All Questions
105 questions
2
votes
0
answers
80
views
Bridging between Rosethal Inequalities and log convex tails
Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$.
Then we have the classical "Rosenthal-type ...
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 ...
2
votes
1
answer
235
views
Kolmogoroff condition for truncated random variables
Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
- 128k
1
vote
1
answer
41
views
Probability that maximal elements has the same position in samples from correlated random variables
Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ ...
- 128k
6
votes
1
answer
2k
views
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
3
votes
0
answers
150
views
Central Limit Theorem for simultaneous sums
Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \...
CommunityBot
- 1
7
votes
4
answers
4k
views
Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 2,821
17
votes
4
answers
2k
views
Good introduction to statistics from a algebraic point of view?
There are already lots of questions on this subject like
Is there an introduction to probability theory from a structuralist/categorical perspective?
Is there a combinatorial/topological treatment ...
- 96
2
votes
1
answer
410
views
References for Hellinger distance/affinity involving mixture distributions
For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...
- 128k
3
votes
2
answers
100
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Left tail of convex combinations of $\chi_1^2$
Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(...
- 128k
32
votes
3
answers
12k
views
What is the Katz-Sarnak philosophy?
It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
- 8,071
37
votes
3
answers
3k
views
On Mathematical Analysis of MathSciNet & MathOverflow
This question has two original motivations: mathematical and social.
The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...
3
votes
1
answer
461
views
Bounding the "spikiness" of a probability distribution
Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"?
I ask this question because I am interested in the families of probability distributions $f(x)$ ...
- 8,071
1
vote
0
answers
66
views
Matrix variate t-distribution and product of Beta distributions
This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then
$$
\frac{\det(U)}{...
- 2,560
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 $\mathcal{F_n}$ ?...
- 8,071
5
votes
3
answers
117
views
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 ...
- 4,529
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.
...
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(...
- 8,071
15
votes
1
answer
1k
views
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 ...
CommunityBot
- 1
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
1
vote
1
answer
140
views
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 ...
CommunityBot
- 1
10
votes
1
answer
210
views
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). ...
- 8,071
5
votes
1
answer
372
views
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
views
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
3
votes
1
answer
941
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 $\...
CommunityBot
- 1
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.,
$$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\...
- 8,071
10
votes
4
answers
645
views
Expected value of Bernoulli quadratic forms
Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
- 3,946
4
votes
0
answers
141
views
Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
- 8,071
3
votes
2
answers
195
views
What is known about the PDFs for the $\ell^2$-norm of these multivariate distributions?
I'm looking for resources giving the PDFs for the $\ell^2$-norm of various spherically symmetric, continuous multivariate distributions.
For instance, the PDF for the $\ell^2$-norm of a multivariate ...
0
votes
1
answer
172
views
constrained optimization problem/proof
Im trying to maximize the probability of a particular outcome occurring subject to a constraint. In particular
$$\max \prod_{i \leq n} 1 - (1 - x_i)^{y_i} \;\;\; \text{ s.t. } \;\;\; i \in \mathbb{N}...
11
votes
1
answer
1k
views
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 ...
CommunityBot
- 1
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?
- 5,063
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 ...
- 6,541
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 ...
- 21
18
votes
1
answer
3k
views
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 ...
CommunityBot
- 1
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
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
1
vote
1
answer
115
views
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 ...
CommunityBot
- 1
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:
$$ p_\infty (x) \propto ...
- 188k
16
votes
1
answer
2k
views
Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\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 $\Phi$ ...
CommunityBot
- 1
1
vote
0
answers
533
views
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–...
CommunityBot
- 1
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 $...
- 253
0
votes
1
answer
200
views
How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
CommunityBot
- 1
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)\...
CommunityBot
- 1
5
votes
0
answers
136
views
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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_{...
- 1,902
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 ...
CommunityBot
- 1
1
vote
0
answers
98
views
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
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 ...
- 7,514