All Questions
18,183 questions
2
votes
0
answers
89
views
How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$
Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
- 63
5
votes
2
answers
731
views
Probabilty measures that are both discrete and continuous
Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...
- 101
-2
votes
1
answer
260
views
On Impossible events
Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...
- 181
2
votes
1
answer
264
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 637
4
votes
0
answers
176
views
Is the test function topology a Mackey topology?
I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...
- 41
2
votes
1
answer
93
views
Why do distributional isomorphisms preserve joint distribution?
Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and
$$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$
be integrable random ...
0
votes
1
answer
92
views
Continuous selectors of a continuous multifunctin on a compact metric space
I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...
- 79
0
votes
0
answers
146
views
Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
- 307
1
vote
1
answer
259
views
Question about the proof of Propp-Wilson algorithm in Olle Häggström's book
Update: Oops! This is a stupid question and should be closed. The definition of the probability space that contains events $A_i$ requires using a single random stream.
I have difficulties ...
- 565
0
votes
1
answer
89
views
$c_0(2^{\kappa})$ does not embed in $\ell_\infty(\kappa)$?
How to prove that $(c_0(2^\kappa),\|\cdot\|_\infty)$ does not embed into $(\ell_\infty(\kappa),\|\cdot\|_\infty)$? Recall that $(c_0(2^\kappa),\|\cdot\|_\infty)$ is the Banach space of all families $(...
3
votes
1
answer
137
views
Thinning of (mixed) binomial point process
Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
- 133
1
vote
1
answer
181
views
Is $\sup_{f\in \mathcal{F}}\left|\int _Xfg \, d\mu\right|<\infty$ true for all $g\in L^\infty _\mathbb{C}(\mu )$?
Suppose that $(X,\mathcal{A},\mu )$ is a finite measure space. Let $\mathcal{F}\subseteq L^1_\mathbb{C}(\mu )$. If $\sup_{f\in \mathcal{F}}\left|\int _Xf\varphi \, d\mu\right|<\infty$ for all ...
- 15
5
votes
2
answers
458
views
Logarithm of a bounded operator
Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...
- 16.2k
1
vote
1
answer
186
views
Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
- 33
1
vote
0
answers
104
views
Amenability of $\textrm{w}_0(L^1(G))$
Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...
- 2,605
1
vote
0
answers
90
views
What do $\gamma$-radonifying operators radonify?
In the second volume of their Analysis in Banach Spaces, Hytönen et al. introduce the notion of $\gamma$-radonifying operator more or less as follow.
Let $(\gamma_j)_{j\in\mathbf N}$ be a sequence of ...
- 407
0
votes
1
answer
142
views
Does weak $H^1$ convergence imply $L^2$ convergence when multiplied with an exponentially decaying function?
I'm trying to see if given a sequence $\{f_n\}_n\in H^1$ which converges weakly in $H^1$ to a function $f_*$, the $L^2$ norm $\|R^2f_n\|_{L^2}^2$ converges to $\|R^2f_*\|^2_{L^2}$, where $R$ is a ...
1
vote
1
answer
95
views
Aggregate function of concave functions
I have two functions $f,g$ form $[0,1]^n$ to $R_{\geq 0 }$ that are concave and monotone.
Given a point $x =(x_1,\ldots,x_n)\in [0,1]^n$, I define $\operatorname{cube}(x) = [0,x_1]\times \ldots \times ...
- 107
2
votes
0
answers
126
views
Random walk with same directions and different step sizes
Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$.
Consider the following two random ...
- 499
7
votes
2
answers
349
views
Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
- 541
0
votes
0
answers
55
views
Sum of Skellam-distributed number of random variables
Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...
- 11
1
vote
0
answers
135
views
Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
- 969
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
1
vote
0
answers
210
views
How to show that every Von Neumann algebra is unital?
I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this :
He first considered the set of all non-empty finite subsets of the set of all projections ...
- 173
3
votes
1
answer
332
views
Sparse representation for continuous function?
I recently came across the field of "Sparse representation".
A talk is given here : https://www.youtube.com/watch?v=2bW4TkfTk-M.
The goal of sparse representation is taking a signal and ...
