All Questions
Tagged with pr.probability st.statistics
1,134 questions
3
votes
0
answers
93
views
Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)
For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
- 6,853
4
votes
0
answers
144
views
Exponential families closed under affine transformations
Let $(\Omega,\Sigma,\mu)$ be a probability space and let $\mathcal{M}$ be an exponential family of probability distributions for $\mu$ of the following form: There are $\varphi_1,\dots,\varphi_n:\...
- 5,405
3
votes
2
answers
348
views
General version of $d$-separation
I find the $d$-separation criterion (see, e.g., Theorem 2 here; note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient ...
- 1,095
1
vote
1
answer
106
views
What is the maximum possible coefficient of variation for data taking values within a specified range?
I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching.
Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be ...
- 708
1
vote
1
answer
613
views
Integral of the product of a gaussian pdf and cdf
I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
- 11
1
vote
1
answer
221
views
Large deviation for empirical median
I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov ...
- 445
2
votes
1
answer
165
views
Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
- 153
2
votes
0
answers
87
views
The covariance of certain random variable
We define two random variables $X_n,Y_n $ on the sample space $\{1,2,3,\cdots,n\}$ with counting measure. We denote by $C_n$ the covariance of theses two random variables: $C_n=Cov(X_n,Y_n)$.
...
- 356
0
votes
0
answers
91
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
1
vote
1
answer
226
views
Orthogonal transformation of multivariate Bernoulli-Gaussian distribution
Actually, I have asked this question in https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution, but I think mathoverflow might be ...
- 21
2
votes
1
answer
377
views
Extension of subcopulas to copulas
This question is about the extension of subcopulas to copulas, shown in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look backward and forward." ...
- 108
1
vote
1
answer
104
views
Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$
Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
$\mbox{trace}(\Sigma_d) = 1$.
$\|\Sigma_d\|_{op} = \mathcal ...
- 6,853
2
votes
1
answer
138
views
Comparison between $\|X\|_2$ and $\|X\|_{2,1}$
For any real random variable $X$, define
$$\|X\|_{2,1}=\int_0^\infty \sqrt{\Pr(|X|>t)}dt.$$
This quantity (it is not a norm) appears in various problems, e.g. the multiplier central limit theorem (...
- 197
1
vote
0
answers
146
views
Using maximum entropy principle for joint probability estimation
Let $X_1, \dots, X_n, Y$ be random variables, each taking values in $\{0,1\}$. Assume that we are interested in estimating, for each $v=(v_1,\dots,v_n)\in \{0,1\}^n$, the probability
$$
p(v) = P[Y=1|...
- 7,182
1
vote
1
answer
157
views
Moments of rescaled Bernoulli random matrix
Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$
Is ...
- 269
1
vote
2
answers
277
views
Distribution of interarrival times for a special class of stochastic point processes
I am interested in Poisson-binomial stationary point processes (here on the real line) defined as follows. Let
$t_k=k/\lambda$, with $k\in\mathbb{Z}$ and $\lambda>0$,
$F_s(x)$ be a symmetric, ...
- 3,098
0
votes
1
answer
133
views
How to demonstrate a correlation inequality? [closed]
If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$.
The correlation between Z, Y is greater than between X, ...
- 11
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
- 6,853
1
vote
1
answer
160
views
Given iid $w_1,\dotsc,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_\text{op}\to0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
- 6,853
7
votes
1
answer
347
views
Expectation for game choosing uniformly number in $[0,1]$ until it decreases
We are playing a game where we keep on choosing a number from the uniform distribution U(0,1). The game goes on until we have the current number less than the previously picked number, i.e. the game ...
2
votes
1
answer
177
views
Matrix-valued cumulant generating function for Wishart matrices
Suppose we have an axis-aligned Gaussian vector $v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$, and consider the Wishart matrix $W = vv^\top$.
Is there a simple closed form/"Lowener order ...
- 101
0
votes
1
answer
83
views
The distribution of number of reverse order pairs in a randomly permuted array
There is an array $a_1,\dotsc,a_n$ whose elements are pairwise distinct. We define a reverse order pair to be an ordered pair $(a_i,a_j)$ such that $i < j$ and $a_i > a_j$. Consider the total ...
