All Questions
26 questions with no upvoted or accepted answers
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
5
votes
1
answer
363
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
- 935
4
votes
0
answers
330
views
Book recommendation in functional analysis and probability
I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...
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
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 5,380
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
4
votes
0
answers
116
views
Log-Sobolev Inequalities for convex bodies
For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form:
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\...
- 519
4
votes
0
answers
269
views
Algebras and $\sigma$-algebras associated to random variables
Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^...
- 1,411
4
votes
0
answers
109
views
How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
- 7,633
3
votes
0
answers
129
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
3
votes
0
answers
158
views
$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
- 1,407
3
votes
0
answers
188
views
Invariant subspaces of Markov operators
I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
- 381
2
votes
0
answers
74
views
References for a class of Banach space-valued Gaussian processes
Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies
\begin{equation}
\mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
- 121
2
votes
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
2
votes
0
answers
69
views
How sensitive are the n-th step transition probabilities of the simple random walk to a small perturbation of an infinite graph?
Suppose that $G$ is an infinite, locally finite, connected graph.
Fix a vertex $o$ in the graph and for each $n$ and $x$ let $p(n,o,x)$ be the probability that a simple random walk (at each step a ...
- 4,304
1
vote
0
answers
90
views
What do $\gamma$-radonifying operators radonify?
In the second volume of their Analysis in Banach Spaces, Hytönen et al. introduce the notion of $\gamma$-radonifying operator more or less as follow.
Let $(\gamma_j)_{j\in\mathbf N}$ be a sequence of ...
- 407
1
vote
0
answers
143
views
Estimator for the conditional expectation operator with convergence rate in operator norm
Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively.
Let $L^2(\pi)$ ...
- 111
1
vote
0
answers
116
views
A formula involving the heat kernel on the universal cover of a punctured plane
I am looking for the earliest reference to the following formula:
$$
\int_0^\infty\tilde{P}(1,e^{i\alpha},t)\frac{dt}{t}=\frac{1}{\pi \alpha^2},\quad \alpha>0,
$$
where $\tilde{P}(x,y,t)$ is the ...
- 8,992
1
vote
0
answers
177
views
A question on Gaussian small ball probability
Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$
where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...
- 119
1
vote
0
answers
123
views
Derivatives of measures of bounded variation on intervals
Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:
Let $\operatorname{BV}[a,b]$ the space of all functions $f:[0, 1] \to \...
- 301
1
vote
0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
1
vote
0
answers
109
views
Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
1
vote
0
answers
83
views
Embedding random variables in infinite-dimensional spaces
Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
- 710
1
vote
0
answers
58
views
Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
- 869
1
vote
0
answers
417
views
Defining density of a random function using Radon-Nikodym Theorem
Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...
- 213
0
votes
0
answers
44
views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
- 11