- 115
1
vote
1
answer
296
views
An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible
Given:
$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$
Can you help me come ...
- 11
2
votes
1
answer
335
views
Hahn-Banach theorem and ultrafilter lemma
I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...
- 95
0
votes
1
answer
235
views
If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\|\cdot \|$ its induced norm. Let $a: H \times H \to \mathbb R$ be a bilinear form. We say that
$a$ is coercive IFF there is $C>...
- 825
0
votes
1
answer
106
views
Weak convergence to product measure form conditional convergence of marginals
$\newcommand\Ac{\mathcal A}$
$\newcommand\BL{\operatorname{BL}}$
$\newcommand\reals{\mathbb R}$
$\newcommand\eps{\varepsilon}$
$\newcommand\pr{\mathbb P}$
$\newcommand\ex{\mathbb E}$
$\newcommand\...
- 1,731
2
votes
0
answers
92
views
Construct a Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
- 503
2
votes
1
answer
156
views
Some identities from graph theory and probability
The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
- 1,305
1
vote
1
answer
157
views
On the additive property of the subdifferential of lower semicontinuous functions
Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by
$$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
- 299
3
votes
0
answers
143
views
Stochastic braids
I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
- 2,152
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
1
vote
0
answers
55
views
functional resembling random variable norm
Let $N\subset\mathbb{R}$ be finite
and define
$$
A(N)
=
\sum_{i
\in\mathbb{Z}
}\min\{
2
^i,
|N\cap[2^i,2^{i+1})|
\},
$$
where
$\mathbb{Z}=\{0,\pm1,\pm2,\ldots\}$
and
$|\cdot|$ denotes set cardinality.
...
- 6,541
3
votes
1
answer
212
views
Distribution of the change in Hamming distance between two sequences
Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$....
- 31
1
vote
1
answer
176
views
Maximization of $\ell^2$-norm
Consider for $r,c>0$ the set
$$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$
Then I can show that $\inf_{x \in X_{r,c}} \|x\|_2 = 0.$
But is it ...
- 11
0
votes
2
answers
130
views
The weak limit of a sequence of argmax functions
I am currently working on a problem related to argmax functions in the context of operations research. I am trying to figure out if the weak limit of a sequence of argmax functions is again a argmax ...
- 79
-2
votes
1
answer
283
views
Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
4
votes
1
answer
143
views
Reflecting Brownian motion in disk
What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it?
The transition density ...
- 177
3
votes
0
answers
111
views
What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
- 2,689
7
votes
1
answer
156
views
Nearest neighbors on random complete graph
Consider the complete graph on $2n$ vertices, where the ${2n \choose 2}$ edges have distinct lengths in uniform random
order. So each vertex $v$ has a nearest neighbor $N(v)$, across the shortest ...
- 236
1
vote
0
answers
193
views
Marcus-SDE to Itô-SDE
In the field of stochastic calculus, everyone knows the Itô and Stratonovich integrals, as well as the conversion from Stratonovich to Itô SDEs.
The Stratonovich integration has the particularity of ...
- 11
0
votes
1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
0
votes
1
answer
338
views
How far does a random walker travel before returning to the origin?
Consider a (continuous time) simple symmetric random walk on $\mathbb Z$, starting from the origin. Let us denote this random walk by $\{X_t: t\geq 0\} $. It is well known that this random walk is ...
- 59
2
votes
0
answers
47
views
From large deviations to finite time probability tails
Cross-Post from Math.SE
Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that
$$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$
One possibility to ...
- 245
2
votes
0
answers
63
views
Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
4
votes
2
answers
199
views
Minimal norm of Fréchet subdifferential for function Lipschitz over its domain
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}$ be an extended real-valued function that is proper, lower semicontinuous, and Lipschitz continuous over its domain $\newcommand{\dom}{\text{dom}...
- 311
4
votes
1
answer
152
views
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
- 300
1
vote
0
answers
122
views
On the derivation of some asymptotic expressions involving combinatorics
My questions come from the supplementary material in a recent preprint Nonequilibrium statistical mechanics of money/energy exchange models. My first question comes from page 35. Specifically, suppose ...
- 730