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0 votes
0 answers
149 views

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 ...
2 votes
1 answer
177 views

Optimization over Poisson-binomial distributions

I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis. Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<...
0 votes
0 answers
85 views

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 ...
2 votes
1 answer
170 views

Law of large numbers for a continuum of Bernoullis

Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
1 vote
0 answers
43 views

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 $...
28 votes
1 answer
6k views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
0 votes
0 answers
99 views

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 ...
4 votes
0 answers
142 views

Algebraic area of Brownian half-plane excursion

Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
3 votes
1 answer
561 views

On the convergence in total variation

$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
7 votes
0 answers
222 views

Projected polar chessboard measure convergence in total variation?

$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set $$F_n:=\...
3 votes
1 answer
407 views

Relative entropy equality for a sequence of Bernoulli random variables

We are given two joint probability distributions, $p$ and $q$, of $n$ Bernoulli random variables $X_1, X_2, \ldots, X_n$. We denote by $p(x_k\mid x^{k-1})$ the probability $\mathbb{P}_p(X_k=x_k\mid ...
4 votes
1 answer
197 views

On a double sum involving binomial coefficients

For natural $n$, let \begin{equation} p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l) \sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1} \end{equation} where $k:=\lfloor(n+1)/...
14 votes
1 answer
416 views

Lipschitz property of the determinant

$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
3 votes
0 answers
141 views

Direct analytic proof of positive definiteness of stable characteristic functions

Is there a direct analytic proof that the function $$ f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \...
5 votes
0 answers
96 views

Is there a name for the set of distributions whose probability generating functions are Mobius transformations?

Consider a discrete random variable $N\in\mathbb N$ with $\mathbb P(N=0) = p$, $\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$. Then the probability generating function of $N$ $$\mathbb E(z^N) = \...
0 votes
1 answer
188 views

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 $$...
1 vote
1 answer
97 views

A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable

Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$? To get the non-strict version of ...
7 votes
1 answer
259 views

Normal distribution by successive approximation?

$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
3 votes
1 answer
266 views

A linearly distributed version of the balls into bins problem

Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
4 votes
1 answer
265 views

Bounds on discrepancy metric of product measures

Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces $$X_1^{q} = (\times_{i=1}^q\...
0 votes
1 answer
153 views

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. $$ \...
2 votes
0 answers
93 views

Approximating a probability density with a point set

Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form? &...
5 votes
1 answer
512 views

Concentration inequality for Hilbert space valued random variables

I have read in a paper about the following result: Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
0 votes
1 answer
79 views

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....
0 votes
0 answers
86 views

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 ...
0 votes
1 answer
209 views

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}...
4 votes
2 answers
480 views

Hitting probability of a line

Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ ...
1 vote
0 answers
74 views

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 \...
1 vote
0 answers
88 views

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 ...
0 votes
0 answers
113 views

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 ...
2 votes
1 answer
476 views

Is total variation distance of normalized sum of random variables to Gaussian monotonic decreasing?

Let $X_1, X_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S_n = (X_1 + \cdots + X_n)/\sqrt{n}$ to be their normalized sum. Define $D_n$ ...
2 votes
0 answers
49 views

What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ...
8 votes
1 answer
2k views

General Fourier inversion formula (Gil-Pelaez)

Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{...
3 votes
2 answers
517 views

CLT for Martingales

I posted this question originally in math stack exchange, but I got no answer. (https://math.stackexchange.com/questions/2604591/clt-for-martingales) In wikipedia, there is a version of a CLT for ...
1 vote
1 answer
149 views

Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$

I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $...
4 votes
1 answer
364 views

Reference for multivariate generalised CLT

I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$, $$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
0 votes
1 answer
428 views

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 $...
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)_{...
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 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 $k$...
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 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 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 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 vote
1 answer
448 views

Law of large numbers for random Dirac measures

Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable. ...
2 votes
1 answer
1k views

Order statistics on the spacings between order statistics for the uniform distribution

For any natural $n$, let $U_1,\dots,U_n$ be independent identically distributed random variables each uniformly distributed on the interval $[0,1]$. As usual, let $U_{n:1}\le\cdots\le U_{n:n}$ ...
1 vote
1 answer
190 views

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 $\mathbb R^2$ ...
6 votes
1 answer
404 views

References for this game

I would like to know how the following game is known in the literature and, possibly, to have references for related papers. Description of the game: Fix a space $X$ and two Borel probability ...
1 vote
1 answer
155 views

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,\...
3 votes
1 answer
364 views

Can anyone give a reference to the proof of this concentration inequality?

The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%...