All Questions
1,779 questions
4
votes
1
answer
275
views
Interesting Grothendieck topologies or coverages on the category Prob
I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck ...
- 91
4
votes
2
answers
4k
views
Expected global clustering coefficient for Erdős–Rényi graph
What is the expected global clustering coefficient $\mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph (ER-graph) $\mathcal{G}(n,p)$ (expectation is over the ensemble of all ER-graphs) as $n \...
- 151
3
votes
1
answer
131
views
Reference request: “A random integral and Orlicz spaces” [duplicate]
I need to find the following paper:
“K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques ...
- 77
3
votes
2
answers
271
views
For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?
Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields
$$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
- 708
3
votes
1
answer
162
views
Weak convergence of Dirichlet distributions to a "multi-Bernoulli" distribution
For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that, if $\sum\...
- 449
3
votes
2
answers
809
views
Moment problem for discrete distributions
Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B \...
user45183
3
votes
1
answer
218
views
Pathwise linearization of diffusion processes
Let $W$ be a standard $n$-dimensional Brownian motion, and $X$ the diffusion process given by the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $\mu: \mathbb R^n \to \...
- 6,273
3
votes
3
answers
203
views
$\mathbf{y}=f(\mathbf{x},\mathbf{z})=g(\mathbf{x})$ if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?
Let $\mathbf{y},\mathbf{x},\mathbf{z}$ are real-valued random vectors with possibly different dimensions. Assume $\mathbf{y}=f(\mathbf{x},\mathbf{z})$ for some function $f$.
If $\mathbf{z} \perp\!\!\!\...
- 193
3
votes
0
answers
144
views
Transitive closure of balanced bounded mass transport
Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
- 19.7k
3
votes
1
answer
206
views
Asymptotic results for smallest gap of Gaussian random matrix
For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.
...
- 288
3
votes
1
answer
243
views
Independent input feature z can be removed: if y=f(x+z,z), then y=g(x)?
Let $y\in \mathbb{R}$ and $\mathbf{x},\mathbf{z}\in\mathbb{R}^p$ be random variable and random vectors. Assume $y=f(\mathbf{x}+\mathbf{z},\mathbf{z})$ for some function $f$.
Is the following statement ...
- 193
3
votes
1
answer
116
views
Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
- 7,545
3
votes
1
answer
740
views
Centralizers in C*-algebra
Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...
- 179
3
votes
1
answer
127
views
Conditions for: (local) lipschitz stability of I-projection
The following post builds on this post; I'll begin by quoting the setting.
Background from Previous Question:
$\newcommand\SS{P}\newcommand\TT{Q}$Call a Gaussian probability measure $\SS$ on $\mathbb{...
- 364
3
votes
1
answer
362
views
Solution of the fractional Laplace equation on a ball
What is the expression of the (non $u \equiv 0$) solutions to
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
...
- 99
3
votes
1
answer
636
views
Non-existence of integral with respect to Poisson Random Measure
Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
- 2,895
3
votes
1
answer
476
views
Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
- 175
3
votes
2
answers
877
views
bound the tail distribution
Suppose that Z_1, ... , Z_n are binomial distributions with E[Z_i]=z_i.
If (Z_i) are pairwise independent, then, It's well known that the Chebyshev inequality can bound the tail distributions.
If (...
- 57
3
votes
1
answer
145
views
Minimization of an entropy type functional
Let $\mathcal P$ be the set of probability densities on $[0,1]$ with mean $1/2$, i.e. $p\in \mathcal P$ iff
$$\int_0^1 p(x)dx=1,\quad \int_0^1 xp(x)dx=\frac{1}{2}\quad \mbox{and}\quad p(x)\ge 0, ~~\...
user128095
3
votes
2
answers
590
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
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
3
votes
1
answer
527
views
Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
- 5,405
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
3
votes
1
answer
197
views
possibility of bounding one functional by another functional
This is a natural follow-up question related to one of my previous questions at here. Assume that $\rho$ is a log-concave probability density function with support $[0,\infty)$ and fixed mean $\mu >...
- 730
3
votes
2
answers
280
views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
3
votes
2
answers
973
views
How much larger than the relaxation time can the mixing time be?
The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer.
Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite ...
- 1,269
3
votes
4
answers
2k
views
Suitable references for the the Stone-von Neumann Theorem
Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...
- 1,719
3
votes
1
answer
348
views
Berry-Esseen bound in 2 dimensions for linear combinations
Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that $\mathbb{P}(...
- 2,288
3
votes
1
answer
142
views
How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
3
votes
0
answers
240
views
Derandomization barriers in complexity theory applicable as barriers to constructive arguments replacing probabilistic method
The probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object while finding that object may take exponential time.
(1) Is there any ...
- 13.9k
3
votes
0
answers
233
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
3
votes
1
answer
315
views
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
- 3,098
3
votes
1
answer
845
views
Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
- 31
3
votes
1
answer
476
views
Kernel of the Radon transform
Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...
- 21.8k
3
votes
1
answer
207
views
Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $\ell_\infty$
In this question (which I think may be interesting in its own) I was asking if we can find a copy of $\ell_\infty$ between a separable subspace $Y$ contained in $\ell_\infty(\Gamma)$ and the whole ...
- 271
3
votes
1
answer
169
views
Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace
To complete a proof I need to know if the following is true:
Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
- 271
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
3
votes
5
answers
986
views
Numerical Solution to Inverse Integral (Pseudo Random Number Generation)
If I have an arbitrary positive monotonically decreasing function $f(x), x \in [0,\infty]$, is there an 'efficient' method for finding $y$ in:
$r = \int\limits_0^y f(x) dx $
for a known $r \in [0, \...
- 144
3
votes
0
answers
78
views
Modify exponential family representation to a semimartingale
Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space.
We ...
3
votes
2
answers
331
views
Extreme couplings
Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...
- 1,655
3
votes
1
answer
192
views
Discrepancy of random bipartite graphs
This is a crosspost from MathStackExchange (original question).
Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$).
Define a ...
- 3,446
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
3
votes
1
answer
788
views
Almost sure convergence vs convergence of probability density functions
Let $X_n, X$ be $[0, 1]$-valued random variables whose laws are absolutely continuous with respect to Lebesgue measure. Suppose $X_n \to X$ a.s. Does this imply that the pdfs of $X_n$ converge to that ...
- 6,273
3
votes
1
answer
425
views
Chernoff style concentration bound for ratio of variables
Suppose I have two random variables $X_1$ and $X_2$. $X_1,X_2$ are both sums of random variables, and I can find Chernoff bounds for both variables independently. That is to say, I have
$$Pr[\mid X_i -...
- 33
3
votes
1
answer
746
views
Is inverse Laplace Transform of a power of $s$ a positive function?
It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
- 123
3
votes
0
answers
160
views
Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras
I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 3,477
3
votes
1
answer
153
views
Randomized version of Turán's theorem II
$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...
- 128k
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159