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.
9,027 questions
-1
votes
1
answer
2k
views
Variance of euclidean norm of Gaussian vectors
Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum Y_i^...
- 11
12
votes
3
answers
782
views
Connectedness of random distance graph on integers
This is not my field, a friend needs the answer for the following question. Suppose we have a decreasing probability function, $p: N \rightarrow [0,1]$ such that $sum_n p(n) = \infty$. Take the graph ...
- 747
3
votes
1
answer
651
views
What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 19.3k
2
votes
0
answers
250
views
Finding the maximum of a parametric function
We are trying to solve the following problem.
Giving a discrete random variable $V$, and a transformation function $\tilde{v}_i=\frac{v_i}{v_i+c\cdot \mathbb{E}(V)}$ we get a new random variable $\...
CommunityBot
- 1
1
vote
0
answers
199
views
Existence of multidimensional Levy process with dependent structure
Levy process is frequently cited recently. When we come to multidimensional Levy process, the components are usually assumed to be independent. Are there any examples on how to construct a Levy ...
- 6,210
4
votes
1
answer
5k
views
Asymptotic behavior of max of chi-squared distribution
Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom.
Since chi squared distribution ...
- 7,559
2
votes
1
answer
297
views
What is the median area of a random n-gon inside a unit square?
A square is bounded by the coordinates (0,0), (0,1), (1,0) and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points. The n points are then randomly connected to ...
- 28k
1
vote
0
answers
464
views
How far away is the maximum of $n$ i.i.d. chi-squared random variables from the rest of the sequence as $n$ gets large?
Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing ...
- 917
9
votes
2
answers
706
views
Measures whose projections are absolutely continuous
Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
CommunityBot
- 1
0
votes
1
answer
369
views
How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X
Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
- 7,559
6
votes
0
answers
487
views
Two sets of independent Bernoulli random variables
There are two sets of random variables $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ satisfying:
Each $X_i$ and each $Y_j$ has a symmetric Bernoulli distribution ($-1$ and $+1$ with probability $\frac12$ ...
- 37.7k
4
votes
1
answer
546
views
Total variation distance between diffusion processes with different volatility coefficient
Preamble:
This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
CommunityBot
- 1
7
votes
2
answers
639
views
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 $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
CommunityBot
- 1
1
vote
2
answers
151
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 ...
- 3,069
0
votes
1
answer
184
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\...
- 188k
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 ...
CommunityBot
- 1
3
votes
1
answer
354
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 ...
- 7,559
2
votes
1
answer
454
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 ...
- 13.9k
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-...
CommunityBot
- 1
13
votes
1
answer
2k
views
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).
...
- 6,210
2
votes
1
answer
324
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 ...
- 7,559
4
votes
0
answers
502
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 ...
- 729
13
votes
1
answer
3k
views
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) \...
CommunityBot
- 1
5
votes
2
answers
985
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 ...
CommunityBot
- 1
3
votes
1
answer
454
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{...
CommunityBot
- 1
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 ...
- 23
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 ...
- 60.6k
5
votes
1
answer
586
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 ...
- 60.6k
5
votes
1
answer
140
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 ...
- 188k
1
vote
2
answers
313
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
346
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,...
CommunityBot
- 1
8
votes
0
answers
212
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 ...
- 866
0
votes
0
answers
145
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 ...
- 25.8k
2
votes
1
answer
439
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}{\...
- 61.9k
1
vote
2
answers
447
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}$. ...
- 37.7k
1
vote
2
answers
239
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 ...
- 23.2k
6
votes
1
answer
883
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 ...
- 1,731
3
votes
1
answer
324
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 ...
- 2,555
0
votes
1
answer
886
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 ...
- 37.7k
0
votes
1
answer
514
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, ...
- 91.8k
4
votes
1
answer
159
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 ...
CommunityBot
- 1
1
vote
1
answer
295
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!
- 200
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: ...
- 326
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$):
$$...
- 7,559
6
votes
2
answers
318
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 $\...
- 17.1k
1
vote
1
answer
588
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 ...
- 7,559
3
votes
1
answer
965
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 ...
- 10.3k
4
votes
0
answers
350
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 ...
- 622
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 ...
- 17.1k
17
votes
2
answers
860
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 ...
CommunityBot
- 1