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11 votes
1 answer
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Uncountable families of measurable sets with pairwise positive intersections

Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$. Is there an ...
Saúl RM's user avatar
  • 10.6k
3 votes
0 answers
129 views

A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
Daan's user avatar
  • 141
2 votes
0 answers
205 views

When should the empirical measure of an infinite sequence be defined?

Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
Tobias Fritz's user avatar
  • 6,406
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 ...
Francesco Bilotta's user avatar
4 votes
1 answer
204 views

How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)

$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
Iosif Pinelis's user avatar
0 votes
0 answers
54 views

Reference request: "doubly empirical" measure associated to a random measure

I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
pseudocydonia's user avatar
0 votes
1 answer
102 views

Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
S.Z.'s user avatar
  • 505
1 vote
1 answer
344 views

Is the Borel-Cantelli Lemma applicable here? [duplicate]

Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
user1234's user avatar
  • 161
1 vote
1 answer
314 views

Proof of the Dunford-Pettis theorem in the context of probability spaces

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
rfloc's user avatar
  • 627
1 vote
2 answers
262 views

Is the Boltzmann entropy lower semi-continuous in the weak topology induced by $C_b (\mathbb R^d)$?

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \...
Akira's user avatar
  • 835
4 votes
2 answers
255 views

Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \...
Akira's user avatar
  • 835
3 votes
1 answer
220 views

Conditional expectation as square-loss minimizer over continuous functions

It is well-known that the conditional expectation of a square-integrable random variable $Y$ given another (real) random variable $X$ can be obtained by minimizing the mean square loss between $Y$ and ...
fsp-b's user avatar
  • 463
2 votes
1 answer
133 views

Can convergence in distribution necessarily be realised by almost-sure convergence?

Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
Julian Newman's user avatar
1 vote
0 answers
168 views

Optimal transport-like problem where the objective depends on conditional probability distribution

$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data. Consider two sets $\...
Francesco Bilotta's user avatar
0 votes
0 answers
161 views

Markov process with time varying transition kernels

I cross post this question from StackExchange as it may be more appropriate. I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
Francesco Bilotta's user avatar
1 vote
0 answers
87 views

Symmetry of the isoperimetric profile

Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as $$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
πr8's user avatar
  • 801
3 votes
2 answers
102 views

Reference for Wiener type measure on $C(T)$ when $T$ is open

I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
Kiyoon Eum's user avatar
2 votes
0 answers
127 views

Measure algebra for a family of probability measures

Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and \begin{align} B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \} \end{align} the ...
Packo's user avatar
  • 285
1 vote
1 answer
96 views

Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials

This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials I am trying to study the asymptotic behavior ...
Francesco Bilotta's user avatar
-1 votes
2 answers
407 views

Conditional expectation: commuting integration and supremum

Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
Vokram's user avatar
  • 109
2 votes
1 answer
198 views

References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent): ...
FD_bfa's user avatar
  • 147
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
Analyst's user avatar
  • 657
5 votes
0 answers
135 views

Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
S.Z.'s user avatar
  • 505
4 votes
1 answer
265 views

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\...
Ludwig's user avatar
  • 2,712
2 votes
0 answers
98 views

Has this "optimal constrained transport" notion of convergence of measures been named and/or studied?

Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$. Fix $L \geq 1$. I will say that $\mu_n$ converges in ...
Julian Newman's user avatar
1 vote
0 answers
191 views

Characterization of Poisson random measure in terms of Laplace transform

Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$. A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if $\pi(B)\sim\operatorname{...
0xbadf00d's user avatar
  • 167
3 votes
1 answer
415 views

Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
Quarto Bendir's user avatar
1 vote
1 answer
137 views

Ergodic theorem on limit of periodic transformations?

Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
Monroe Eskew's user avatar
  • 18.6k
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 \...
ABIM's user avatar
  • 5,405
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
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
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
4 votes
2 answers
614 views

Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant

Setting Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that $$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...
Pablo Lessa's user avatar
  • 4,304
5 votes
2 answers
589 views

Properties of measures that are not countably additive but have countably additive null ideals

This is a very naive question, maybe more of a reference request than anything else. Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has ...
aduh's user avatar
  • 869
1 vote
1 answer
88 views

Convergence of probability measures which (asymptotically) concentrate along a submanifold

Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define \begin{align} P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\ z (\beta) &= \...
πr8's user avatar
  • 801
5 votes
1 answer
363 views

Inverse marginal property of a collection of $\sigma$-algebras

In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras. Let $(\...
Ivan Feshchenko's user avatar
2 votes
0 answers
61 views

Measurable extensions of probability measures

Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\...
aduh's user avatar
  • 869
2 votes
0 answers
261 views

Reference for Borel $\sigma$-algebra of topology of convergence in probability

I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before. So I'm wondering if there are any papers/...
Julian Newman's user avatar
1 vote
0 answers
83 views

Embedding random variables in infinite-dimensional spaces

Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
JohnA's user avatar
  • 710
1 vote
1 answer
448 views

Law of large numbers for random Dirac measures

Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable. ...
Learning math's user avatar
0 votes
1 answer
189 views

Visualization of the disintegration theorem [closed]

Where can I find a picture that gives a visualization of the disintegration theorem? If such reference does not exist, what would a nice visualization of this fundamental result look like?
Jay's user avatar
  • 109
1 vote
1 answer
206 views

Almost identical $\sigma$-algebras and measurability

Let $(X,\mathscr X,\mathbb P)$ be a probability space, $(Y,\mathscr Y)$ a measurable space, and $h:X\times Y\to\mathbb R$ a real-valued function measurable with respect to the product $\sigma$-algebra ...
triple_sec's user avatar
1 vote
1 answer
120 views

Are there well-established notions of convergence of measures that take into account differentiable structure?

All the notions of convergence of measures that I know of are either in the purely measure-theoretic category (e.g. strong convergence, total variation), or in the topological category (e.g. weak ...
Julian Newman's user avatar
5 votes
2 answers
791 views

What is this disintegration-like theorem?

This is cross-posted at MSE. I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any ...
user435571's user avatar
1 vote
0 answers
91 views

Probability space with countable subset such that every subset of positive measure meets the subset

Let $(X, \mathcal F, P)$ be a probability space. Question What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that $\forall$ measurable $A \subseteq X$, $P(A) >...
dohmatob's user avatar
  • 6,853
0 votes
1 answer
133 views

Product of sets with the Radon-Nikodym Property (RNP)

I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP. Does the above result ...
BigbearZzz's user avatar
  • 1,245
7 votes
1 answer
1k views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
JohnA's user avatar
  • 710
2 votes
1 answer
235 views

Kolmogoroff condition for truncated random variables

Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
Maximilian Janisch's user avatar
1 vote
0 answers
162 views

Does the law of a Feller process depend continuously on the initial condition?

Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing ...
Potato's user avatar
  • 11
0 votes
0 answers
424 views

Bounding the total variation distance between two measures from a given set

I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ : $d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
YZ22's user avatar
  • 31