ Let $X^n \to N(0,1)$ in distribution, and $... probability-theory conditional-expectation weak-convergence uniform-integrability- 51
Show that law of sum of sums converges weakly to law integral of brownian motion
Let $(X_k)_{k=1}^\infty$ be a sequence of independent and identically distributed random variables with zero mean and unit variance. Define, for $n\geq1$, $$S_n=\sum_{k=1}^nX_k.$$ Prove that the law ... probability-theory brownian-motion martingales weak-convergence random-walk- 1,011
Find a sequence of functions that converge weakly to 1/2
I'm trying solve the following problem: Find an example of a sequence of functions $f_n:[0,1]\to[0,1]$ such that $$(f_n)_\#\mathscr{L} = \mathscr L$$ but $f_n\rightharpoonup 1/2$ (weak convergence). ... functional-analysis analysis weak-convergence- 91
Let $X$ be a separable Banach space. Then every pointwise bounded $(f_n) \subset X^*$ has a subsequence that converges uniformly on compact sets
This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ... functional-analysis normed-spaces banach-spaces weak-convergence weak-topology- 10.3k
Weak convergence imply bounded sequences at all sample points?
Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space, with a sequence of random variables $\{Y_n\}_{n=1}^\infty$ defined on it. Suppose $Y_n\xrightarrow{\ d\ }Y$. Can we say that for (almost) ... probability-theory weak-convergence- 14.2k
How to prove that a random variable sequence converges in distribution (weak convergence)?
Given a random variable sequence $X_t$, I wonder how to prove it converges in distribution? For example, if $$X_{t+1}=4+0.5(2r_1-1)(8r_2-X_t)$$ (where $r_1,r_2~i.i.d.\sim U(0,1)$), I can derive the ... probability probability-theory probability-distributions weak-convergence- 93
Weak convergence against upper invariant measure
Setting I am studying invariant measures and their weak limits. In a book about probability on graphs the following setting is presented in chapter 6.3 (this is a short form of the actual presentation)... markov-process weak-convergence invariant-theory- 31
Setwise convergence of measures implies weak convergence under special hypothesis
I'm struggling with producing a proof of the following result: Let $X = \overline{\mathbb{C}}$ be the Riemann sphere, and consider $M(X)$ the space of finite Borel measures on $X$ with norm given by ... measure-theory weak-convergence weak-topology- 1,557
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