All Questions
105 questions
1
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0
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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
1
vote
1
answer
251
views
Expand the pdf of Wishart distribution into power series via orthogonal polynomials
In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related ...
- 163
2
votes
1
answer
269
views
Square integrable conditional expectations as projections
I see this page Ordinary least square and random projection, and I am thinking that how $L^2$ integrable random variables be regarded as projections over a defined filtration sequence $\mathcal{F_n}$ ?...
- 23
5
votes
3
answers
117
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Looking for a certain kind of a distribution
Is there any probability distribution supported on a compact or a half-open interval (of $\mathbb{R}$) such that if a vector $\vec{x} \in \mathbb{R}^n$ is sampled by sampling its coordinates like that ...
- 2,246
2
votes
2
answers
351
views
Weak convergence for discrete-time processes using characteristic functions
I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem
for probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology.
...
39
votes
3
answers
4k
views
Manifold of probability measures: connections between two types of metrics
The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
- 1,127
4
votes
0
answers
188
views
Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
5
votes
1
answer
372
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What are some of results in low dimensional statistics that do not hold in high dimensions?
This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional ...
- 8,071
3
votes
0
answers
98
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Asymptotic results on statistical graph models
This post is partly inspired by this post.
Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix
While it is well-known that two basic ...
- 8,071
4
votes
0
answers
141
views
Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
- 8,071
14
votes
1
answer
3k
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How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
3
votes
2
answers
195
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What is known about the PDFs for the $\ell^2$-norm of these multivariate distributions?
I'm looking for resources giving the PDFs for the $\ell^2$-norm of various spherically symmetric, continuous multivariate distributions.
For instance, the PDF for the $\ell^2$-norm of a multivariate ...
0
votes
1
answer
172
views
constrained optimization problem/proof
Im trying to maximize the probability of a particular outcome occurring subject to a constraint. In particular
$$\max \prod_{i \leq n} 1 - (1 - x_i)^{y_i} \;\;\; \text{ s.t. } \;\;\; i \in \mathbb{N}...
- 109
2
votes
1
answer
2k
views
Bounds on the eigenvalues of the covariance matrix of a sub-Gaussian vector
Suppose that $\boldsymbol{x}\in\mathbb{R}^n$ is subgaussian random vector of variance proxy $\sigma^2$, i.e.,
$$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\...
- 127
9
votes
2
answers
878
views
Is there a combinatorial/topological treatment of statistical independence?
Is there any reference which studies sets of random variables as independence systems, a type of combinatorial object (see below)?
Motivation:
In particular, since independence systems are abstract ...
- 2,680
11
votes
1
answer
1k
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What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
- 395
1
vote
1
answer
140
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Reference request: Cover times, Mixing Times and DGFF applied in statistics?
I am trying to find if in active research in statistics, there is interest in mixing times, cover times of graphs, and/or the discrete Gaussian free field?
I haven't found anything so far for the ...
- 11
4
votes
3
answers
273
views
Concentration inequalities for random sets
$\newcommand{\abs}[1]{\left|#1\right|}$
There is a population $O$ with a countable (finite or infinite) number of subjects. The population is colored randomly: for each subject, an unbiased coin-toss ...
- 3,549
3
votes
1
answer
940
views
What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
- 8,071
3
votes
1
answer
253
views
Can we find an Stein operator characterizing a distribution without density function?
It is known that Stein operator characterizes a probability distribution and there are a lot of ways of find a Stein operator.
For example, if $Z$ is the standard normal distribution, with pdf(...
- 329
2
votes
0
answers
54
views
Literature on transformed Gaussian matrices
I am considering real $n$-by-$m$ matrices of the following type:
$$
M=SM^\prime,\\
M^\prime_{ij}\sim^{iid}N(0,1).
$$
Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
- 121
10
votes
4
answers
645
views
Expected value of Bernoulli quadratic forms
Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
- 363
1
vote
0
answers
69
views
Norm-averaging reference request
(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...
- 123
2
votes
1
answer
268
views
Monotonicity of the Hellinger integral/distance
Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
- 128k
1
vote
1
answer
124
views
"Convergence speed" results for the Langevin process
The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...
1
vote
0
answers
438
views
Chain rule for maximal correlation
Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
- 1,109
16
votes
1
answer
2k
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Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
- 497
1
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0
answers
533
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Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula
Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–...
- 497
5
votes
2
answers
368
views
Reference to iterated logarithm law and Smirnov law of empirical CDF
I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let $...
- 162
1
vote
1
answer
115
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Supremum of centered jointly generalized chi-square random variables
Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
- 1,533
6
votes
1
answer
2k
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Kullback Leibler "variance": does that divergence have a name?
If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...
1
vote
1
answer
142
views
Subclass of semimartingales for which all characteristics can be estimated?
I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.
An Ito semimartingale is a martingale for which the ...
- 273
5
votes
1
answer
365
views
power laws emerging from the sandpile model
Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
- 6,962
5
votes
0
answers
136
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What's the variance in the Six Degrees model?
Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...
- 30.3k
15
votes
1
answer
1k
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Table with the most seated customers in Chinese restaurant process
Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...
- 151
4
votes
1
answer
804
views
Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)
I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
- 123
2
votes
2
answers
268
views
Distribution of a random walk on a directed line
Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and $x_{...
3
votes
1
answer
306
views
Mutual information decrease with coarse-graining
Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus B|C=0,D=0)\...
- 585
1
vote
0
answers
98
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Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
- 111
2
votes
1
answer
591
views
Concentration rates for the posterior distribution
Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution.
What is known about the ...
- 591
10
votes
2
answers
590
views
"Fractional sampling" from a probability distribution
My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
- 8,501
4
votes
0
answers
153
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A simplified MCMC / MH algorithm. Are there known convergence results?
Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
- 241
0
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0
answers
160
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
18
votes
1
answer
3k
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Distribution of maximum of random walk conditioned to stay positive
I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
- 747
6
votes
2
answers
2k
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Interesting thesis topic on statistical inference that is sufficiently mathematical
Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on ...
- 61
18
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1
answer
1k
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Applications of the Giry monad in probability and statistics
In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...
32
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4
answers
7k
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Bayesian statistics for pure mathematicians
Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. ...
10
votes
1
answer
210
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Distribution of the maximum of the norm of k-averages of n i.i.d. d-dimensional random vectors
Suppose $X_1, ... X_n$ are i.i.d. random vectors in $d$-dimensional space (i.e., $R^d$) with continuous centrally symmetric density function $f(\cdot)$ (i.e., symmetric with respect to the origin). ...
- 121
4
votes
1
answer
151
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Mean occurrences of letters in complete strings given by a Bernoulli scheme
Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
- 418
5
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0
answers
1k
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Multidimensional Berry–Esseen for probability density functions
This is a follow up to this recent question: Berry Esseen type result for probability density functions
There exists a multidimensional version of the usual Berry–Esseen theorem (for cumulative ...
