All Questions
Tagged with ca.classical-analysis-and-odes pr.probability
146 questions
2
votes
0
answers
43
views
A distribution defined via an ODE for its Laplace trnsform
Fix a parameter $0 < c < \infty$.
As the solution to a certain problem,
there is a probability density function $f_c(t)$ on $0 < t < \infty$ with mean $1$ and
whose Laplace transform $L(\...
- 236
2
votes
0
answers
85
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,195
2
votes
1
answer
208
views
Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors
Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
- 349
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
1
vote
1
answer
148
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
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
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
2
votes
1
answer
131
views
Gradient flows and particle representations
I was looking into gradient flows and their particle representations, mostly in the context of probability.
A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
- 123
1
vote
0
answers
137
views
Relating $f(x)$ to its Laplace Transform for values other than $x=0$?
Suppose $X\in (0,1]$ is a random variable where $f(x)$ is its CDF and $g(t)$ is the Laplace Transform of $f(x)$. Tauberian theorems (Theorem 2.3 in Coqueret's "Approximation of probabilistic ...
- 2,442
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
0
votes
1
answer
59
views
Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$
A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...
- 6,853
2
votes
1
answer
373
views
Bakry-Emery criterion
The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class.
Consider $u(x)=|x|^2 + ...
- 41
3
votes
1
answer
442
views
$\cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots$ as $x \rightarrow \infty$
It is known that
$$
\cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1})
$$
Is it true that
$$
f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\...
- 1,608
1
vote
1
answer
299
views
Bounding $2$-Wasserstein distance and the $L^1$ distance
My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
- 730
3
votes
2
answers
226
views
Continuity of Radon transform w.r.t the angle
Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by
$$
R[f](w,b) := ...
- 6,853
1
vote
0
answers
107
views
$L^p$ inequality for "positively correlated" random variables
Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
- 7,193
1
vote
1
answer
95
views
Monotonicity of the top eigenfunction of the generator of a diffusion
Consider in 1D the operator given by
$$
\mathcal{L} = \frac{d^2}{dx^2} - V'(x)\frac{d}{dx},
$$
where $V(x)$ is a convex, sufficiently quickly growing potential, so that $\mathcal{L}$ has a complete ...
- 2,449
3
votes
2
answers
297
views
Does my construction always result in a stationary Poisson point process of intensity $1$? How so?
My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
- 3,098
0
votes
1
answer
162
views
Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
2
votes
0
answers
115
views
Least positive value of a random polynomial
Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...
- 3,062
2
votes
1
answer
329
views
Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
user44143
2
votes
1
answer
198
views
Bounds for the beta CDF
This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...
- 4,049
2
votes
1
answer
102
views
Approximation of $\Phi (p)$
I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
- 23
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
1
vote
3
answers
236
views
Can we calculate the probability that $f(x)$ is positive for a random $x\in(0,m)$ as $m\to\infty$? (uniform distribution)
Following my previous question here, I have this function
$$f(x)=10+3 \cos (ax-bx)+13 \cos (ax+bx)+2 \cos (\frac32 a x)+17 \cos (b x),$$
with $\frac ab \notin \mathbb{Q}$.
What is the limit
$$ \lim_{m\...
user215601
0
votes
1
answer
134
views
Integral bound for square of log derivative
I am currently facing the following problem:
Given a polynomial $f(x) = \sum_{s \in S_f} u_s x^s$, $f(0)\neq 0$, $\lvert S_f \rvert \leq t$ (i.e. $f$ is $t$-sparse) with $u_s$ coming as samples from i....
- 13
4
votes
1
answer
387
views
Asymptotic formula for fractional Laplacian
For the solution of
$$
\begin{cases}
\lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\
u^\epsilon=1 & \text{on } \partial \Omega
\end{cases}
$$
Varadhan ...
- 303
3
votes
2
answers
71
views
Optimal scaling of Lipschitz estimates in generalized geometric series
If we did not know it before, then wikipedia teaches us the generalized geometric series
$$\sum_{n \ge 0} \binom{n+k}{n} (1-\mu)^{n} \mu^k = \frac{1}{\mu}.$$
We can then study for $0 <\varepsilon &...
- 173
18
votes
0
answers
571
views
Fundamental Theorem of Algebra via multiple integrals
Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
- 60.5k
3
votes
0
answers
60
views
Existence, Uniqueness, and "ODE Characterization" of Minimizers for Variational Functionals from Large Deviations
A [classical result][1] of E. Lieb is that the functional
$$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dx~dy$$
for $\phi\in W^1(...
- 891
1
vote
1
answer
493
views
Sufficient and necessary conditions for decomposing the sum of random variables
Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and
$\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
- 550
1
vote
1
answer
266
views
Decomposition of the sum of nonnegative random variables [closed]
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
- 550
5
votes
2
answers
243
views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 804
1
vote
2
answers
88
views
Non-uniqueness of loop-erasure for continuous-time curves
Question. Is there a continuous curve in the plane that has a non-unique loop-erasure?
Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve ...
- 378
2
votes
0
answers
176
views
Uniqueness for measure valued ode
Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(\mathbb{N})$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's ...
1
vote
2
answers
136
views
Uniform boundedness of integral?
I have perhaps a very simple question where I lack some inutition at the moment: Is the expression
$$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
- 536
20
votes
1
answer
1k
views
Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
- 178
8
votes
2
answers
484
views
Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
- 60.5k
1
vote
0
answers
60
views
A determinantal mixture of probability densities
I came up with this operation after playing with determinantal point processes:
Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set
$$
f\star g(x)...
- 2,135
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
0
votes
1
answer
163
views
Sphere inversion in Riesz potential
I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554).
On page 546, the authors ...
- 3
2
votes
1
answer
286
views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
- 97
0
votes
1
answer
301
views
Is there a Gaussian process for the solutions of the wave equation?
Call a Gaussian process $g$ a prior for a topological space $X$ if the realizations of $g$ are (a.s.) contained in $X$ and dense.
Consider the 1D wave equation
$\frac{\partial^2}{\partial t^2}u(t,x)=...
1
vote
1
answer
74
views
Joint density of a quadratic function of entries of orthogonal matrix
$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
- 1,495
2
votes
1
answer
248
views
Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
- 1,495
5
votes
1
answer
170
views
Ratio of integrals with increasing dimension over Euclidean balls
Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
- 1,495
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
4
votes
1
answer
1k
views
Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization
The following result is well-known:
Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
$$H \left( (1 - \lambda)(x_1,y_1) + \...
- 536
2
votes
1
answer
675
views
Moment generating function of random unit vector
Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of
$$\mathbb{E}[\exp(X^Tv)]$$
for any $v$?
- 1,495
5
votes
1
answer
258
views
Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$
Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...
- 1,959