All Questions
Tagged with pr.probability probability-distributions
1,384 questions
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,467
2
votes
1
answer
318
views
Distribution of a stopped random sum, with subexponential stopping time
I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X_n)_{...
- 1,372
8
votes
1
answer
2k
views
Scalar product of random unit vectors
Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
$X,X'$ independent with uniform distribution on the sphere $...
- 83
0
votes
1
answer
66
views
Minimum mean over all random variables subject to logarithm constraint
Does the following problem have a solution?
$$
\min_X \mathbb{E} X
\quad\text{subject to}\quad
\mathbb{E} \log X = C.
$$
Here, the minimization is with respect to all integrable random variables $X$ ...
- 3
0
votes
1
answer
209
views
Distribution of the direction of Gaussian random variable
Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
- 1,069
4
votes
1
answer
214
views
Rates of convergence to Tracy-Widom?
$\renewcommand{\!}{\mathbf}
\renewcommand{\Ai}{\operatorname{Ai}}$
One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where
$$\mathbf A(x, y)=\...
- 833
-2
votes
1
answer
84
views
Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]
I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...
- 137
5
votes
1
answer
1k
views
Explicit constant for Carbery–Wright inequality
The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables.
See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
- 111
3
votes
0
answers
132
views
Probability that a Voronoi cell contains exactly k random points
Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a ...
- 137
1
vote
1
answer
335
views
Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
- 31
0
votes
1
answer
199
views
Is this probability inequality true?
This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
- 195
3
votes
2
answers
1k
views
Expected value of a truncated binomial
Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
- 4,049
1
vote
1
answer
144
views
A uniform mixture of order statistics
Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
- 4,049
4
votes
2
answers
856
views
Disintegration, conditional probabilities, and conditional expectation
On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
- 5,405
0
votes
1
answer
103
views
Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?
I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
- 309
0
votes
0
answers
250
views
Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables
I asked this on MSE, but got no answer, hence asking here now. Help appreciated!
My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
- 1,467
3
votes
1
answer
829
views
The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$
In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
user157884
0
votes
1
answer
583
views
Find a conditional expectation of a difference of two independent Poisson process
Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
user157884
2
votes
1
answer
195
views
Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S
Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
0
votes
1
answer
519
views
Lyapunov condition for CLT for asymptotically independent sequence
Suppose I have some triangular array $\{X_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that
$$Var\left(\sum_{j=1}^n X_{n,j}\right)\to \...
- 203
1
vote
0
answers
113
views
Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
- 2,246
0
votes
1
answer
378
views
Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
6
votes
1
answer
291
views
Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group
Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables :
Let $X$ and $Y$ be $G$-valued ...
- 2,101
0
votes
0
answers
141
views
What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
- 1,467
1
vote
0
answers
64
views
Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?
Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
- 6,853
3
votes
1
answer
431
views
Inequality on the Hellinger distance between Poisson and mixture of Poisson
Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$,
$$
H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n)}...
- 1,372
0
votes
1
answer
86
views
Integral rising from difference of chi-squared random variables
Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...
- 109
2
votes
1
answer
900
views
Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$
Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following:
...
- 1,467
6
votes
1
answer
264
views
Which orthant probabilities are the largest? (For a multivariate normal distribution)
I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
2
votes
1
answer
331
views
Prove or disprove the linearity of expectiles
For expectation (mean), there are many useful properties such as Linearity of Expectation:
$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$
$\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$
(The two equations ...
- 139
1
vote
1
answer
100
views
An elementary question on probability distributions
I have encounter the following problem, but after trying a little I did not arrive to a good conclusion.
Suppose that $X$ is a positive random variable for which we only know that $E[X] = 2$ and $E[1/...
- 1,561
0
votes
0
answers
132
views
Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
- 109
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
5
votes
1
answer
3k
views
Eigenvalues and eigenvectors of Gaussian random matrices
Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues?
...
- 899
1
vote
2
answers
212
views
Random variable corresponding to sum of density functions [closed]
The distribution of functions of random variables is well-studied for various different and general cases, but I didn't find much result for the reverse.
Suppose that $X_1, X_2$ are (probably ...
- 133
0
votes
0
answers
54
views
A distribution of maximum of sums if add to the minimal
Consider a vector of $n$ integer variables with initial values of 0. Each step we take random $w_i\thicksim NB(q, l)$ (independent randon values with the same negative binomial distribution) and add ...
- 11
2
votes
1
answer
1k
views
Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data
Let $Z=[z_1, \dots z_n]$ be a $d \times n$ matrix, where the $z_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $...
- 1,467
1
vote
0
answers
124
views
Law of large numbers and Central Limit Theorem for eigenvalues of perturbed matrices
I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely,
Let $ \forall n \in \mathbb{N},...
- 1,467
2
votes
1
answer
210
views
Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,467
4
votes
0
answers
143
views
A possible generalization of Solomonoff's theorem
Assume that $P$ and $Q$ are probability distribution on the binary tree,
i.e. $P$ and $Q$ are functions $\{0,1\}^{*} \to \mathbb{R}$ such that:
for every $x$: $P(x)=P(x0)+P(x1)$ and $P( \text{empty ...
- 2,576
16
votes
2
answers
1k
views
How often two iid variables are close?
Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a ...
- 1,352
1
vote
1
answer
131
views
Large scale analysis of matrix multiplications
Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with ...
- 287
1
vote
0
answers
83
views
Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
- 1,467
1
vote
1
answer
140
views
Conditional density for random effects prediction in GLMM
I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...
- 13
0
votes
1
answer
552
views
Hypothesis testing for not identically distributed random variables conditioned on the outcome of a subset
I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar ...
- 9
2
votes
1
answer
760
views
Show the coordinate distribution has a very large sub-gaussian norm
Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors ...
- 23
0
votes
1
answer
1k
views
Expectation of inverse of random matrices
Assume that $\mathbf{X}$ is a random positive-definite matrix. Then, is there any upper or lower bound on the expectation of the following expression
$$\mathbb{E}[\mathbf{X}^{-1}]-\alpha\mathbb{E}[\...
- 287
0
votes
0
answers
275
views
Is there any relation between moments of random matrix and its eigenvalue distribution?
Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation ...
- 287
2
votes
2
answers
854
views
Eigenvalue distribution of a random matrix
Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables ...
- 287
1
vote
1
answer
104
views
Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?
Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...
- 1,467