All Questions
Tagged with pr.probability fa.functional-analysis
616 questions
5
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2
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491
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Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
2
votes
1
answer
194
views
Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ?
Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?
BACKGROUND
The background of asking the value of $...
CommunityBot
- 1
0
votes
1
answer
915
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Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
- 5,721
6
votes
1
answer
355
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Why is the dimension of Gaussian variables is bounded by the dimension of the space?
I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."
For each $\epsilon >0$ there is a ...
- 1,302
5
votes
1
answer
577
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Does generator of continuous time random walk map heat kernel from L^2 to L^2?
Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...
- 269
1
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0
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466
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Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
3
votes
1
answer
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Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
CommunityBot
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5
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1
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781
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Does a log-concave function on a convex set extend continuously to the boundary?
Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
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7
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1
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If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
CommunityBot
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14
votes
5
answers
4k
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Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 279
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 11.4k
5
votes
1
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666
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Question regarding divergence
Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e.,...
- 498
1
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0
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309
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 44.4k
94
votes
1
answer
11k
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The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 2,097
4
votes
1
answer
985
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weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 3,069
3
votes
3
answers
2k
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
