Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
8,632
questions
7
votes
2
answers
637
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Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
1
vote
2
answers
149
views
Distribution similar to PPP
According to the definition of Poisson Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed ...
0
votes
1
answer
170
views
Can I obtain the limit value of a linear spectral statistics using Stieltjes transform?
I would like to calculate the limit value of a linear functional
\begin{equation}
\lim_{n\rightarrow\infty}\mathcal{I}_n=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n f(\lambda_i)=\lim_{n\...
22
votes
4
answers
5k
views
Eigenvalues of permutations of a real matrix: can they all be real?
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
3
votes
1
answer
338
views
Determining the asymptotic behavior of random matrices with vanishing ratio dimensions
Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...
2
votes
1
answer
448
views
metric for signal to noise ratio in communication systems
I'm not quite sure about how to define a good measure of the quality of a communication channel with fading and interference. Let us assume the simplest case, where a node in a network receives the ...
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-...
13
votes
1
answer
2k
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Counting subtrees of a random tree ("random Catalan numbers")
Given a rooted tree $T$ and an integer $k \geq 1$, let $N_k(T)$ be the number
of subtrees of $T$ containing the root and having exactly $k$ nodes (take $N_k(T)=0$ if $T$ has less than $k$ nodes).
...
2
votes
1
answer
301
views
Tail bound for $L_2$ norm of top $k$ singular values of a random matrix
Let $Y=X^\top W$ , with $X, W \in \mathbb{R}^{d \times d}$ are random matrices with standard normal entries. Let $\lambda_j$ be the $j^{th}$ singular value of $Y$. Is there a way to bound the tail ...
4
votes
0
answers
498
views
Implications of a recent result on Benford's law
I want to the discuss the implications of a theorem by J. Morrow (2010) regarding Benford's law.
There are many papers written about Benford's law with a comprenhensive discussion of the advantages ...
12
votes
1
answer
3k
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What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?
$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) \...
5
votes
2
answers
949
views
Expected distance of a random point to the convex hull of N other points
Let $X_1, \cdots, X_N$ be i.i.d. $d$-dimensional random vectors, the exact distribution of $X$ is not very important in my application (as long as it continuous) so pick the one that works best, but ...
3
votes
1
answer
440
views
Uniform bound on the rate of convergence of the renewal measure
Consider a renewal process whose holding times are given by a continuous random variable $X$ supported on $[0,1]$. It is known (e.g. Stone '65) that the renewal function $m(t)$ converges to $t/\mathbb{...
0
votes
1
answer
2k
views
Markov Chain: state reduction
Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...
6
votes
2
answers
2k
views
Absolute moments of symmetrical distributions
Suppose $F~$ is a probability distribution symmetrical about 0, for which all moments exist. Let $\mu_i~$be the $i$-th moment (of course $\mu_i=0$ if $i~$ is odd).
We know there are some conditions ...
5
votes
1
answer
559
views
Existence of a probability measure with "confined" zero measure sets
Hi, I am struggling with the following question that is tangentially arising from a paper I'm working on. It is not at all essential for the revision but it would be nice to know if there is a ...
5
votes
1
answer
137
views
a measure of difference for arrangements of sphere points
Suppose one has a distribution of $N$ points on the sphere. Is there an agreed upon metric for the difference of this distribution and $N$ equidistant points on the sphere? To me entropy seems like ...
1
vote
2
answers
310
views
Apparently simple probability
Hello,
Let $x\in[0;1]$ and $(B_i)_i$ be events defined by $P(B_i)\leq x, \forall i$.
Furthermore, this inequality is independent of the other events $B_i$ but the events are not necessarily ...
7
votes
1
answer
345
views
Probability density that minimizes the sample range
Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in [0,...
8
votes
0
answers
210
views
Qualitative weakenings of probabilistic independence
In probability theory, independence of random variables is characterised by
$$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$
where $P_{(X,Y)}$ is the joint probability ...
0
votes
0
answers
143
views
multivariate integral calculation in closed form
I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
2
votes
1
answer
427
views
Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure
Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define
$$\Delta(u)= \frac{\int u(h) \exp(-\eta u(h))\exp(-\frac{\lambda}{2}h^2)~\mathrm{d}h}{\...
1
vote
2
answers
434
views
Local limit theorem for Bernoulli sums
Let $p_i\in (c,1-c)$ for some fixed $c\in(0,1)$ . Consider a sum $X=\varepsilon_{1}+\cdots+\varepsilon_{n}$ where $\varepsilon_{i}$ are independent Bernoulli random variables with parameters $p_{i}$. ...
1
vote
2
answers
226
views
Can averaging order statistics produce independent Gaussian random variables?
If I average two independent realizations of $N(0,1)$, I get a random variable with distribution $N(0,1/2)$.
Now suppose $X_1,\ldots,X_n$ are independent realizations of $N(0,1)$. Sort them in ...
