All Questions
Tagged with pr.probability measure-theory
823 questions
1
vote
1
answer
112
views
Smallest ensemble of sets stable by any intersections and finite union
Let $E$ be a set, and let $S$ be any set of subsets of $E$, such that $S$ contains the empty set. Can you identify the smallest set of subsets $T$ of $E$ such that $T$ contains all elements of $S$, ...
- 128k
2
votes
2
answers
2k
views
The probability distribution of random variable of random variable
In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...
5
votes
2
answers
679
views
distance in terms of the variance between two absolutely continuous probability measures
Consider two probability measures $\mu_0$ and $\mu_1$ on $\mathbb{R}^n$, such that $\mu_0\ll \mu_1$. Then I can define a "distance" like quantitiy
$$
\mathrm{Var}_{\mu_1}\left(\frac{\mathrm{d}\mu_0}{\...
- 1,446
3
votes
2
answers
250
views
Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 10.6k
3
votes
2
answers
994
views
measurability of integrated functions
DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as ...
- 4,713
0
votes
0
answers
38
views
Reference on multifractal complex measures?
This is a cross-post of this physicsSE post; I am also posting it here since this question lies at the boundary of both physics and math.
I am learning about multifractal formalism recently. It seems ...
- 715
3
votes
1
answer
271
views
Expectation on a Polish space
I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...
- 128k
0
votes
0
answers
73
views
Criteria giving sufficient conditions for a Borel measure to have compact support
I am interested in criteria that guarantee that a Borel probability measure has compact support.
I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient ...
- 860
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 ...
- 869
3
votes
1
answer
156
views
Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
- 2,830
0
votes
0
answers
22
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
1
vote
0
answers
35
views
Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
- 715
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(\...
- 128k
2
votes
0
answers
54
views
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
1
vote
0
answers
124
views
Density of absolutely continuous measures on a Polish Space
Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
- 185
1
vote
0
answers
87
views
$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?
Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
- 309
10
votes
2
answers
5k
views
Approximate a probability distribution by moment matching
Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
- 1,446
18
votes
3
answers
1k
views
Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 4,713
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
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
0
votes
0
answers
114
views
Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
6
votes
1
answer
168
views
Weak convergence of random measures generated by non-negative martingales?
If I have a sequence of non-negative continuous martingales $(M_n(x))_{n\ge 1}$ on $x\in[0,1]$, i.e. for each fixed $n$, $M_n:[0,1]\to[0,\infty)$ is a continuous process, and for each fixed $x\in[0,1]$...
- 3,808
2
votes
2
answers
328
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
- 1,043
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 ...
- 128k
18
votes
1
answer
452
views
Is defining measures as functionals ever insufficiently general in practice?
Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here.
The way I learned measures was as set functions on a $\sigma$-algebra with certain ...
- 3,768
0
votes
1
answer
206
views
Some continuity issues of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
0
votes
0
answers
49
views
Weak convergence of Gibbs measures with converging energy functions
Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$
and let $Q$ be a fixed probability measure on $A$.
For each $\theta>0$, define the probability ...
- 503
1
vote
1
answer
128
views
Is a bounded convex function $g$ that is non-negative on this particular convex set also non-negative on the its closure?
Setup :
Let $S$ be the simplex on $\mathbb N$, i.e. the set of probability distribution on the natural numbers. Suppose we have $p\in S$ such that, for all $n\geq 1$, $p_n > 0$. For any $\emptyset\...
- 204
1
vote
1
answer
123
views
Properties of the relatively bounded probability distributions on the simplex over the natural numbers
Setup :
Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...
- 24.2k
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
$$
\...
- 5,380
3
votes
1
answer
192
views
Does $L^1$ boundedness and convergence in probability imply convergence in probability of the Cesaro sums?
Let $X_n$ be a sequence of random variables with uniformly bounded $L^1$ norm. Suppose $X_n$ converges in probability to $X \in L^1$.
Is it true that the Cesaro sums $Y_n := \frac{1}{n} \sum_{i = 1}^n ...
- 3,958
3
votes
1
answer
201
views
Continuity of conditional expectation
Let $X$ be a compact metric space, $\mu$ a Borel probability measure on $X$ and $f: X \to \mathbb{R}$ a continuous function. Consider an increasing sequence of $\sigma$-algebras $A_n$ so that for all $...
0
votes
1
answer
116
views
Existence and uniqueness of a posterior distribution
I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
12
votes
3
answers
891
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
- 1,446
0
votes
1
answer
450
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 724
0
votes
0
answers
73
views
Computationally efficient solution for the measure of central tendency minimizing Lp loss for p > 1
We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$
For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of ...
- 109
0
votes
1
answer
88
views
Can we lower bound this entropy by $\int_{\mathbb R^d} \rho^k (x) \, \mathrm d x$ and $\int_{\mathbb R^d} |x|^2\rho (x) \, \mathrm d x$?
We define $U : [0, \infty) \to [0, \infty)$ by $U(0) := 1$ and $U (s) := s \log s + (1-s)$ for $s >0$. Then $U$ is strictly convex. The minimum of $U$ is $0$ and is attained at $s=1$. Let $\mathcal ...
- 128k
4
votes
2
answers
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 ...
- 2,101
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
$$
\...
- 128k
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 ...
- 128k
2
votes
1
answer
158
views
Definition of average $\langle \langle \cdot \rangle \rangle$
I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
- 188k
1
vote
1
answer
100
views
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 $\...
- 128k
0
votes
0
answers
107
views
How to show that the map $ R $ here is measure-preserving
Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...
- 575
3
votes
2
answers
253
views
Number of atoms of a probability measure
Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$. Is the function
\begin{align*}
P\mathbb{R} &\to \mathbb{N} \cup \{\infty\}\\
\mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\...
- 29.8k
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 ...
- 5,474
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 34.7k
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 $\...
2
votes
2
answers
823
views
Kolmogorov-Smirnov distance and expectation
Let $P$ and $Q$ be two probability measures over $R^n$, with CDF denoted by $F_P,F_Q$, respectively (that is, $F_P(x)=P(\{x'\in R^n:x'\leq x\})$, where $\leq$ is taken componentwise. The Kolmogorov-...
- 128k
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 ...
4
votes
2
answers
378
views
A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, ...
- 17.2k