All Questions
Tagged with measure-theory probability-distributions
150 questions
1
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0
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72
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Do almost all Gibbs' measures satisfy the weak-Poincare Inequality?
I am trying to interprete the discussion given in Section 3 of this paper,
https://core.ac.uk/download/pdf/82217936.pdf
Lets suppose we restrict to considering Gibbs's measures of the form $\sim e^{-...
- 617
2
votes
1
answer
231
views
Is Boltzmann entropy well-defined for arbitrary probability density function?
$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \...
- 835
1
vote
2
answers
101
views
Ratio of Gaussian measure over Euclidean balls
Let $\nu\in\mathcal P(\mathbf R^d)$ be the standard Gaussian distribution $\mathcal N(0,I_d)$.
Denote by $\mathscr B$ the class of Euclidean balls $B_r(x)$ (centered in $x\in\mathbf R^d$ with radius $...
- 11
2
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1
answer
837
views
Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb R^d} (1+|x|) |\ell (x)|^{1-\alpha} \, d x < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that
$$
\|...
- 835
2
votes
1
answer
170
views
Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
3
votes
1
answer
122
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When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?
This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
- 568
2
votes
1
answer
86
views
From convergence of sequences to uniform convergence in probability
For $n=1, 2,\ldots$ consider a sequence of sets of ascending integers $I_n=\{\underline{i}_n,\underline{i}_n+1, \ldots, \overline{i}_n\}$, with $\underline{i}_n \to \infty$ and $\underline{i}_n=o(\...
- 195
2
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0
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54
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If a probability measure is a mixture of products of its marginals, does it have finite moments?
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is ...
- 716
0
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0
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53
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Existence of derivative of distribution of exponential family?
Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
5
votes
2
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729
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Probabilty measures that are both discrete and continuous
Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...
- 101
2
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0
answers
98
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Understanding simple point processes (part 2)
This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods
by Daley ...
- 181
3
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0
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94
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Understanding simple point processes
Background
I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes.
A simple point process is a random variable whose ...
- 181
2
votes
2
answers
227
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Weak convergence of measures on continuous function spaces
Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion.
I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by
$\mu_r(A):=P\Big(\frac{...
- 21
4
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2
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274
views
Does strong stochastic ordering exist?
For two probability measure $\mu$ and $\nu$ on $\mathbb{R}$, we call $\mu$ is stochastically smaller than $\nu$ (i.e., $\mu\leq\nu$) , if $\int f \, d\mu\leq\int f \, d\nu$ for any nonnegative bounded ...
- 43
2
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2
answers
211
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Limit of a integral whose integrand diverges under the limit
I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
- 43
2
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0
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174
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Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
1
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1
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100
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Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 13
1
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1
answer
168
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
- 11
2
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1
answer
152
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A different way to try to define a measure on the unit-circumference circle
Let C denote the Lie group (ℝ/ℤ, +), and let ℤn denote the subgroup of C generated by [1/n].
Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of ...
- 2,858
2
votes
1
answer
89
views
p.d.f. of exponential family
I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{...
- 53
4
votes
1
answer
2k
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Examples of convergence in distribution not implying convergence in moments
It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.
Let $\{X_n\}$ be a ...
- 227
1
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0
answers
43
views
Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
- 167
2
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0
answers
164
views
Convex ordering of measures that are obtained by different push-forwards of a same measure
Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...
- 21
0
votes
1
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105
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Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 5
4
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1
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265
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Bounds on discrepancy metric of product measures
Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...
- 2,712
1
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2
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235
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Connection between invariant measure and positive recurrence for continuum state space markov chain
Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ...
8
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0
answers
422
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
- 399
5
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1
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261
views
Infimum of Fourier transform of singular measure
Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\...
- 93
2
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1
answer
297
views
Examples of "almost" Ahlfors regular measures
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...
- 5,405
1
vote
1
answer
332
views
Extension of measurable function from dense subset
Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$...
2
votes
0
answers
192
views
Convergence of Gibbs distribution to Dirac measure [closed]
Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$:
$$
q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right).
$$
Denote $x^* = \arg \...
- 31
4
votes
1
answer
206
views
Existence of measures with given 1d marginals
This is a question about marginals of probability measures, which seems unrelated to previous questions.
Let $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ be the unit sphere. Assume that for each $\theta\in \...
0
votes
1
answer
274
views
Upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$
Let $\xi$ be a random vector taking values in $\mathbb{R}^d$. Is there an upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$?
I have seen that if $p=...
1
vote
0
answers
74
views
Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 5,405
0
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1
answer
133
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Convoluted Cantor-like measure which has a continuous component [duplicate]
Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
\sum_{k\ge 1}3^{-k}X_k
$$...
- 1,352
0
votes
1
answer
86
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Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
1
vote
1
answer
134
views
KL-divergence and sub-$\sigma$-algebras
I am trying to understand if the following claim is true:
Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\...
- 13
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
2
votes
1
answer
1k
views
measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
- 23
1
vote
1
answer
240
views
Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
- 5,405
2
votes
1
answer
268
views
Union bound probability of random union
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E_i\}_{i = 1}^N,$ with $E_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$
Is there ...
- 245
0
votes
1
answer
268
views
Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?
From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...
- 657
2
votes
1
answer
241
views
Weak continuity of law
Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
- 5,405
2
votes
1
answer
101
views
If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?
$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let
\begin{equation*}
\exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...
- 128k
-2
votes
1
answer
108
views
If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent
Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
- 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 ...
- 23k
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,\...
- 167
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)\ \...
- 57
1
vote
1
answer
246
views
Scaling behavior of Wasserstein distances
Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...
- 83
1
vote
2
answers
113
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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 ...
- 463