All Questions
950 questions
2
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0
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80
views
Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
- 5,407
2
votes
1
answer
533
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
- 51
2
votes
1
answer
299
views
Can this particular random matrix model be converted/related to any existing graph theory model?
Context:
This a sequel to the question: Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value
$1$ with probability $p$ ...
user123818
2
votes
0
answers
286
views
Probability of 4 Points being in Convex Configuration
Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane;
the idea is to process 4-...
- 13.2k
2
votes
1
answer
378
views
Distribution of the Gram matrix
Let $\mathbf{X}$ be an $m\times k$ random matrix ($m>k$) of rank $k$, having the density function $f_\mathbf{X}(X)$. What is the distribution of $\mathbf{Y}=\mathbf{XX}^T$? Basically my question is ...
- 141
2
votes
1
answer
450
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 167
2
votes
1
answer
437
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
- 167
2
votes
2
answers
119
views
Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
- 947
2
votes
3
answers
998
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
2
votes
1
answer
237
views
Is the following set compact w.r.t. the Wasserstein distance?
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \...
- 291
2
votes
1
answer
273
views
Two kinds of invariance of full conditional probabilities
Given a field $F$ of subsets of $\Omega$, we can define full conditional probabilities to be a function $P:F\times (F-\{ \varnothing \}) \to [0,1]$ such that:
$P(-|B)$ is a finitely-additive ...
- 2,555
2
votes
1
answer
470
views
Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing
Imagine I have a point source $p_0 = (x_0,y_0,z_0)$ that releases a point-like Brownian particle with a lifetime given by an exponentially distributed rate parameter $\lambda$. When the particle's ...
- 43
2
votes
1
answer
198
views
Average cluster size of a n-size vector
Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector.
My goal is to calculate the average cluster size for all ...
- 47
2
votes
1
answer
3k
views
Empirical estimator fot the total variation distance on a finite space
I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$):
$$...
- 1,655
2
votes
1
answer
170
views
Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
2
votes
2
answers
331
views
what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
2
votes
1
answer
810
views
Set of distributions that minimize KL divergence,
Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence,
is there a ...
- 167
2
votes
1
answer
453
views
Weak convergence of probability measures on weak versus strong dual
The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
2
votes
3
answers
335
views
Choosing $n$ times from $n$ objects
I am given $n$ objects and for $n$ times, I pick one of them with uniform probability and put it back after picking it.
For $k\in\{1,\ldots,n\}$ let $f_k$ denote the number of times that I have ...
- 48.8k
2
votes
2
answers
736
views
Submartingales bounded in $L^p$, $p>1$
Let $p>1$ be a real number. It is known that if $(X_n)_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X_n|^p), n\geq 0\} < +\infty$ ), then $(X_n)_{n\geq 0}$ converges a....
user102214
2
votes
0
answers
491
views
Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,026
2
votes
1
answer
157
views
Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?
Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
user75246
2
votes
1
answer
144
views
Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
2
votes
2
answers
485
views
Bayes statistics precisely formulated
I am trying to learn something about Bayesian statistics, however, I am struggling already with the simplest equations and, moreover, with the very basic questions: What are we given? What is our goal?...
- 334
2
votes
1
answer
534
views
Concentration inequality for maximum of gaussians
Let $Z_1,\ldots, Z_n$ be standardized Gaussian random variables and denote $\rho_{ij}=\mathbb{E}Z_iZ_j$. Can one give an asymptotically sharp bound for
$$\mathbb{P}\,(\max_{1\leq i\leq n}Z_i>x), \...
- 2,288
2
votes
1
answer
360
views
Large deviations for maximizer of random walk with drift
Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?
Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a ...
- 1,069
2
votes
1
answer
154
views
Reference Request for Couplings with Conditions
I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several ...
- 329
2
votes
2
answers
403
views
Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
- 23
2
votes
1
answer
638
views
$L^p$-convergence of submartingale
Let $p\geq1.$ Consider a $\mathcal{F}_k$-submartingale $(X_k)_k$ in $L^p.$ We can prove easily that $(X_k)_k$ converges in $L^p$ if and only if $(|X_k|^p)_k$ is uniformly integrable.
