All Questions
Tagged with ca.classical-analysis-and-odes pr.probability
146 questions
191
votes
34
answers
81k
views
What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
99
votes
28
answers
14k
views
Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
- 1,695
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
3
answers
6k
views
Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.
What is the point of $\pi$-systems and
$\mathcal{D}$ / Dynkin /
$\lambda$-systems?
I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newer ...
- 1,771
21
votes
1
answer
3k
views
Intuitive Proof of Cramer's Decomposition Theorem
Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
- 4,952
21
votes
1
answer
725
views
What is the best probabilistic estimate from below for a random polynomial on an arc?
I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k x^...
- 61.9k
20
votes
1
answer
1k
views
Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
- 178
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
18
votes
0
answers
571
views
Fundamental Theorem of Algebra via multiple integrals
Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
- 60.5k
18
votes
0
answers
310
views
Profiles of very high dimensional functions
This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
- 96.4k
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
17
votes
2
answers
1k
views
The Bruss-Yor conjecture about an iterated integral
Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$?
This is a conjecture of F. ...
- 16.4k
16
votes
1
answer
2k
views
Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
- 497
15
votes
2
answers
3k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,135
14
votes
2
answers
783
views
Are two probability distributions uniquely constrained by the sum of their p-norms?
Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, ...
- 2,649
13
votes
1
answer
761
views
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
- 5,470
13
votes
0
answers
1k
views
Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839
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
11
votes
3
answers
4k
views
erfc lower bound
I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(-x^...
- 5,227
11
votes
1
answer
745
views
Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)
Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$.
Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$.
Find ...
- 3,527
11
votes
1
answer
1k
views
Has anyone seen this series?
I come across the following infinite series.
$$
\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
2
answers
2k
views
Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
11
votes
2
answers
1k
views
Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?
The expected distance $d$ of randomly selected points within a unit square to the square's center is
$d = \frac{1}{6} P$
where P is the universal parabolic constant
$P = \sqrt{2} + \ln{\left(1+\...
- 1,571
11
votes
1
answer
1k
views
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result.
The background for this problem comes from the composition of Brownian motion and ...
- 265
11
votes
3
answers
2k
views
Number of lattice points in a random disk of radius r
Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
- 19.7k
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$ ...
- 1,839
10
votes
4
answers
966
views
What m minimizes E(|m-X|^3) for a random variable X?
Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...
- 14k
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
1
answer
527
views
Analogue of Wick formula for orthogonal polynomials
n-point correlations of Gaussian random variables can be simplified with Wick expansion.
$$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots x_{i_{2n}}...
- 22.8k
9
votes
1
answer
396
views
Log-concavity of repeated convolution
Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
- 452
9
votes
2
answers
670
views
Product rules are local and covariance identities are global
Start with the simple identity:
$$(f(x) - a)(g(x) - b) + a(g(x) - b) + b(f(x) - a) = f(x)g(x) - ab.$$
If $a$ and $b$ are the respective values of $f$ and $g$ at some point, then, after dividing both ...
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
2
answers
484
views
Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
- 60.5k
8
votes
1
answer
449
views
What do smooth signatures give you?
My background is in rough paths theory.
In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
- 1,904
7
votes
5
answers
682
views
Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
- 93
7
votes
2
answers
988
views
Missing mass conjecture
Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture:
$$
\sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my ...
- 6,541
7
votes
1
answer
357
views
maximal coordinate on a sphere
What is the easiest (preferably without calculations) way to see that the mean value of $\max(x_1,x_2,\dots,x_n)$ on the sphere $\mathbb{S}^{d-1}= \{ (x_1,\dots,x_n):\ x_1^2+\dots+x_n^2=1 \}$ behaves ...
- 109k
7
votes
4
answers
4k
views
Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,512
7
votes
4
answers
854
views
Laplace transform on the cone of positive-definite matrices
The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.
One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
- 2,600
7
votes
1
answer
482
views
Continuous dependence of the expectation of a r.v. on the probability measure
$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by $\newcommand{\eA}{\...
- 34.7k
7
votes
0
answers
393
views
Fixed radius mean value property implies harmonicity?
Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
$f$ is harmonic.
$f$ satisfies the ball mean value property
$$
f(x)=\frac{1}{|B(x,r)...
- 527
6
votes
2
answers
5k
views
Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
- 185
6
votes
2
answers
927
views
Regularity of random Fourier series
The following two statements appear to be true (but do correct me if I am wrong):
The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm on ...
- 96.4k
6
votes
3
answers
423
views
Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
- 85.6k
6
votes
1
answer
575
views
Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel
I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
- 1,495
6
votes
3
answers
4k
views
Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes
This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...
- 5,935
6
votes
0
answers
398
views
semiclassical proof of Wigner semicircle
In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...
- 22.8k
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
5
votes
3
answers
707
views
Is there an analytic solution for this partial differential equation?
The Fokker-Planck equation for a probability distribution $P(\theta,t)$:
\begin{align}
\frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\...
- 73
5
votes
2
answers
641
views
Percolation Model and Complex Probabilities
Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...
- 2,044