6
votes
1
answer
822
views
Random walk with positive uniformly distributed steps
Let $U_1,U_2,\ldots$ be iid random variables distributed uniformly on $[0,1]$. I am interested in the random walk $X_i = \sum_{j \leq i} U_j$. In particular,
What is the expected number of points ...
3
votes
1
answer
307
views
A stronger version of supramenability?
A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
0
votes
1
answer
821
views
Large deviations for bernoulli sums
Let $p\in (0,1)$ be fixed and let $X$ be a binomial random variable with parameters n and p. Consider a related normal random variable $N$ with mean $np$ and variance $np(1-p)$. Is it true that for ...
0
votes
1
answer
495
views
Relating percentiles to moments [closed]
There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, ...
4
votes
1
answer
154
views
diffusions corresponding to estimators
I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
1
vote
1
answer
293
views
Equivalent Markov Random Fields
Hi,
Is it possible to have topologically different Markov Random Fields (few different edges) and yet yielding the same inference results ?
Thanks!
13
votes
1
answer
1k
views
Does $P(X_1>X_2)$ and $P(X_1=X_2)$, where $X_1$ and $X_2$ are independent and Poisson distributed, uniquely determine the parameters?
Let $X_1$ and $X_2$ be independent Poisson distributed random variables with parameters $\lambda_1$ and $\lambda_2$, respectively.
Let $a = P(X_1 > X_2)$ and $b = P(X_1 = X_2)$.
Question: ...
2
votes
1
answer
2k
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$):
$$...
6
votes
2
answers
309
views
Borel kernel over an analytic set implies existence of a Borel map
Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel $\...
1
vote
1
answer
575
views
On the expectation of a path integral involving Brownian motion up to a random time
Let $W$ be a one-dimensional standard Brownian motion and denote $$X_t=-\mu t + \sigma W_t, \quad t\ge 0,$$ where $\mu$ and $\sigma$ are positive constants. For $b<0$ denote the first passage time ...
3
votes
1
answer
909
views
Is there a Feynman-Kac formula applicable to Dirichlet problems for Schrödinger operators?
On pg. 133 of Heat Kernels and Spectral Theory, Davies is studying the heat kernel $K(x,y,t)$ of the operator $H = -\Delta + |x|^{\alpha}$ for $\alpha > 0$. He wishes to prove a lower bound, and ...
4
votes
0
answers
324
views
Lyapunov function of exponential growth for existence of a solution of an SDE
Let
$$dX_t = a(X_t) dt + b(X_t) dW_t$$
be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...
23
votes
1
answer
3k
views
Bochner integral of stochastic process = path by path Lebesgue integral?
After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
17
votes
2
answers
845
views
A moment problem on $[0,1]$ in which infinitely many moments are equal
Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$. Let their $n$-th moments be denoted by $\mu_n$ and $\nu_n$, respectively, for $n \in \mathbb{N}$.
If we know that $\mu_n=\nu_n$ for ...
0
votes
1
answer
4k
views
Calculate the probability of winning for a selected tic-tac-toe player
I am not a mathematician, I am a programmer. Sorry, if formulation of the problem is inexact.
I want to calculate the probability of winning for a selected tic-tac-toe player.
I have a directed graph ...
1
vote
1
answer
446
views
A set of positive-measure not being a countable union of cylinder sets and zero-measure sets?
Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ ...
1
vote
0
answers
182
views
Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts
I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
0
votes
1
answer
88
views
Is a $CD(K,\infty)$ space a length space?
Let $(X,d)$ be a complete and separable metric space endowed with a nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray}
\mu(B(x,r))<\infty,\quad\mbox{for every }x\in X\...
1
vote
0
answers
190
views
Nontransitive dice: the least number of faces?
Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity?
Here is ...
13
votes
1
answer
3k
views
random walk and Brownian motion on Riemannian manifold
As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy ...
4
votes
1
answer
274
views
Approximation of an integral over the unit ball of L_1
For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
1
vote
1
answer
274
views
A measure of closeness to a discrete set in a metric space
Consider a metric space $(M,d)$ and consider a collection of points $X_n := \{x_1,\dots,x_n\} \subset M$. Let
$$
N_\epsilon(y;X_n) := | \{ x \in X_n: d(x,y) \le \epsilon \}|
$$
where the RHS is ...
3
votes
1
answer
752
views
Expected number of random binary vectors so that the form a basis
I would like to compute the expected number of vectors in $\mathbb{F}_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}_2^n$, i.e., that we have $n$ ...
6
votes
1
answer
264
views
Proof of a statement from Steele's "Probability theory and combinatorial optimization"
I am reading "Probability theory and combinatorial optimization" by J.M. Steele and am hung up on a statement made in Section 2.2 of Chapter 2, "Easy size bounds", in which it is stated (paraphrasing ...
2
votes
1
answer
117
views
Expected length of a certain kind of nearest-neighbor graph
Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...
7
votes
2
answers
637
views
What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?
The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...