If $(X_k)_k$ was ...
- 249
2
votes
1
answer
834
views
Jacobian of changing of variables to singular value decomposition
It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues.
Now suppose I have a ...
- 884
2
votes
1
answer
235
views
Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution
$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...
- 1,826
2
votes
2
answers
564
views
Expected maximum of Laplacian random variables
Let $X={x_1, x_2,...,x_n}$ be Laplacian iid random variables with mean zero and scale $b$. Let $Y = \max(X)$.
Is there a closed-form expression for the expectation $E(Y)$? If not, is there any non-...
- 105
2
votes
1
answer
137
views
Design constraint systems over the reals
This question is inspired by the discussion at this problem.
Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
- 30.1k
2
votes
2
answers
185
views
Independence depth of linearly dependent random variables
Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise ...
- 2,618
2
votes
2
answers
710
views
Runs in coin flips
Let $P(j,k,n)$ be the probability of getting $j$ uniform runs of length $k$ from $n$ fair coin flips. What's the best way to compute $P$? I have no idea how difficult it might be; if it's a very ...
- 1,769
2
votes
1
answer
2k
views
The expected minimum Hamming distance within a set of randomly selected binary strings
If I randomly sample with replacement $P$ times from a set of all possible binary strings of length $L$, what is a good lowerbound on the expected minimum Hamming distance between any two of my $P$ ...
- 67
2
votes
1
answer
346
views
Relate the solid angle and surface measure of a surface
Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...
- 167
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
2
votes
2
answers
294
views
Imprecise Definition of a $\sigma$-algebra
I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
- 3,515
2
votes
1
answer
161
views
Expected value of global functions in renormalization group
This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance ...
- 3,515
2
votes
1
answer
189
views
Let $a\in S^d$, $b\in S^{d-1}$ be uniform on the spheres. How to show $\mathbb E[\frac{||a||_1}{\sqrt {d+1}}] \le\mathbb E[\frac{||b||_1}{\sqrt d}]$?
Let $a\in S^d$ and $b\in S^{d-1}$ be points chosen uniformly at random on the $(d+1)$ and $d$-dimensional spheres.
I'm interested in showing some inequalities regarding their norms, the simplest being:...
- 55
2
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0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
2
votes
2
answers
2k
views
The probability distribution of random variable of random variable
In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...
- 131
2
votes
1
answer
386
views
How balanced can abc triples be?
I was looking at the $241$ known "good" abc triples (i.e. with quality $\geqslant1.4$), wondering how frequently $a$ and $b$ would have more or less the same order of magnitude. The outcome is not ...
- 13.4k
2
votes
0
answers
159
views
Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit sphere
This is a follow-up question to the one asked here (the unit circle case). What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^2}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The ...
- 3,209
2
votes
1
answer
154
views
Smooth conditional expectation with nonsmooth "reverse"
I am looking for a concrete example of the following: $(X,Y)$ are real-valued random variables such that:
$E[Y|X]$ is smooth
$E[X|Y]$ is discontinuous
Even better, I'd like to see an example where ...
- 23
2
votes
1
answer
218
views
Probability distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
- 1,677
2
votes
1
answer
128
views
Is $\mathbb E\left[\frac{d}{||x||_1^2}\right]=O(1)$ for all $d\in\mathbb R^+$, where $x\in S^{d-1}$ is a random $d$-dimensional unit vector?
Let $x\in S^{d-1}$ be chosen uniformly at random from the $d$-dimensional unit sphere.
I want to show that there exists a universal constant $c\in\mathbb R$ such that $\mathbb E\left[\frac{d}{||x||_1^...
- 55
2
votes
1
answer
161
views
Trying to bound one functional by another functional
In my little research project, I faced the following problem: Assume that $\rho$ is a probability density function with support $[0,\infty)$ and mean $\mu >0$. Let $$H[\rho] = \iiint_{y,v,w\geq 0} \...
- 730
2
votes
1
answer
773
views
On the continuity of map $\Gamma$
Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>...
- 1,334