All Questions
Tagged with reference-request pr.probability
792 questions
2
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0
answers
62
views
On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 505
3
votes
0
answers
176
views
A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
- 5,380
5
votes
1
answer
2k
views
Mathematics research relating to machine learning
What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
- 173
1
vote
0
answers
115
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Modulus of "set"-continuity for Wiener Field
My question concerns some "set-wise" continuity properties of Gaussian random fields, more specifically of Wiener fields (see definition here: https://encyclopediaofmath.org/wiki/...
- 73
3
votes
1
answer
229
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Maximum cardinality of separated sets in the Hamming distance
This question is motivated by section 15.1 (Codes) of Alon and Spencer's The probabilistic method.
Fix $\alpha<\frac{1}{2}$ and for each $n\in\mathbb{N}$ let $\{0,1\}^n$ be the length $n$ binary ...
- 10.6k
3
votes
2
answers
478
views
Random spanning trees probability problem
We are given a simple connected graph $G(V,E)$ with vertex and edge set $V$ and $E$ respectively. For any vertex $v\in V$, let $D_T(v)$ the degree of $v$ in a uniformly generated random spanning tree $...
- 1,677
5
votes
0
answers
117
views
U-statistics with infinite moment
Let $X_1,X_2,\ldots$ be i.i.d. in a probability space $(\Omega,\mu)$, $F(X,Y)$ a non-negative measurable function on $(\Omega\times \Omega,\mu\times \mu)$ such that $\mathbb{E}\, F(X,Y)=\infty$. What ...
- 109k
2
votes
1
answer
291
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Joint distribution for sticky Brownian motion
$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by
\begin{gather*}
dX_t=1_{[X_t\neq 0]}dB_t\\
L_t(X)=\int_0^t 1_{[X_s=0]}ds,...
- 5,380
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
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
1
vote
1
answer
91
views
Density of eigenvalues of empirical covariance matrix of vectors uniform on the sphere
Is anyone able to point me to a reference for this?
Let the rows of $X \in \Re^{n\times d}$ be i.i.d. uniform on the sphere of radius $\sqrt{d}$ in $\Re^d$. What is the density of the eigenvalues of $...
- 337
1
vote
1
answer
124
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References: error and stability estimates for information projection
$\newcommand\SS{P}\newcommand\TT{Q}$I will call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ isotropic if its covariance matrix is diagonal with non-vanishing determinant; i.e. $\Sigma_{i,i}&...
- 364
-3
votes
1
answer
154
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Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$
I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that ...
- 147
10
votes
2
answers
1k
views
Simple proof of sharp constant in DKW inequality
The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, ...
- 390
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
0
votes
1
answer
188
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Equality cases in a certain case of Jensen's inequality
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
$$...
- 128k
1
vote
1
answer
97
views
A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 128k
2
votes
1
answer
153
views
Interpolation theorem for general rough paths
In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...
- 1,904
6
votes
1
answer
112
views
What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map?
A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. ...
- 53
0
votes
1
answer
115
views
Reference request: Gaussian branching processes
(Q1) Are there known constructions of branching general Gaussian processes (preferably in continuous time)? Something like branching fractional Brownian motion or OU.
Also, (Q2) what are the modern ...
- 620
2
votes
1
answer
198
views
References on tilting distributions
I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent):
...
- 147
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
2
votes
2
answers
206
views
Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions
Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).
Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...
- 457
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
3
votes
1
answer
266
views
A linearly distributed version of the balls into bins problem
Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
- 1,677
0
votes
1
answer
178
views
Ito-Levy decomposition for $\alpha$-stable processes?
The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?
- 19
3
votes
1
answer
152
views
Mutual information in large deviation theory
Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...
- 1,608
2
votes
1
answer
309
views
Upper bound Wasserstein distance by $\chi^2$ distance
Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (...
- 730
5
votes
1
answer
323
views
Percolation: at what length scale do we see it?
Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p_c(d) \in (0,1)$ such that $p>...
- 2,155
4
votes
0
answers
219
views
Conditional distribution of steps of random walk given the sum
Set-up. Consider a random walk $S_n=\sum_{i=1}^n X_i$, where $\{ X_i, 1\leq i < \infty \} $
is a sequence of i.i.d. random variables with distribution $\mu$, $\mathbb{E}X_1 = 0$. Let $a > 0$.
...
- 724
5
votes
0
answers
135
views
Criteria for tightness of Gaussian measures on Banach spaces
In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
- 505
4
votes
0
answers
134
views
Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 5,380
3
votes
1
answer
257
views
Triangle equality for cosine similarity in high dimensions
I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$:
$$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$
Where $\cos(x,y)$ gives cosine of the angle ...
- 2,442
3
votes
1
answer
466
views
Equivalence between two fractional Sobolev spaces
For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...
- 161
6
votes
2
answers
813
views
Kolmogorov's approach to probability theory
Question:
Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s?
Context:
In 1965, Andrey Kolmogorov considered three approaches to ...
- 3,871
7
votes
1
answer
186
views
$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary
Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. ...
- 1,677
0
votes
1
answer
153
views
Maximum of a certain Gaussian field
Let $S_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z_x)_{x \in S_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e.
$$
\...
- 1,295
7
votes
1
answer
391
views
Idempotent splitting for Markov kernels
Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation,
$$e(A|x) = \int_X e(...
- 6,406
0
votes
0
answers
69
views
Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface
Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
- 269
4
votes
1
answer
265
views
Bounds on discrepancy metric of product measures
Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...
- 2,712
2
votes
1
answer
268
views
An oversimplified model for optimal distribution of wealth
Consider the following, overly simplified, model for determining an optimal wealth distribution for society:
Let $X$ be a random variable, which will model the distribution of wealth in a society.
The ...
- 1,326
1
vote
1
answer
425
views
Invariance principle: Brownian bridge and random walk conditioned on end point
Let $\{X_i, i \in \mathbb{N}\}$ be a sequence of non-lattice i.i.d. centered random variables, $\mathbb{E} |X_1| ^3 < 0$. Let $S_n = \sum\limits _{i=1} ^n X_i$ be the corresponding random walk and ...
- 724
1
vote
0
answers
54
views
Idiosyncratic measure on a space of nondecreasing PL paths from the unit interval to itself
The image below shows a piecewise linear nondecreasing path from (0,0) to (1,1) produced via a simple random process. First, we pick a point on the diagonal from (0,1) to (1,0) uniformly at random. ...
- 15.4k
2
votes
0
answers
93
views
Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...
- 4,049
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
$$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
- 708
3
votes
1
answer
220
views
Carne-Varopoulos bound and stationary measure
Let $\Gamma$ denote the Cayley graph for a finitely generated group $G$, and let $p_n(x, y)$ denote the transition probability that a random walk starting at $x$ reaches $y$ at time $n$. A famous &...
- 31
1
vote
0
answers
163
views
A basic formula for the falling factorial
Whis is a question I asked on Math.SE, but didn't get any response.
Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e.
$$
X(\omega): = \sum_{A \in \...
- 1,533
10
votes
1
answer
492
views
(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$
Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits_{i=0}^{\infty} A^{2i}=\left( ...
- 767
2
votes
1
answer
222
views
Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?
Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...
- 6,853
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
3
votes
0
answers
114
views
Book and article recommendations with the purpose of studying the intersection between probability theory and lattice theory
Lately, I have been studying probability theory and lattice theory separately and I would like to investigate ideas which relate both subjects together. Having said that, I would like to know if ...
- 161