All Questions
73 questions
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104
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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
1
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1
answer
190
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Hitting time estimates
In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T_n)$ taking values in the reals (or sometimes a process $(T_n,X_n)$ taking values in $\mathbb R^2$ ...
- 23.2k
1
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1
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155
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Reference request concerning order statistics from the uniform distribution
Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\...
- 128k
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43
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Definition of "interval of continuity" for function defined on sets
At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
- 12.8k
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74
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Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 5,405
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88
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Berry-Esseen type bounds for functions of almost Gaussian random variables
Suppose that I have $n$ dependent random variables $X_1,\ldots,X_n$ with $\mathbb{E}[X_i]=0, \mathbb{E}[X_i^2]=1$, where we have the following bounds on the Kolmogorov distance from a normal ...
- 141
1
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64
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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
1
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124
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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
1
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0
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83
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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
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72
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Large Deviation of Triple Poisson Product
Let $X_i$ with $i=1,\ldots,n$ be independent Poisson variables, $X_i$ with parameter $\lambda_i.$
Let $\circ$ be a group operation on a group of size $n.$
I would like to obtain a large deviation ...
- 10.4k
1
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66
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Matrix variate t-distribution and product of Beta distributions
This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then
$$
\frac{\det(U)}{...
- 2,560
0
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1
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428
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First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
- 1,467
0
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1
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188
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Equality cases in a certain case of Jensen's inequality
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
$$...
- 128k
0
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1
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153
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Maximum of a certain Gaussian field
Let $S_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z_x)_{x \in S_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e.
$$
\...
- 1,295
0
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1
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209
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Factorisation of Gaussian random matrix into random Hermitian and correction factor
By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...
user274127
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149
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Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
- 1,561
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85
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When is a family of distributions "closed" with respect to minimal sufficient statistics?
As in the title, I am interested in understanding how to express the idea that a parametric family of distribution is "closed" with respect to minimal sufficient statistics. Before giving ...
0
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99
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Random walks on groups
I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
- 9
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1
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79
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Visualization PDF of distribution defined by quantiles
How can I visualise PDF of distribution defined by quantiles, that I predict with my neural network? Now I'm passing quantiles to the histogram, but I don't think it is the correct way for visualising....
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86
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Expected diameter of a random point set
General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...
- 19
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113
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How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
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141
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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
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424
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Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
- 31