All Questions
Tagged with fa.functional-analysis pr.probability
596
questions
3
votes
1
answer
422
views
Spectral Radius and Spectral Norm for Markov Operators
My question concerns differences between the spectral radius $\rho$ and norm $\| \cdot \|$ of Markov operators in infinite-dimensional Banach spaces. This is far from my area of expertise, that is ...
- 560
3
votes
1
answer
752
views
Lipschitz function of subgaussian random variable
The similar and more general question is asked here, whose setting is random vectors.
Let $X$ be $\sigma$-sub-Gaussian and $f$ is a Lipschitz function w.r.t. constant $L$. How to prove can the ...
- 71
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
97
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 ...
- 305
1
vote
0
answers
59
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 ...
- 11
0
votes
0
answers
124
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) ...
- 460
0
votes
2
answers
96
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:...
- 109
0
votes
1
answer
58
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 ...
- 6,824
4
votes
2
answers
343
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 ...
- 647
2
votes
2
answers
225
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 ...
- 6,824
0
votes
1
answer
306
views
Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
- 131
1
vote
1
answer
163
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 $...
- 6,824
3
votes
0
answers
161
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
126
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 ...
- 555
4
votes
0
answers
125
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 ...
- 5,223