All Questions
Tagged with pr.probability fa.functional-analysis
616 questions
2
votes
1
answer
77
views
Measurability of random function with values in $C(K,E)$
Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random ...
3
votes
1
answer
100
views
Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 6,367
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 ...
- 63.9k
1
vote
0
answers
64
views
Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 149
0
votes
2
answers
101
views
Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
- 18.7k
0
votes
0
answers
128
views
When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 420
6
votes
3
answers
2k
views
Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 63.9k
4
votes
1
answer
218
views
Schauder basis of the Hardy space of semi-martingales
Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm
$$
\...
- 5,474
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 ...
- 128k
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 ...
- 271
2
votes
2
answers
228
views
Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
- 5,474
1
vote
1
answer
176
views
Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball
Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $...
- 6,853
3
votes
0
answers
175
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
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 ...
- 63.9k
2
votes
1
answer
197
views
$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?
Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
- 10.3k
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
0
votes
1
answer
105
views
Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 716
5
votes
1
answer
331
views
Gaussian-to-Gaussian transformations are affine a.e.?
Let $\mathcal{G}_n = \{ N(\mu,\Sigma) ; \mu \in \mathbb{R}^n, \Sigma > 0\}$ be the collection of Gaussian distributions on $\mathbb{R}^n$ with full support.
If $f : \mathbb{R}^n \to \mathbb{R}^k$ ...
- 6,045
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
1
answer
185
views
Cameron-Martin space of product space
Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
- 150
2
votes
0
answers
57
views
Right spectral gap of vector of two independent Markov chains
Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
- 203
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
1
vote
0
answers
96
views
Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
1
vote
0
answers
99
views
Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 63.9k
8
votes
5
answers
685
views
Distributions of distance between two random points in Hilbert space
Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$.
So $D$ has some probability ...
- 128k
2
votes
1
answer
374
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 + ...
- 1,081
1
vote
1
answer
241
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
- 128k
1
vote
0
answers
37
views
If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?
Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...
- 167
5
votes
2
answers
2k
views
Relationship between KL, chi-squared, and Hellinger
There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
- 128k
2
votes
2
answers
1k
views
Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
- 103
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,074
0
votes
1
answer
296
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 128k
3
votes
1
answer
155
views
What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?
The Tracy–Widom distributions admit many interpretations.
One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...
- 1,765
2
votes
0
answers
112
views
Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?
I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...
0
votes
1
answer
284
views
Is there a "smooth Kantorovich-Rubinstein duality" for Wasserstein distances on smooth/Euclidean space?
Let $X$ be a compact metric space, and fix an arbitrary point $x_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W_1$ on the set of Borel probability measures ...
- 128k
1
vote
0
answers
417
views
Conditions for equivalence of RKHS norm and $L^2(P)$ norm
Let $K$ be a psd kernel on an abstract space $X$ and let $H_K$ be the induced Reproducing Kernel Hilbert Space (RKHS). Let $P$ be a probability measure on $X$ such that $H_K \subseteq L^2(P_X)$ and ...
- 6,853
1
vote
1
answer
113
views
Distinction between $\mathcal{C}\left([0,T],\mathcal{P}(\mathcal{X})\right)$ and $\mathcal{P}\left(\mathcal{C}\left([0,T],\mathcal{X}\right)\right)$
Suppose that $\{X^i_t\}_{1\leq i\leq n;\,t\in [0,T]}$ is an interacting particle system prescribed by certain SDEs, with each $X^i \in \mathcal{X}$ (the state space). Define the associated empirical ...
- 128k
1
vote
0
answers
135
views
Description of state space of $C(K,M_n)$?
Edit: closed convex hull added.
I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space.
My guess would be that these are the closed convex hull of states on $C(...
- 598
2
votes
0
answers
131
views
Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
3
votes
2
answers
203
views
Computing moments of discrete probability distribution
I am wondering whether or not there is a computationally efficient way to compute the first $N$ moments $$m_k=\sum_{n=1}^{N}p_nx_n^k,\;\;\;\;k=1,...,N$$ of a probability mass function with mass $p_1,.....
- 1,171
1
vote
0
answers
254
views
Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 5,405
2
votes
0
answers
69
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
5
votes
1
answer
284
views
Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 5,474
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
1
vote
0
answers
152
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
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
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 ...
- 63.9k
3
votes
1
answer
350
views
Talagrand's inequality for L1 norm
I have a series of $n$ independent random variables $X_1,\ldots, X_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 ...
- 63.9k
2
votes
0
answers
122
views
Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation
Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
- 6,853