All Questions
18,179 questions
2
votes
0
answers
43
views
A distribution defined via an ODE for its Laplace trnsform
Fix a parameter $0 < c < \infty$.
As the solution to a certain problem,
there is a probability density function $f_c(t)$ on $0 < t < \infty$ with mean $1$ and
whose Laplace transform $L(\...
- 236
4
votes
1
answer
162
views
Topology on $O_M$, the space of slowly increasing smooth functions?
A smooth function on $\mathbb{R}^n$ is called slowly increasing if each of its derivatives is polynomially bounded. It seems that the collection of such functions is denoted as $O_M$.
Obviously, $O_M$ ...
- 3,477
2
votes
1
answer
124
views
Choice of the eigenbasis for the Dirac operator on $S^d$
This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.
Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with ...
- 3,477
3
votes
0
answers
122
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,649
9
votes
1
answer
257
views
Higher or lower? (#2)
$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
- 6,215
9
votes
0
answers
163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
1
vote
1
answer
100
views
Is Nelson-Symanzik positivity compatible with fermionic statistics?
Let $\{ S_n \}_{n =0}^\infty$ be a sequence of tempered distributions where $S_n \in \mathcal{S}'(\mathbb{R}^{nd})$ where $d \in \{2,3,4\}$ is fixed. Moreover, we put three additional conditions:
$...
- 3,477
0
votes
0
answers
50
views
Self-adjoint operators and index of quadratic form associated to it
Let $B$ a bounded self-adjoint operator on a real Hilbert space $H$ with an associated inner product $(\cdot,\cdot).$ Take $V=\operatorname{span}\{f_1, f_2, \ldots, f_n\}$ a finite dimensional ...
- 101
1
vote
0
answers
104
views
Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
5
votes
2
answers
256
views
On the closed convex hull of a weakly compact set
Let $H$ be an infinite-dimensional real Hilbert space and let $B$ be the closed unit ball of $H$. Let $K\subset B$ be a weakly compact set whose closed convex hull agrees with $B$. Question: does $K$ ...
- 283
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 5,405
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
3
votes
1
answer
435
views
What is the connection between these three methods of generating this sequence?
I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
- 39
1
vote
0
answers
71
views
Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
- 109
4
votes
1
answer
286
views
The gacha stamp collector’s problem
Let $N \gg n \geq 2$ be fixed natural numbers. In the Gacha stamp game, players are given an $N \times N$ square grid, with each point occupied by a unique stamp.
On every turn, they may choose a ...
- 6,215
1
vote
0
answers
41
views
Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
2
votes
0
answers
67
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,399
9
votes
0
answers
242
views
Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 128k
2
votes
1
answer
156
views
Measurability of $X$ with respect to $Y$ in conditional probability distributions
Let $\pi$ be a probability measure on $\mathbb{R}^2$ with respective marginals $\mu$ and $\nu$ such that $(X,Y) \sim \pi$.
Notation:
$\pi_{X=x}$ be the conditional distribution of $Y$ given $X=x$,
$\...
0
votes
1
answer
231
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 349
2
votes
0
answers
51
views
Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
- 143
19
votes
2
answers
2k
views
Higher or lower?
Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
- 6,215
1
vote
0
answers
176
views
If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.
Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
- 641
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 654
1
vote
1
answer
91
views
Positive definite kernels on compact interval $[0,1]$
From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ ...
- 216
5
votes
1
answer
188
views
Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
- 203
1
vote
1
answer
62
views
MGF relevant to modified 2nd kind Bessel
Given the moment-generating function
$$
m_{0}(t)=\frac{1}{\sqrt{1-t^2}}\,\text{ for }t<1,
$$ which corresponds to a distribution with density
$$
f(u) = \frac{1}{\pi}K_{0}(\frac{u}{\pi })
$$ where $...
4
votes
1
answer
66
views
Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
1
vote
1
answer
330
views
Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?
I am trying to study the converge of the series
$$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$
But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
- 287
3
votes
1
answer
91
views
Conditional Expectation in Diffusion Process
Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:
$$\left\{\begin{matrix}
...
- 143
0
votes
1
answer
57
views
Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 6,215
0
votes
0
answers
105
views
Generalizing the property of linear independent set in infinite dimensional TVS
Given a infinite dimensional Hilbert space $H$, and a countable set of vectors $\{v_{i}\}_{i=1}^{\infty}$. I want to study the following property of $\{v_{i}\}_{i=1}^{\infty}$:
There exists sequences $...
- 523
20
votes
1
answer
2k
views
How rich is the richest person in a society satisfying the Pareto principle?
The Pareto Principle roughly states that in many societies, the top 20% of people hold over 80% of the wealth. Suppose we had a society that satisfied this principle in every stratum of society - how ...
- 6,215
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
2
votes
1
answer
152
views
Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one.
I will write everything in detail for the sake of completeness.
Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
- 3,477
0
votes
1
answer
88
views
Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
- 305
4
votes
1
answer
132
views
Direct characterization of finite-dimensional $1$-injective Banach spaces
It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
- 2,793
2
votes
3
answers
338
views
Sum of RVs satisfying Bernstein condition on moments
Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that
$$
|\mathbb{E}[(X - \mu)^k]| \le \frac ...
- 367
3
votes
0
answers
145
views
Distribution of Brownian motion conditional on linear growth
Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.
Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event
$$ E_T := \{|B_s| \geq \lambda s\ \...
- 6,215
3
votes
1
answer
181
views
A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 6,215
5
votes
1
answer
379
views
Nuclear vs Banach spaces: compactness properties
A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.):
While normed spaces, ...
- 6,028
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
3
votes
0
answers
53
views
Can one parameterize transition rate matrices such that the stable distribution becomes independent of the transition rates?
I am trying to model a problem in which I need to describe a set of continuous time markov chains that depend on some parameter $v$. Thus, for each $v$, let $K(v)$ be $n\times n$ transition matrix ...
- 131
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
0
votes
0
answers
68
views
Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
1
vote
0
answers
67
views
A functional equation coming from a distribution function
Currently, I am working on a random series as follows. Let $\{Y_k\}$ be a sequence of i.i.d. Bernoulli random variables with expectation $p$. Then we define
$$
S = \sum_{k=1}^\infty \prod_{\ell=1}^k 2^...
- 33
3
votes
1
answer
182
views
Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
- 3,477