All Questions
Tagged with pr.probability fa.functional-analysis
222 questions with no upvoted or accepted answers
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
12
votes
0
answers
196
views
UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
- 408
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
9
votes
0
answers
305
views
Convergence in $L^2$ of iterated expectations
Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...
- 1,068
8
votes
0
answers
182
views
Distribution domination for sums of independent random variables in Banach spaces
Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying
$$
\sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A),
...
- 131
8
votes
0
answers
211
views
Superharmonic functions and amenability
Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.
Assume that there is a set of non-...
- 4,705
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 109
7
votes
0
answers
549
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,124
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
6
votes
0
answers
281
views
Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
- 519
6
votes
0
answers
189
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 8,512
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
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
6
votes
0
answers
715
views
What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,512
6
votes
0
answers
295
views
Is there an idempotent measure on the free LD system?
This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...
- 3,547
5
votes
0
answers
159
views
Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
- 873
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
0
answers
74
views
Concentration bound on additive functions with constraints
Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...
- 107
5
votes
1
answer
774
views
Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
user128095
5
votes
0
answers
242
views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
- 3,669
5
votes
0
answers
166
views
Fourier basis for sub-Gaussian spaces?
Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with ...
- 519
5
votes
0
answers
120
views
L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension
For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
(...
- 4,010
5
votes
0
answers
244
views
Tensorization of Orlicz norm?
Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...
- 1,269
5
votes
0
answers
178
views
Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?
Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
- 2,306
5
votes
0
answers
161
views
$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?
Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \...
- 4,010
5
votes
0
answers
569
views
Argmax of random walk vs of Brownian motion
Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is,
$$
\begin{cases}
RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\
RW_{t-1} = RW_t - d + \...
- 1,069
5
votes
0
answers
221
views
Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
5
votes
0
answers
200
views
Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
- 8,512
5
votes
0
answers
537
views
Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
- 8,512
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
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
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
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 5,380
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
4
votes
0
answers
656
views
Eigenvalues of Matérn covariance function
Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
- 397
4
votes
0
answers
116
views
Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
4
votes
0
answers
116
views
Log-Sobolev Inequalities for convex bodies
For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form:
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\...
- 519
4
votes
0
answers
162
views
Are sums extremal for subgaussian concentration?
Bobkov and Houdre https://projecteuclid.org/euclid.bj/1178291721
showed that among all $f:R^n\to R$ that are $1$-Lipschitz
with respect to the $\ell_1$ metric,
the variance is maximized by sums. ...
- 6,541
4
votes
0
answers
322
views
Compactness of semigroups of one-dimensional diffusions
I have a question about semigroups of one-dimensional diffusions.
Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as
$$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$
It is ...
- 721
4
votes
0
answers
238
views
Does Novikov condition imply BMO martingale?
Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
- 195
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
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
271
views
Concentration of infinite-dimensional Gaussian measure
I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely:
$\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-...
- 537
4
votes
0
answers
107
views
Is Wiener's Tauberian theorem true in Wiener space?
Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.
Is the following true?
...
- 4,010
4
votes
0
answers
309
views
Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,287