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Marginal distribution of $I$-projection

I am reading this paper by Csiszar. Given a probability measure $R$ and a convex subset $\mathcal{E}$ of probability distributions, it defines ‘I-projection of R on $\mathcal{E}$’ (provided there ...
Raghav's user avatar
  • 371
1 vote
0 answers
191 views

Almost sure convergence and asymptotic measurability

Let $(\Omega,\mathcal{A}, P)$ be a probability space and $X$ be a Borel measurable and separable map. (i) $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ and $\left(d\left(X_{n},X\right) \right)$ is ...
Geor11's user avatar
  • 11
1 vote
1 answer
137 views

Embeddings of spaces of probability measures

What is the relationship between the spaces $X_1\triangleq \mathscr{P}(C([0,1],\mathbb{R}))$ and $X_2\triangleq C([0,1],\mathscr{P}(\mathbb{R}))$; where $\mathscr{P}(\cdot)$ denotes the Borel ...
ABIM's user avatar
  • 5,405
1 vote
1 answer
301 views

Definition of Wasserstein distance through cumulative distribution

Let $X$ and $Y$ be random variables on the same probability space. The $\infty$-Wasserstein distance between $X$ and $Y$ is defined as $$d_{\infty}(X, Y) = \inf \|X_1 - Y_1\|_{L_{\infty}},$$ where the ...
Seven9's user avatar
  • 565
0 votes
0 answers
59 views

Examples of strongly continuous measure-valued functions

Let $X$ be a compact geodesic metric space and let $P_p(X)$ be the set of all finite Borel measure on X with finite $p^{th}$ moment. We equip $P_p(X)$ with the total variation topology metric. What ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
71 views

Conditions for existence of a semi-martingale representing a system of probability measures

Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$. Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
268 views

Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?

From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem: Tightness: Let $\Pi$ be a ...
Mark's user avatar
  • 657
0 votes
0 answers
151 views

Definition of conditional expectation for singleton

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Furthermore, let $X, Y$ be two random variables from our probability ...
timudk's user avatar
  • 33
2 votes
1 answer
241 views

Weak continuity of law

Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
ABIM's user avatar
  • 5,405
3 votes
1 answer
226 views

Expected measure of a ball in a probability space with a metric

Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms ...
eonaran's user avatar
  • 33
0 votes
1 answer
134 views

How can we show this estimate for the convolution of two probability measures?

Let $(\delta_k)_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta_k\xrightarrow{k\to\infty}0$ and $(\varepsilon_k)_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{k\in\mathbb N}\...
0xbadf00d's user avatar
  • 167
2 votes
1 answer
101 views

If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?

$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let \begin{equation*} \exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...
Iosif Pinelis's user avatar
1 vote
1 answer
165 views

If $\mu_t\to\mu$ weakly, then $\limsup_t|\mu_t|(A)\le|\mu|(A)$ for all closed $A$

Let $E$ be a metric space, $\mathcal M(E)$ denote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$, $(\mu_t)_{t\in I}$ be a net ...
0xbadf00d's user avatar
  • 167
2 votes
2 answers
322 views

If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?

Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$. I would like to know ...
0xbadf00d's user avatar
  • 167
-2 votes
1 answer
108 views

If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent

Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$. If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
0xbadf00d's user avatar
  • 167
0 votes
2 answers
167 views

Equidistributed sequence wrt exponential/Gaussian measure

For an arbitrary probability space $(X,\mu)$, a sequence $(x_n)$ in $X$ is said to be equidistributed with respect to $\mu$ if the measures $\frac 1 n \sum_{1\le k\le n} \delta_{x_k}$ converges weakly ...
Tartrate's user avatar
  • 341
0 votes
0 answers
139 views

How many moments determine a normal distribution?

I know that a Gaussian distribuion is determined by its moments. I was wondering if there is a result of the form: if we know that the first thousand moments of a random variable are Gaussian, then is ...
Dr. Pi's user avatar
  • 3,062
0 votes
0 answers
302 views

Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem

In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
0xbadf00d's user avatar
  • 167
4 votes
3 answers
2k views

Duality of finite signed measures and bounded continuous functions

Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
142 views

Radon-Nikodým-like theorem for Radon measures

Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$. I'm searching for a Radon-Nikodým-like ...
0xbadf00d's user avatar
  • 167
1 vote
1 answer
151 views

Lower-bound on Sobolev norm of function on $(d-1)$-dimensional sphere, whose sign has been fixed at $n$ points

Let $\mathbb S_{d-1} := \{x \in \mathbb R^d \mid x^\top x = 1\}$ be $(d-1)$-dimensional sphere in $\mathbb R^d$ and let $\sigma_d$ be the uniform distribution on $\mathbb S_{d-1}$. Let $x_1,\ldots,x_n$...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
97 views

Wigner semicircle law and random measures

tl;dr: the proof of the Wigner semicircle law seems to confuse measures with random measures. I do not understand why. Scroll down until 'QUESTION' if you are fine with the theoretical stuff. T. Tao ...
gangrene's user avatar
3 votes
1 answer
278 views

Is this statement of the Lévy–Khintchine formula ill-posed?

