3
votes
0answers
22 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
3
votes
2answers
62 views

The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
1
vote
1answer
34 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
1
vote
2answers
47 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
2
votes
1answer
54 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
3
votes
0answers
61 views

Topological interpretation of the following equivalence.

We assume $\{X_n\}_{n\in\mathbb{N}}$ and $X$ are random variables from $\{\Omega,\mathcal{F},\mathbb{P}\}$ to $(S,d_s)$, wehre $S$ a separable metric space. One can establish the following ...
1
vote
0answers
24 views

Regularity of measures proof

A probability measure $\mathbb P$ on a metric space $(S,d)$ is closed regular if $$ \mathbb P(A) = \sup \{ \mathbb P(F) : F \subseteq A, F \text{ - closed} \} \text{*}$$ with $A\in ...
1
vote
0answers
37 views

A basic question on converges in distribution

The distance $d(F,G)$ between two distribution functions is the infimum of those $\epsilon > 0$ such that $G(x-\epsilon) - \epsilon \leq F(x) \leq G(x+\epsilon) + \epsilon \quad\forall x $. Now I ...
1
vote
0answers
157 views

is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
3
votes
2answers
79 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
3
votes
1answer
44 views

Is the strong convergence of Borel probability measure metrizable?

In a metric space $(X,e)$, a sequence of Borel probability measure converges strongly, $\mu_i \xrightarrow{s} \mu$, iff for each Borel subset $S \in X$, we have $\lim_{i \to \infty}\mu_i(S) = \mu(S)$. ...
2
votes
0answers
143 views

Bounded Lipschitz Metric on Space of Positive Measures

The bounded Lipschitz metric ($d_{BL}$) metrizes the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ $$d(\mu, \nu) = ...
2
votes
0answers
70 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
1
vote
0answers
48 views

Computationally efficient means of determining distance in the Skorohod Topology?

I have two functions f and g in a computer. Domain 1...N. I'd like to compute their distance using the Skorohod Topology in an efficient manner. (I first ran across this metric many years ago in ...
3
votes
2answers
201 views

A question on Tight probability measures (regular measure)

This is somewhat a basic question, but I'm having difficulty proceeding with a certain part of the proof. I was reading Billingsley "Convergence of Probability Measures", and I encountered the ...
3
votes
2answers
55 views

Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ Borel measurable?

Let $(X,d)$ a metric space and $\mathcal{X}$ its Borel sigma-algebra, i.e. the sigma-algebra generated by the open sets of $X$. Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ $\mathcal{X} ...
2
votes
1answer
110 views

convergence of functions on probability measure

I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence. $\mathfrak{F}([0,1])$ is the set of distribution functions over ...
8
votes
2answers
718 views

Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
4
votes
1answer
89 views

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
8
votes
1answer
393 views

Algorithms for computing or numerically approximating the Prokhorov metric?

I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them? The general ...
1
vote
1answer
260 views

Cylindrical sigma algebra and continuous functions.

Consider the space $\mathbb R^{[0,1]}$ of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. I know how to prove that $C[0,1]\not \in \mathcal B$. My ...
1
vote
1answer
200 views

Levy-Prokhorov metric question

I have a question relating to the Lévy-Prokhorov metric and its description on wikipedia. The metric is defined for measures on $\mathbb R^d$ and is defined by $$ ...
4
votes
1answer
709 views

About the Wasserstein “metric”

I've just encountered the Wasserstein metric, and it doesn't seem obvious to me why this is in fact a metric on the space of measures of a given metric space $X$. Except for non-negativity and ...
5
votes
0answers
289 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of ...
2
votes
0answers
106 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...