Intereting Posts

Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$
Mathematical Induction and “the product of odd numbers is odd”
The free-space Green's function for the Stokes flow
Associates in Integral Domain
Sequence Limit: $\lim\limits_{n \rightarrow \infty}{n\,x^n}$
Prokhorov metric vs. total variation norm
Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
Why is there a difference between a population variance and a sample variance
A combinatorial proof that the alternating sum of binomial coefficients is zero
The correspondence theorem for groups
Nice proofs of $\zeta(4) = \pi^4/90$?
$\sum \limits_{n=1}^{\infty}n(\frac{2}{3})^n$ Evalute Sum
Subway & Graphs
Evaluate $\int_0^1\frac{x^3 – x^2}{\ln x}\,\mathrm dx$?
Is there a homology theory that counts connected components of a space?

Let $X_n$ and $X$ be random variables taking values in the metric space $(S,d)$.

The sequence $(X_n)_n$ is *convergent* to $X$ *in distribution (or weakly)* if

$E[f(X_n)] \to E[f(X)]$ for all $f:S\to R$ continuous and bounded.

- Can I understand Egorov's theorem in this way?
- Sample variance converge almost surely
- Convergence of Roots for an analytic function
- How to find the sum of the following series
- When does the $f^{(n)}$ converge to a limit function as $n\to\infty$?
- Decimal representation series

I read somewhere that it’s equivalent to consider only *uniformly continuous* and bounded $f$.

Could you give me a proof of this fact?

- Explicit characterization of dual of $H^1$
- Using Hahn-Banach in proving result about operators and their adjoints on Banach Spaces
- Linear isometry between $c_0$ and $c$
- matrix derivative of gradients
- Positive bounded operators
- Does weak convergence in Sobolev spaces imply pointwise convergence?
- How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$
- Reconciling several different definitions of Radon measures
- Is the composition function again in $L^2$
- Is this an inner product on $L^1$?

We denote $\Bbb P_n$ and $\Bbb P$ the measures associated with $X_n$ and $X$ respectively.

Assume that $E[f(X_n)]\to E[f(X)]$ for all $f$ uniformly continuous and bounded. Fix $F$ a closed set and let $O_n:=\{x\in S,d(x,F)<n^{-1}\}$. Then the map $f_n\colon x\mapsto \frac{d(x,O_n^c)}{d(x,O_n^c)+d(x,F)}$ is uniformly continuous and bounded.

**Claim:** $\limsup_{n\to +\infty}\Bbb P_n(F)\leqslant \Bbb P(F)$.

Indeed, $f_n(x)=1-\frac{d(x,F)}{d(x,O_n^c)+d(x,F)}$ is monotone, bounded by $1$ and converges pointwise to the characteristic function of $F$. We have for each $n$ and $N$,

$$\Bbb P_n(F)\leqslant \int f_N(x)d\Bbb P_n,$$

so for all $N$,

$$\limsup_{n\to +\infty}\Bbb P_n(F)\leqslant \int f_N(x)dP,$$

and we conclude by monotone convergence.

Now, fix $f$ a continuous function such that $0\leqslant f\leqslant 1$. Let $F_{n,j}:=\{x\in S,f(x)\geqslant \frac jn\}$.

\begin{align*}

\int_S fd\Bbb P_N-\int_S fd\Bbb P &\leqslant \sum_{k=0}^n\frac kn\left(\Bbb P_N\left(\frac kn\leqslant

f(x)<\frac{k+1}n\right)-\Bbb P\left(\frac kn\leqslant f(x)<

\frac{k+1}n\right)\right)+\frac 1n\\\

&=\sum_{j=0}^n\frac jn\Bbb P_N(F_{n,j})-\sum_{j=1}^{n+1}\frac{j-1}n\Bbb P_N(F_{n,j})

-\sum_{j=0}^n\frac jn\Bbb P(F_{n,j})\\&+\sum_{j=1}^{n-1}\frac{j-1}n\Bbb P(F_{n,j})+\frac 1n\\

&=\frac 1n\sum_{j=1}^n\left(\Bbb P_N(F_{n,j})-\Bbb P(F_{n,j})\right)+\frac 1n.

\end{align*}

Taking $\limsup_{N\to +\infty}$ and doing the same for $1-f$ instead of $f$, we get the wanted result.

It’s a part of portmanteau theorem.

A good reference for questions about weak convergence is Billingsley’s book *Convergence or probability measures*.

- Axis of rotation of composition of rotations (Artin's Algebra)
- Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
- What exactly is a Kähler Manifold?
- The direct sum of two closed subspace is closed? (Hilbert space)
- Definition of a Functor of Abelian Categories
- Characterizations of Prüfer Group
- How To Know in a Application Sequence/Series Problem Which Variable is $a_{0}$ or $a_{1}$?
- Finding integer solutions for $6x+15y+20z=1$
- Proving that if $f>0$ and $\int_E f =0$, then $E$ has measure $0$
- Integral of $\int e^{2x} \sin 3x\, dx$
- Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.
- How to express the function $\mathbb{N} \to \mathbb{N}\times \mathbb{N}$ as a mathematical statement?
- Why do we say “radius” of convergence?
- As shown in the figure: Prove that $a^2+b^2=c^2$
- Un-curl operator?