All Questions
1,778 questions
12
votes
1
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2k
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Mean of i.i.d Random Variables With No Expected Value
Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
- 23k
12
votes
2
answers
1k
views
A variation of the Ryll-Nardzewski fixed point theorem
Is there a fixed-point theorem that implies the following result?
Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
- 45k
12
votes
1
answer
885
views
bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
12
votes
3
answers
2k
views
Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
- 3,699
12
votes
2
answers
2k
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Can we do better than Azuma-Hoeffding when the variance is small?
The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
- 1,955
12
votes
1
answer
927
views
On an Inequality of Lars Hörmander
Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...
- 640
12
votes
0
answers
284
views
Star-shaped Folner sequence
Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that $...
- 4,432
12
votes
0
answers
825
views
Eigenvalues of permutations of a real matrix: how complex can they be?
This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
- 13.4k
12
votes
1
answer
1k
views
Smoothness of distance function to a compact set
Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
$K$ is regular : ...
- 5,405
12
votes
2
answers
2k
views
What is the probability a random Turing machine is isomorphic to a DFA?
This is a sort of Chaitin/Omega constant type question, and so I do not expect this probability to be computable to arbitrary precision. However, it is also a very practical thing to know from the ...
- 2,392
12
votes
3
answers
1k
views
How to sample a uniform random polyomino?
A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...
- 7,857
12
votes
4
answers
4k
views
Mixtures of Gaussian distributions dense in distributions?
It seems that a mixture of Gaussians can approach any probability distribution, as the number of mixture components approaches infinity. Is this true? And if so, is it precise and correct to say ...
- 141
12
votes
3
answers
3k
views
Gaussian distribution, maximum entropy and the heat equation
I have asked this question on MathSE, but I got no replies, so I thought of trying here.
Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the ...
- 577
12
votes
3
answers
666
views
An "inchworm-like" random walk on an integer interval
Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...
- 121
12
votes
2
answers
1k
views
Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$
For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$.
Question
Given $\epsilon> 0$, find a "low-degree" ...
- 6,853
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
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
2
answers
3k
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Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
- 8,512
12
votes
1
answer
908
views
Equivalence of σ-convex hull and closed convex hull
Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
- 123
12
votes
3
answers
4k
views
How many proofs of the Polya's recurrent theorem are there?
Polya's famous theorem states that a simple random walk on $\mathbb{Z}^d$ is transient if $d>2$ and recurrent if $d=1,2$. This theorem is included in almost every textbook on probability theory.
...
- 121
12
votes
1
answer
2k
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Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
- 6,853
12
votes
2
answers
2k
views
Reference on Minty's trick
I am searching for a precise reference for the following result:
Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.
Assume that a sequence of nonnegative functions $(u_n)_n$ ...
- 3,425
12
votes
2
answers
1k
views
lower-bound for $Pr[X\geq EX]$
Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...
- 121
12
votes
2
answers
878
views
The ground state is signed and symmetric
Background
In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$...
- 39.1k
11
votes
2
answers
506
views
Minimization problem for convolution
Let $g(x)$ be a non-negative function supported on $[0,1]$. Let $g \ast g$ denote the convolution of $g$ with itself. Question: What is the smallest possible $L^1(0,1)$ norm of $g$, if I require that $...
- 2,207
11
votes
1
answer
950
views
Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
- 1,655
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
11
votes
2
answers
545
views
Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?
Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by $(zf)(g):...
- 1,550
11
votes
5
answers
2k
views
Coin flipping and a recurrence relation
How can one solve the following recurrence relation?
$f(n) = 1 + \frac{1}{2^n} \sum_{k = 0}^n {{n}\choose{k}} f(k)$
$f(0) = 0$
As it happens, I can show $f(n) = \Theta(\log n)$ through other means (...
- 501
11
votes
2
answers
714
views
Pursuit-Evasion type game on graph ("Flyswatter game")
An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...
- 113
11
votes
1
answer
451
views
Comparison of the absolute value of an operator with its positive parts
It is well known that the absolute value on operators does not satisfy the triangle inequality.
My question is whether for all positive operators $P,Q \in B(\mathcal H)$ is there a universal ...
- 3,984
11
votes
3
answers
9k
views
Time integral of an Ornstein-Uhlenbeck process
Let $X_t$ be an Ornstein-Uhlenbeck process solving $dx_t = \theta (\mu-x_t)\,dt + \sigma \,dW_t$.
The solution is known and given by:
$$ x_t = x_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \...
- 667
11
votes
1
answer
668
views
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
- 7,789
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 60.5k
11
votes
1
answer
3k
views
A sum of two binomial random variables
Let $p\in(0,1)$, $n$ a positive even integer, $k,l\in\{0,\dots,n\}$, and $X_k\sim \text{Binomial}(k,p)$, $Y_{n-k}\sim \text{Binomial}(n-k,1-p)$ independent random variables. I would like to prove that
...
- 947
11
votes
0
answers
364
views
Carleson's Theorem on Manifolds
Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
- 1,124
11
votes
2
answers
2k
views
Green's function of the Ornstein-Uhlenbeck operator
The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
11
votes
10
answers
1k
views
Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?
A huge amount of financial mathematics assumes Gaussian distributions of risks and Brownian movement of prices. What efforts have there been to replace these with heavy-tailed distributions? For ...
- 2,383
11
votes
7
answers
29k
views
Resultant probability distribution when taking the cosine of gaussian distributed variable
I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the ...
- 113
11
votes
1
answer
964
views
Quotients of l^infty
Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
11
votes
2
answers
2k
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Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $
In physics papers, the massless free boson has a definition involving an action:
$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$
The random functions $X(z)$ are ...
- 22.8k
11
votes
2
answers
1k
views
Heuristic lower bounds on small sums of roots of unity
Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...
- 4,589
11
votes
1
answer
2k
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Bounded operator on a normed space with empty spectrum
A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a ...
- 4,461
11
votes
1
answer
229
views
The set of boundary vectors of compact convex body has empty interior
Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$.
Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
- 41.9k
11
votes
1
answer
1k
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What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
- 395
11
votes
1
answer
636
views
A simple proof for a theorem of Szekeres and Turán
Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
- 24.7k
11
votes
1
answer
1k
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Show that these vectors are linearly independent almost surely
So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question.
Problem: I have $m<n$ real $...
- 344
11
votes
2
answers
2k
views
Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
11
votes
1
answer
336
views
Notions in the literature capturing the "symmetric" or "homogeneous" flavour of $L_p$?
This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise.
For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of ...
- 25.8k
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495