Please take a look at the following statement of the Lévy–Khintchine formula given in Probability Theory: A Comprehensive Course (2nd edition)$^1$: Am I missing something or is this an ill-posed ...
0xbadf00d's user avatar
  • 167
2 votes
1 answer
183 views

Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?

I am trying to better understand a condition that appears in Theorem 1 of this paper. Let $K$ be a convex and compact subset of a locally convex tvs. The condition is: $K$ embeds linearly into a ...
aduh's user avatar
  • 869
0 votes
0 answers
150 views

Define the convolution root of probability measures on a measurable group

Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$. Remember that a probability ...
0xbadf00d's user avatar
  • 167
0 votes
0 answers
85 views

If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
0xbadf00d's user avatar
  • 167
3 votes
0 answers
179 views

Probability terminology

This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
Steven Landsburg's user avatar
2 votes
1 answer
329 views

Projective limit of spaces of probability measures

Consider a projective system $\dots X_{n+1} \to X_n \to \dots \to X_1$ of completely regular Hausdorff spaces with projective limit $X$. Then the linking mappings $f_n$ induce a projective system (in ...
yada's user avatar
  • 1,773
1 vote
1 answer
1k views

Closed-form upper-bounds for Wasserstein distance between finite measures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
ABIM's user avatar
  • 5,405
-1 votes
1 answer
92 views

Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?

Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...
0xbadf00d's user avatar
  • 167
1 vote
1 answer
154 views

If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$

Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
0xbadf00d's user avatar
  • 167
3 votes
3 answers
244 views

Example of a (strictly) proper scoring rule on a general measurable space?

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...
aduh's user avatar
  • 869
4 votes
0 answers
160 views

Can we show equivalence of two distributions based on their statistics?

Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
Zhifeng Kong's user avatar
0 votes
2 answers
222 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
James Propp's user avatar
  • 19.7k
6 votes
1 answer
343 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
aduh's user avatar
  • 869
1 vote
1 answer
193 views

Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$

Let $Y$ be a symmetric random variable, $(X_n)_n$ be a sequence of nonnegative random variables, and set $p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if $c$ is a constant ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
156 views

Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
ABIM's user avatar
  • 5,405
3 votes
1 answer
77 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
52 views

A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
Error 404's user avatar
  • 111
8 votes
4 answers
775 views

Self-contained formalization of random variables?

I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
user21820's user avatar
  • 2,912
2 votes
1 answer
95 views

Is the set of almost surely continuous points dense?

Denote by $D(0,T)$ the space of right continuous functions with left limits defined on $[0,T]$. Let $\mathbb P$ be a probability measure on $D(0,T)$. Define $$cont(\mathbb P):=\Big\{t\in [0,T]:~ \...
user avatar
5 votes
1 answer
774 views

Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
user avatar
0 votes
1 answer
115 views

Average over spheres finite

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$ I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...
Pritam Bemis's user avatar
5 votes
1 answer
319 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
Pritam Bemis's user avatar
1 vote
2 answers
113 views

If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent?

Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$. I was ...
fsp-b's user avatar
  • 463
1 vote
2 answers
194 views

Continuity of the densities of a stochastic process

Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
fsp-b's user avatar
  • 463
1 vote
1 answer
164 views

Is this (somewhat specific) moment problem treated somewhere?

Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as : $$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faà di Bruno's formula, I can ...
lrnv's user avatar
  • 686
3 votes
0 answers
77 views

Reference Request: Is every interval-valued probability measure consistent?

Short version: Does every interval-valued probability measure contain a conventional probability measure? I have a sense that this is a basic result about an obscure topic but I am having trouble ...
Owen Biesel's user avatar
  • 2,356
0 votes
1 answer
187 views

Does the finitely additive integral preserve convergence for non-negative measurable functions?

Let $(X, \mathcal X)$ be a measurable space. Say that a net $(\mu_\alpha)$ of finitely additive probability measures converges to a finitely additive probability measure $\mu$ if and only if $\mu_\...
aduh's user avatar
  • 869
2 votes
1 answer
181 views

Conditional entropy - solve example

Given a random variable $X$ that is uniformly distributed on $[-b,b]$ and $Y=g(X)$ with $$g(x) = \begin{cases} 0, ~~~ x\in [-c,c] \\ x, ~~~ \text{else}\end{cases}$$ Now I want to compute the ...
Phobos's user avatar
  • 131

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