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Results tagged with metric-spaces
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user 36721
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
2
votes
Density of subsequences in Bolzano-Weierstrass
$\newcommand{\N}{\mathbb N}\newcommand{\R}{\mathbb R}\newcommand{\md}{\ (\operatorname{mod}2)}$This is to present a formalized version of the nice answer by Christian Remling.
Suppose that a function …
- 128k
1
vote
Which posets arise from closed, transitive relations?
It is easy to see that the posets $(U,\le)$ that can arise in this way are exactly such that for all $u$ and $v$ in $U$
$$u\le v\ \&\ v\le u\implies u=v. \tag{1}\label{1}$$
That is, any poset can aris …
- 128k
3
votes
Accepted
Is completion of measures equivalent to completion of sigma algebras as metric spaces with r...
In Theorem 1.4 of this paper or of its preprint version, it is shown that, for any measure $m$ on any algebra $\mathcal A$ of subsets of a set $X$, a subset $E$ of $X$ is locally approximable by sets …
- 128k
4
votes
Accepted
Are there any statistical metrics that satisfy this kind of condition?
According to Proposition 7 on p. 236 of Givens and Shortt, the $L^2$ Wasserstein distance between $N(\mu_1,\sigma_1^2)$ and $N(\mu_2,\sigma_2^2)$ is the Euclidean distance between the points $(\mu_1,\ …
- 128k
3
votes
Accepted
Linear process close to a Gaussian process
$\newcommand{\R}{\mathbb R}\newcommand{\Z}{\mathbb Z}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$Let $\psi_j:=0$ for $j=-1,-2,\dots$. Then
\begin{equation*}
X_t=\sum_{j\in\Z}X_{t,j}
\end …
- 128k
3
votes
Accepted
Inferring the modulus of continuity
$\newcommand\om\omega\newcommand\R{\mathbb R}$In general, even the inequality
$$\om_g\le\om\circ\om_f^{-1}\tag{0}$$
will not hold, for the right inverse $\om_f^{-1}$ of $\om_f$ defined by
$$\om_f^{-1} …
- 128k
2
votes
Accepted
Can a measure on a finite metric space be Alhfors regular?
Take any $x\in X$ with $m:=\mu(\{x\})\in(0,\infty)$ (since $X$ is finite and $\mu$ is a probability measure, such a point $x$ exists). Let $R:=\min\{d(y,x)\colon y\in X\setminus\{x\}\}$. Then $R\in(0, …
- 128k
1
vote
Accepted
Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
$\newcommand\R{\mathbb R}$By rescaling, without loss of generality $\lambda=1$.
So, the question becomes the following: is there a real $C$ not depending on $u=(u_1,\dots,u_n)\in\R_+^n$, $v=(v_1,\dots …
- 128k
1
vote
Accepted
A property for maps between metric spaces
" How is it related to $f$ being an isometry?"
Obviously, any isometry $f$ has the property in question.
On the other hand, not every $f$ having this property is an isometry. Indeed, let $X=Y=\{1,2,3\ …
- 128k
3
votes
Accepted
Expected measure of a ball in a probability space with a metric
Let $S:=\mathbb X$. Let $X$ and $Y$ be iid random elements of $S$ each with distribution $\mathbb Q$. It is more transparent to rephrase the question as follows:
Can one give a good lower bound on $P …
- 128k
1
vote
Metric for measuring linearity of finite set of points in $R^2$
$\newcommand\R{\Bbb R}\newcommand\la{\lambda}$One good "metric" is as follows. Let $S$ be your finite set of $n\ge2$ distinct points in $\R^2$. Let $C(S)$ be the covariance matrix of the uniform proba …
- 128k
3
votes
Accepted
Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
$\newcommand{\vpi}{\varphi}\newcommand\R{\mathbb R}$
Claim 1: The map $F$ is not Lipschitz if $p>1$.
Claim 2: The map $F$ is $1$-Lipschitz if $p=1$: For all $f,g$ in $D_p$,
\begin{equation*}
W_ …
- 128k
4
votes
Accepted
Equivalent definition for Skorokhod metric
$\newcommand\ep\varepsilon\newcommand\la\lambda\newcommand\La\Lambda\newcommand\tla{\tilde\lambda}\newcommand\tLa{\tilde\Lambda}\newcommand\ga\gamma\newcommand\Ga\Gamma\newcommand\de\delta$Your conjec …
- 128k
4
votes
Accepted
Does complete and separable Wasserstein space imply a complete base space?
$\newcommand\de\delta\newcommand\ep\varepsilon$The conjecture is true.
Indeed, let $(z_n)$ be a Cauchy-convergent sequence in $(Z,d)$. Then
$$d_{W_p}(\de_{z_m},\de_{z_n})=d(z_m,z_n)\to0$$
(as $m,n\to\ …
- 128k
4
votes
Accepted
Vague convergence: confusion about the regularity of a signed Radon measure and that of its ...
$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\O …
- 128k