All Questions
Tagged with real-analysis pr.probability
388 questions
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
89
votes
1
answer
21k
views
Is the largest root of a random polynomial more likely to be real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO.
The number of real roots of a ...
- 5,470
54
votes
4
answers
3k
views
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...
- 11.4k
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
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
30
votes
1
answer
2k
views
Have any numbers been proven to be normal that weren't constructed to be?
It's easy to construct an example of a number that's normal in a given base, but for most given numbers it's notoriously hard to prove that they're normal.
Has any number ever been proven to be normal ...
- 1,311
26
votes
4
answers
2k
views
$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?
Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex ...
- 271
21
votes
2
answers
981
views
What is the optimal speed to approach a red light?
Suppose from distance $d$, while driving at speed $v_0$, I notice that there's a red traffic light in front of me. Suppose that there are no other vehicles, my vehicle has perfect brakes, my maximum ...
- 18.9k
20
votes
1
answer
2k
views
How rich is the richest person in a society satisfying the Pareto principle?
The Pareto Principle roughly states that in many societies, the top 20% of people hold over 80% of the wealth. Suppose we had a society that satisfied this principle in every stratum of society - how ...
- 6,215
20
votes
3
answers
2k
views
Do convex and decreasing functions preserve the semimartingale property?
Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
- 17.1k
18
votes
2
answers
1k
views
An Entropy Inequality (generalized)
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability measure ...
- 351
16
votes
6
answers
3k
views
A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 2,239
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
14
votes
1
answer
416
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
13
votes
2
answers
1k
views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality
$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$
holds for any continuous convex function $g$ and any probability ...
- 303
13
votes
0
answers
710
views
Minimizing total variation under constraint
For $p\in[0,1]$, we write $\mathrm{Ber}(p)$
to denote the Bernoulli measure on $\{0,1\}$;
that is, $\mathrm{Ber}(p)(0)=1-p$,
$\mathrm{Ber}(p)(1)=p$.
For $n\in\mathbb{N}$ and $p=(p_1,\ldots,p_n)\in[0,1]...
- 6,541
12
votes
3
answers
2k
views
Looking for sufficient conditions for positive Fourier transforms
I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, ...
- 178
12
votes
2
answers
812
views
Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
- 6,694
12
votes
1
answer
525
views
An inequality about unit vector orthogonal to $(1,1,...,1)$
Does there exist a constant $\alpha>0$ such that the following holds?
$$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
- 1,495
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
12
votes
1
answer
694
views
History of the Jaccard distance $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$
I'm wondering where the relative probabilistic distance or Jaccard distance was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$...
- 24.8k
11
votes
2
answers
841
views
Computing the sum of an infinite series as a variant of a geometric series
I came across the following series when computing the covariance of a transform of a bivariate Gaussian random vector via Hermite polynomials and Mehler's expansion:
$$
S = \sum_{n=1}^{\infty} \frac{\...
- 984
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
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 12.8k
11
votes
1
answer
1k
views
Has anyone seen this series?
I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...
- 1,649
11
votes
1
answer
3k
views
A sum of two binomial random variables
Let $p\in(0,1)$, $n$ a positive even integer, $k,l\in\{0,\dots,n\}$, and $X_k\sim \text{Binomial}(k,p)$, $Y_{n-k}\sim \text{Binomial}(n-k,1-p)$ independent random variables. I would like to prove that
...
- 947
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
11
votes
2
answers
505
views
An inequality for copulas
Suppose that $f$ from $[0,\infty]$ onto $[0,1]$ is completely monotonic on $(0,\infty)$, and let $g$ be the inverse of $f$. For $(u,v)$ in $[0,1]^{2}$, define $C(u,v) = f(g(u)+g(v))$, and let $a = (u+...
- 3,443
11
votes
0
answers
381
views
Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
- 4,201
10
votes
2
answers
1k
views
Does a conditionally convergent sum with random signs converge almost surely?
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
- 6,215
10
votes
2
answers
9k
views
When do maximum and expectation commute?
Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [...
- 111
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
10
votes
2
answers
344
views
A moment problem
Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...
- 205
10
votes
2
answers
488
views
A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
- 128k
10
votes
1
answer
330
views
(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals
Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$.
In the case of bounding $E(XY)$...
- 181
9
votes
4
answers
952
views
What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?
My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples.
Let me start the discussion with ...
- 3,245
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
9
votes
1
answer
958
views
Quantitative bounds for multivariate central limit theorem
For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
https://...
- 93
9
votes
1
answer
556
views
A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
- 128k
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
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
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
9
votes
1
answer
380
views
Two dice yielding uniform distribution, part 2
Since this question is on the front page again, a generalization.
Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides ...
- 156k
9
votes
2
answers
519
views
The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
- 91
8
votes
2
answers
891
views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
- 205
8
votes
1
answer
391
views
On the limit of partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n_k\}_{k=1}^\infty$ such that
$$ \lim_{k\to\infty}\frac{1}{n_k}\sum_{i=1}^{n_k}\sum_{j=1}^{n_k}...
- 171
8
votes
1
answer
694
views
A generalization of Jensen's Inequality
Jensen's inequality is well known as
$$E\big[f(X)\big]\le f\big(E[X]\big)$$
where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also http://en.wikipedia.org/...
- 1,835
8
votes
1
answer
449
views
What do smooth signatures give you?
My background is in rough paths theory.
In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
- 1,904
8
votes
2
answers
330
views
q-Means and the mode of a distribution
Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and ...
- 640