All Questions
87 questions
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 ...
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 ...
- 708
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):
...
- 147
2
votes
1
answer
133
views
Optimal-score partitions
The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
- 128k
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 ...
- 6,406
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 ...
- 285
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 ...
- 708
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. $\...
- 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/...
- 708
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
$$
\...
- 825
1
vote
1
answer
448
views
Law of large numbers for random Dirac measures
Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.
...
- 1,467
1
vote
1
answer
137
views
Ergodic theorem on limit of periodic transformations?
Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
- 18.6k
1
vote
1
answer
96
views
Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials
This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials
I am trying to study the asymptotic behavior ...
1
vote
1
answer
164
views
Is this (somewhat specific) moment problem treated somewhere?
Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as :
$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$
Using Faà di Bruno's formula, I can ...
- 686
1
vote
1
answer
88
views
Convergence of probability measures which (asymptotically) concentrate along a submanifold
Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...
- 801
1
vote
1
answer
206
views
Almost identical $\sigma$-algebras and measurability
Let $(X,\mathscr X,\mathbb P)$ be a probability space, $(Y,\mathscr Y)$ a measurable space, and $h:X\times Y\to\mathbb R$ a real-valued function measurable with respect to the product $\sigma$-algebra ...
- 405
1
vote
1
answer
120
views
Are there well-established notions of convergence of measures that take into account differentiable structure?
All the notions of convergence of measures that I know of are either in the purely measure-theoretic category (e.g. strong convergence, total variation), or in the topological category (e.g. weak ...
- 708
1
vote
1
answer
344
views
Is the Borel-Cantelli Lemma applicable here? [duplicate]
Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
- 161
1
vote
0
answers
168
views
Optimal transport-like problem where the objective depends on conditional probability distribution
$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
1
vote
0
answers
87
views
Symmetry of the isoperimetric profile
Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
- 801
1
vote
0
answers
191
views
Characterization of Poisson random measure in terms of Laplace transform
Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$.
A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if
$\pi(B)\sim\operatorname{...
- 167
1
vote
0
answers
74
views
Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 5,405
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 ...
- 710
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) >...
- 6,853
1
vote
0
answers
163
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 ...
- 11
1
vote
0
answers
58
views
Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
- 869
1
vote
0
answers
120
views
Existence of Time-Reversed Markov Kernels
Suppose I have a probability measure $\pi$ and a Markov kernel $q$ which leaves $\pi$ invariant, in the sense that
\begin{align}
\int_x \pi(dx) q(x \to dy) = \pi(dy)
\end{align}
Then, a (the) time-...
- 801
1
vote
0
answers
192
views
References about distances between singular probability measures
I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total ...
- 1,334
1
vote
0
answers
417
views
Defining density of a random function using Radon-Nikodym Theorem
Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...
- 213
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 ...
- 1,245
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 ...
- 505
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 ...
- 341
0
votes
1
answer
190
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?
- 109
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 ...
- 660
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 ...
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]- \...
- 31
-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 ...
- 109