All Questions
1,766 questions
4
votes
1
answer
222
views
Memory game inspired problem
Motivation. As I was playing the pairs-matching game "Memory" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling ...
- 48.9k
4
votes
0
answers
617
views
Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
- 1,444
4
votes
2
answers
407
views
Can we define smooth diffeomorphisms on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature?
Can we define smooth diffeomorphisms on the Hilbert cube$[0,1]^\mathbb{N}$ ? Has it been done in the literature?
For example, let $E$ be the set of bijections $f$ of the Hilbert cube, such that $f((...
- 403
4
votes
1
answer
830
views
Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
- 141
4
votes
1
answer
424
views
Hilbert space representation of a vector in terms of a continuous eigenbasis
Let $\mathscr{H}$ be a complex Hilbert space and $A$ be an Hermitian operator $A: \mathscr{H}\to \mathscr{H}$. Suppose, for a moment, that $A$ has a set of discrete eigenvalues $\{\lambda_{n}\}_{n\in \...
- 1,305
4
votes
1
answer
447
views
Area enclosed by Brownian motion (without winding number)
The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
- 29.8k
4
votes
1
answer
387
views
Asymptotic formula for fractional Laplacian
For the solution of
$$
\begin{cases}
\lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\
u^\epsilon=1 & \text{on } \partial \Omega
\end{cases}
$$
Varadhan ...
- 303
4
votes
2
answers
434
views
A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 433
4
votes
4
answers
797
views
On Köthe sequence spaces
I asked this a week ago at math.stackexchange, but without success.
As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
- 7,426
4
votes
2
answers
410
views
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$
The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree ...
- 1,199
4
votes
0
answers
970
views
Expected operator norm of inverse Wishart matrix
Let $ W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum ...
- 41
4
votes
2
answers
447
views
Heat equation and evolution of number of critical points
Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
4
votes
1
answer
225
views
Is the inequality of the random matrices correct?
I am not familiar with random matrices but I need to confirm the correctness of the inequality below.
Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let
$A_1,\ldots, A_n$ be $m\times m$ ...
- 131
4
votes
2
answers
16k
views
Minimum of exponential distributions
Consider $n$ independent random variables $X_i \sim \exp(\lambda_i)$ for $i = 1,\dots,n$. Let $\lambda = \sum_{i=1}^n \lambda_i$. Of course, the minimum of these exponential distributions has ...
- 231
4
votes
3
answers
161
views
Find distribution that minimises a function of its moments
Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...
- 223
4
votes
0
answers
589
views
Optimal transport between two distributions in a Markov chain
In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...
- 6,853
4
votes
0
answers
756
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
4
votes
1
answer
228
views
Continuity upgrade for nonlinear maps
Let $E,F,G$ be topological vector spaces such that $F\subset G$ with continuous embedding.
By continuity upgrade I mean the following phenomenon: In some circumstances a continuous linear map $f:E\...
- 779
4
votes
0
answers
182
views
Determine the minimal elements of a Dynkin system generated by a finite set of finite sets
(This is a refined version of https://cs.stackexchange.com/q/144371)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
- 5,822
4
votes
0
answers
291
views
trace-class embeddings
There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
- 3,388
4
votes
2
answers
367
views
Fokker-Planck equation for a truncated process
Let $X_t = x + bt + \sigma W_t$ be an arithmetic Brownian motion, where $x$ is a random variable independent to $W$, and $\sigma>0$. Suppose the initial distribution is given by $\mathbb P(X_0 \in ...
- 1,399
4
votes
0
answers
149
views
Cyclic vectors for the translation operator
Let $b\in \mathbb{R}\neq 0$, and consider the translation operators:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known ...
- 5,405
4
votes
1
answer
1k
views
Where to learn about parabolic Hölder spaces and when to use them
Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...
- 45
4
votes
1
answer
280
views
Reference request: Baire's theorem for operator ranges
Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
- 12.6k
4
votes
1
answer
2k
views
Bounding Entropy in terms of KL-Divergence
Let $h(X)$ be the differential entropy of a continuous random variable $X$ with density $f$, and let $Y$ be another continuous random variable with density $g$. If $KL(X\mid\mid Y)$ is the Kullback-...
- 49
4
votes
1
answer
520
views
Compactly generated Banach spaces
Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (in the original topology) subset $K$. Do we know anything "nice" about $X$, ...
- 3,115
4
votes
2
answers
267
views
Grand-canonical Gibbs measure for continuous systems
Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
- 3,535
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 53
4
votes
1
answer
365
views
Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
4
votes
2
answers
565
views
A relation between the second moment of a distribution and one of its particular probability
I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...
user318428
4
votes
1
answer
197
views
On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 128k
4
votes
1
answer
412
views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 617
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
4
votes
1
answer
275
views
Interesting Grothendieck topologies or coverages on the category Prob
I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck ...
- 91
4
votes
0
answers
509
views
Good reference for noncommutative $L^p$ spaces
I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
- 41
4
votes
1
answer
637
views
Characterizations of the GOE/GUE family of distributions
This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
- 4,952
4
votes
2
answers
255
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 825
4
votes
1
answer
2k
views
Covariance of points distributed in a n-ball
Is there a closed form expression for the covariance of a uniform distribution in a n-ball? I would like to develop a test for vector sums of points sampled from a uniform distribution in a n-ball. I ...
- 41
4
votes
0
answers
502
views
Every convex sequentially closed set is closed
Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed.
Is there some description ...
- 105
4
votes
0
answers
143
views
For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?
Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
- 1,955
4
votes
1
answer
228
views
Haar-null union of dense subsets
Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...
- 63
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
4
votes
1
answer
245
views
Probability of a vertex being a "degree-celebrity" in a random graph
If $G(n,p)$ is a random graph of the Erdös-Rényi model,
what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$
Please feel free to relate answers to other ...
- 13.2k
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
4
votes
2
answers
280
views
An extremal type problem on segments
I am interested in the following extremal-type problem.
Let us define $\Psi$ by
$$\Psi(x)=\max_{f\in L^2[0,x] \,\,\text{with}\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$
on $(0,\...
- 41
4
votes
1
answer
317
views
Infinite Tree with Poisson Clocks
Let $\mathcal{T}$ be the infinite countable $3$-regular tree graph. Pick a vertex in this graph, call it the root. Let the root carry the value $0$.
Next, assign $1$ to the neighbours of the root. ...
- 403
4
votes
1
answer
521
views
What is the category of algebras for the finitely supported measures monad?
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...
- 1,313
4
votes
2
answers
730
views
Finite dimensional approximations of operators on Hilbert spaces
Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n \...
- 614
4
votes
0
answers
309
views
When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for zero-centered Gaussian $x_i$?
Suppose $x_i\in \mathbb{R}^d$ is sampled IID from $\mathcal{N}(0,H)$. Let $A_i=(I-x_i x_i^T)$ and assume $d$ is large. What are necessary conditions for the following to converge with probability 1?
$...
- 2,252
4
votes
2
answers
378
views
Basic properties of expectation in non-separable Banach spaces
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear ...
- 931