All Questions
73 questions
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
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
16
votes
1
answer
2k
views
Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
- 497
14
votes
1
answer
416
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
8
votes
1
answer
2k
views
General Fourier inversion formula (Gil-Pelaez)
Gil-Pelaez (1951) proves the Fourier inversion formula
\begin{align*}
F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\
&= \frac{1}{2} - \frac{...
- 255
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 128k
7
votes
0
answers
222
views
Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
6
votes
1
answer
773
views
Stein's Lemma for Discrete Distribution
Stein's Lemma in its standard form states that $X \sim N(0,1) \Leftrightarrow E[f'(X) - X f(X)] =0 $ for all bounded one-time differentiable functions $f$ (I am ignoring the exact conditions on $f$ ...
- 519
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 ...
user11618
5
votes
2
answers
289
views
Finding joint probability from double marginals
Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$.
When does a global joint probability $p(x,y,z)$ (possibly not unique) exist?
The first compatibility condition to ...
- 2,129
5
votes
1
answer
942
views
Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
- 519
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 ...
- 2,246
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 $V$-...
- 115
5
votes
1
answer
231
views
CLT for Bernoulli RV with negative correlation
Suppose $X_1,X_2,...$ are Bernoulli random variables with $P(X_i=1)=p_i$ and $X_i$ have negative correlation. Is there a CLT in this case, i.e. does $\frac{Z_n-(\Sigma^n_{i=1}p_i)}{\sqrt{n}}$ converge ...
- 339
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 $N$
$$\mathbb E(z^N) = \...
- 241
4
votes
2
answers
480
views
Hitting probability of a line
Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ ...
- 17k
4
votes
1
answer
197
views
On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 128k
4
votes
1
answer
364
views
Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
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
$$X_1^{q} = (\times_{i=1}^q\...
- 2,712
4
votes
0
answers
142
views
Algebraic area of Brownian half-plane excursion
Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
- 3,927
4
votes
0
answers
261
views
Tight bounds for finite de Finetti's theorem
de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following ...
- 763
3
votes
1
answer
561
views
On the convergence in total variation
$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
- 128k
3
votes
1
answer
407
views
Relative entropy equality for a sequence of Bernoulli random variables
We are given two joint probability distributions, $p$ and $q$, of $n$ Bernoulli random variables $X_1, X_2, \ldots, X_n$.
We denote by $p(x_k\mid x^{k-1})$ the probability $\mathbb{P}_p(X_k=x_k\mid ...
- 1,677
3
votes
1
answer
266
views
A linearly distributed version of the balls into bins problem
Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
- 1,677
3
votes
1
answer
129
views
Reference Request: Simple Random Walk on $\mathbb Z$ is Unimodal
I am looking for a reference to the following claim. Let $X = (X_t)_{t\ge0}$ be a continuous time simple random walk. Then
$$
m \mapsto P(|X_t| = m) : \mathbb N \to [0,1]
$$
is (weakly) decreasing (or ...
- 560
3
votes
2
answers
517
views
CLT for Martingales
I posted this question originally in math stack exchange, but I got no answer.
(https://math.stackexchange.com/questions/2604591/clt-for-martingales)
In wikipedia, there is a version of a CLT for ...
- 339
3
votes
1
answer
723
views
Random weighted selection without replacement
I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$.
Selection procedure
...
- 837
3
votes
1
answer
364
views
Can anyone give a reference to the proof of this concentration inequality?
The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%...
- 123
3
votes
1
answer
461
views
Bounding the "spikiness" of a probability distribution
Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"?
I ask this question because I am interested in the families of probability distributions $f(x)$ ...
- 4,049
3
votes
1
answer
520
views
Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.
Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...
- 577
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
3
votes
0
answers
303
views
Exchangeable or iid random variables and linear conditioning
Let $X_1,\ldots ,X_N$ be independent identically distributed random variables (or, more generally, exchangeable random variables,
but let's assume independence for simplicity). Then
$$
E(X_i\mid X_1+\...
- 1,765
3
votes
1
answer
476
views
distribution discretization
Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
- 640
2
votes
2
answers
407
views
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?
Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...
- 139
2
votes
1
answer
154
views
Reference Request for Couplings with Conditions
I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several ...
- 329
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 ...
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)_{...
- 1,372
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 $D_n$ ...
- 763
2
votes
1
answer
598
views
Cantelli's inequality: the original source
Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
- 128k
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<...
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 $...
- 1,467
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 $U_{n:1}\le\cdots\le U_{n:n}$ ...
- 128k
2
votes
1
answer
401
views
Reference on Probability theory on functional spaces (in special Hilbert spaces)
Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
2
votes
2
answers
348
views
Suggestions for dealing with the "timed" balls-into-bins model
Definitions: Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D \...
- 139
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?
&...
- 4,049
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 ...
- 223
2
votes
0
answers
100
views
Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes
Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
- 560
2
votes
0
answers
301
views
Schilder's theorem for brownian bridges
I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the ...
- 5,380
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.
...
- 1,467
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 ...
- 128k
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 $...
- 1,467