All Questions
Tagged with pr.probability fa.functional-analysis
616 questions
94
votes
1
answer
11k
views
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 ...
- 23.5k
65
votes
9
answers
12k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
40
votes
5
answers
5k
views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
30
votes
1
answer
1k
views
Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
- 6,853
27
votes
5
answers
3k
views
Nice applications for Schwartz distributions
I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and ...
26
votes
3
answers
11k
views
L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 833
25
votes
2
answers
4k
views
Understanding of rough path
A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...
- 1,399
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 8,512
23
votes
1
answer
1k
views
How do mathematicians and physicists think of SL(2,R) acting on Gaussian functions?
Let $\mathcal{N}(\mu,\sigma^2)$ denote the Gaussian distribution on $\mathbb{R}$:
$$ \mathcal{N}(\mu,\sigma^2)(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$
A Gaussian ...
- 1,101
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
- 361
21
votes
2
answers
2k
views
Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
- 24.9k
21
votes
2
answers
2k
views
A strange variant of the Gaussian log-Sobolev inequality
Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
- 562
20
votes
2
answers
922
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
- 3,946
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
18
votes
4
answers
1k
views
Reference for a strong intermediate value theorem for measures
Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
- 5,407
18
votes
1
answer
996
views
Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 8,512
17
votes
2
answers
953
views
Convexity of spectral radius of Markov operators, Random walks on non-amenable groups
Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$.
Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication.
Question
Under which conditions can we show that ...
- 171
16
votes
3
answers
708
views
An inequality for two independent identically distributed random vectors in a normed space
Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$?
Some background information on ...
- 128k
16
votes
3
answers
918
views
What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$
Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...
- 311
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
15
votes
2
answers
2k
views
Intuitive explanation of Dvoretzky's theorem
I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...
- 96.4k
15
votes
2
answers
3k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,135
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
5
answers
4k
views
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 ...
- 943
14
votes
2
answers
892
views
Do distance functionals separate probability measures?
Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
- 459
14
votes
2
answers
2k
views
Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?
Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...
- 3,968
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
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
13
votes
2
answers
656
views
Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 1,495
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
13
votes
1
answer
3k
views
Does this metric have an official name? Lévy metric? Ky Fan metric?
Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...
- 6,287
12
votes
3
answers
870
views
Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
- 8,512
12
votes
3
answers
2k
views
Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
- 243
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
12
votes
0
answers
196
views
UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
- 408
11
votes
1
answer
320
views
Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$
For $x\in \mathbb{R}^d$, an elementary computation yields that
$$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=-\frac{1}{p^2}\operatorname{...
- 775
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
11
votes
1
answer
642
views
Random walk origin return monotinicity
Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...
- 4,952
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
10
votes
2
answers
2k
views
Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
10
votes
2
answers
559
views
Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
10
votes
1
answer
652
views
Extending state space to make a process Feller
Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...
- 29.8k
10
votes
2
answers
926
views
Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
10
votes
2
answers
2k
views
When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
- 8,512
10
votes
1
answer
253
views
Approximation via finite rank Cameron-Martin projections
Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...
- 29.8k
10
votes
1
answer
274
views
Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric
Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.
Denote by $\gamma_{A}$...
- 1,034