1
vote
1
answer
88
views
tail probability of max of Gaussians
I'm trying to follow an argument in C. Giraud's "High Dimensional Statistics" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books here but the formatting is awful....
- 13
1
vote
0
answers
78
views
Canonical representation of the a probability distribution for Hammersley Clifford Theorem
I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
- 903
1
vote
0
answers
348
views
Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
- 420
2
votes
2
answers
322
views
Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
- 6,853
1
vote
1
answer
415
views
Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
1
vote
1
answer
410
views
Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
3
votes
1
answer
416
views
Well-definedness of maximum likelihood estimation
Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
- 2,247
2
votes
1
answer
90
views
Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix
Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
- 6,853
1
vote
1
answer
84
views
Jeffreys' priors as coefficients of a linear estimator
I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question:
I have three random variables, $X_1$, $X_2$, and $...
user386878
1
vote
1
answer
202
views
A problem related to stochastic ordering
Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray*}
\boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...
- 211
2
votes
1
answer
116
views
A question on the applicability Chebyshev inequality for sequence of random quantities
Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.
...
- 6,853
2
votes
1
answer
110
views
Lower bound on likelihood of binary outcomes
I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...
- 341
1
vote
1
answer
123
views
Stochastic ordering of absolute multivariate normal random variables
Let $X\sim\mathcal{N}(\boldsymbol{\mu}_1,\mathrm{\Sigma}_1)$ and $Y\sim\mathcal{N}(\boldsymbol{\mu}_2,\mathrm{\Sigma}_2)$. Then it is know that $\mathbb{P}(X>\boldsymbol{t})\leq\mathbb{P}(Y>\...
- 211
4
votes
0
answers
656
views
Eigenvalues of Matérn covariance function
Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
- 397
2
votes
1
answer
294
views
An inequality in the optimality of Bayes' theorem
$\DeclareMathOperator\Ent{Ent}\newcommand{\prior}{\mathrm{prior}}\newcommand\Data{\mathrm{Data}}$I came across this paper on the optimality of Bayes' theorem
https://sinews.siam.org/Portals/Sinews2/...
- 23
3
votes
1
answer
318
views
Divergence-free Gaussian vector field with given mean magnitude and correlation function
My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector ...
- 147
2
votes
1
answer
304
views
An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere
Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
- 6,853
2
votes
1
answer
187
views
Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
- 6,853
1
vote
0
answers
45
views
Is there a local limit theorem for functions of Gaussian random vectors?
Assume that $\sqrt{n} (\boldsymbol{Z}_n - \boldsymbol{\mu}) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0},\Sigma)$, as $n\to \infty$, for some $\boldsymbol{\mu}\in \mathbb{R}^d$ ...
- 219
4
votes
1
answer
332
views
Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
- 6,853
3
votes
1
answer
170
views
Donsker class and law of the iterated logarithm
Let $P$ be a probability measure on a measurable space $(E, \mathcal {E})$, and let
$\mathcal {F}$ be a countable collection of measurable functions $f : E \to \mathbb {R}$
which is a Donsker class ...
- 61
0
votes
0
answers
95
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
0
votes
2
answers
341
views
Conditions for existence of a distribution with full support
Consider a $6\times 1$ continuous random vector
$$
\eta\equiv (\eta_1,\eta_2,..., \eta_6)
$$
satisfying the following property:
$$
\underbrace{\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix}}_{\...
- 108
0
votes
2
answers
534
views
Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
- 6,853
0
votes
1
answer
101
views
Realizations of alternative configurations
Consider a discrete distribution $P(\mathbf{X},Y)$ with $\mathbf X = \{ X_1, \dotsc, X_N \}$. I use the shorthand notation $p(\mathbf{x}, y)$ for $P(\mathbf{X}=\mathbf{x}, Y=y)$. Consider $P_\text{ind}...
- 189
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
1
vote
1
answer
217
views
How to normalize an Inverse Wishart random matrix?
Background:
Let $d\in \mathbb{N}$.
Define the space of (real symmetric) positive definite matrices of size $d\times d$ as follows:
\begin{align}
\mathcal{S}_{++}^d := \big\{\mathbb{M}\in \mathbb{R}^{d\...
- 219
3
votes
1
answer
1k
views
Convergence of empirical measures in Wasserstein distance
Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let
$\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
- 31