All Questions
74 questions
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
0
votes
1
answer
172
views
Taking away the "almost sure" [closed]
Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a ...
- 87
2
votes
1
answer
363
views
Integration against Borel measures on compact Hausdorff spaces
I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...
- 165
4
votes
3
answers
712
views
Measure of intersections in probability spaces
Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
...
- 167
2
votes
1
answer
144
views
Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
1
vote
1
answer
166
views
Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
- 1,835
3
votes
1
answer
304
views
Question abouth Skorokhod representation of random variables
It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...
- 1,835
0
votes
1
answer
557
views
Is the limsup or liminf of n-wise independent events independent?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
9
votes
1
answer
950
views
Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
- 247
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
6
votes
1
answer
1k
views
About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
- 8,071
1
vote
0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
0
votes
0
answers
145
views
Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
- 215
2
votes
0
answers
160
views
Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?
Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ \...
- 3,245
4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
- 1,649
4
votes
1
answer
1k
views
General version of Skorokhod representation of random variables
Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
- 941
1
vote
0
answers
416
views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 61
6
votes
2
answers
720
views
Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 293
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,200
11
votes
2
answers
2k
views
Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 12.8k
5
votes
0
answers
369
views
Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
- 4,952
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
- 1,839