All Questions
Tagged with pr.probability probability-distributions
1,384 questions
7
votes
1
answer
305
views
Why does non-decreasing entropy imply actual convergence to that max entropy distribution?
Let $X_n$ be i.i.d with finite variance. Let $\bar X_n=\frac 1n \sum_{i=1}^nX_i$. It is a famous result that the continuous/differential entropy of the normalized average is non-decreasing. $$\mathrm ...
- 10.5k
1
vote
1
answer
109
views
A question related to the CDFs of multivariate normal distribution
Let $\boldsymbol{\xi} = (\xi_1,\xi_2,\xi_3)$ such that $\xi_i\geq 0$and $\xi_1+\xi_2+\xi_3 = 1$. Let $Y\sim N_3(\boldsymbol{\mu}(\boldsymbol{\xi}), \mathrm{\Sigma}(\boldsymbol{\xi}))$, where the ...
- 211
4
votes
1
answer
136
views
Decreasing tail integrals for nonnegative random variable $X$
Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also
$$\ell(x):=\frac{1}{...
- 5
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
436
views
Is the limit of compound Poisson random variables a compound Poisson r.v.?
Let $Y$ be an infinitely divisible (I.D.) random variable.
Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...
- 13
1
vote
1
answer
362
views
An inequality involving the Wasserstein distance and chi-squared distance
$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
- 128k
4
votes
1
answer
338
views
Minimum of random walks
Let $M$ independent and identical random walks that follow the chi-squared distribution, i.e. in each step, a $X^2_1$ random variable is added. I am interested in the minimum random walk at each step. ...
7
votes
2
answers
392
views
On a von Bahr–Esseen-type inequality for pairwise independent zero-mean random variables
For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr–Esseen-type inequality
\begin{equation}\label{eq:pair}\tag{1}
E\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1}...
- 128k
0
votes
1
answer
82
views
WLLN for bootstrap means of stationary ergodic processes?
Setup:$\quad$
Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$.
Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
- 127
0
votes
0
answers
72
views
A moment-based stochastic order
Given that a Borel probability measure $\mu$ on [0,1] is characterized by its moments, it seems natural to consider the following stochastic order: say that $\mu\le\mu'$ if $$\forall k\in\mathbb N,\...
- 234
3
votes
1
answer
210
views
Probabilistic Taylor theorem for concave functions
This paper proves a probabilistic version of Taylor's theorem
\begin{equation*}
\mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...
- 45
3
votes
0
answers
610
views
High-dimensional uniform distribution
Let $\mathcal X$ be either a subset of $\mathbb R$ equipped with Lebesgue measure or a countable set with counting measure.
The Gibbs' principle in statistical physics asserts that if $(X_1 , \dots, ...
- 1,608
1
vote
1
answer
879
views
Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable?
(I asked this question on MSE 10 days ago, but I got no answer.)
Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)...
- 13
0
votes
1
answer
105
views
Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 5
3
votes
0
answers
164
views
Random permutations constructed via randomly chosen transpositions
Given positive integers $k$ and $n$, we define the probability distribution $p_{n,k}$ on $S_n$ as:
$$
p_{n,k}(\sigma):=\frac{\#\{(\tau_1,\dots,\tau_s)\mid \sigma=\tau_1\dots\tau_s, \text{ each }\tau_i\...
- 3,599
4
votes
1
answer
489
views
CLT convergence rate for sum of uniforms (in TV distance)
Suppose $X_1, \cdots, X_n \sim_{\mathrm{iid}} U([-1,1])$, where $U([-1, 1])$ denotes the continuous uniform distribution over the interval $[-1, 1]$ (so $E[X_i] = 0$ and $\text{Var}[X_i]= 1/3$). Let $...
- 43
1
vote
1
answer
125
views
Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
- 63
1
vote
1
answer
241
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
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.
$$
\...
- 1,295
4
votes
1
answer
212
views
Uniform iid sequence
The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?
- 9,007
3
votes
0
answers
334
views
Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix
Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
- 223
3
votes
1
answer
269
views
Discrimination between set of binary distributions
Suppose we know two sets of distributions
$A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$.
We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$.
...
- 1,503
1
vote
0
answers
96
views
If all moment of X are greater than all moment of Y, can we said something about their probability?
Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables.
Let's say one can show that $E[X]^k \...
- 149
4
votes
1
answer
265
views
Bounds on discrepancy metric of product measures
Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...
- 2,712
1
vote
1
answer
215
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
2
votes
0
answers
138
views
Optimal Monte Carlo Trace Estimator
For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...
3
votes
1
answer
614
views
An inequality relating the Kullback-Leibler divergence of two discrete distributions with constant reference distribution
Suppose that $D_{KL}(p_1\parallel q)<1$ and $D_{KL}(p_2\parallel q)<1$. I'm trying to show that either $D_{KL}(p_1\parallel p_2)$ or $D_{KL}(p_2\parallel p_1)$ will have an upper bound close to ...
3
votes
2
answers
667
views
Is every discrete compound Poisson distribution a mixed Poisson distribution?
I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...
- 1,311
0
votes
1
answer
296
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
3
votes
1
answer
442
views
$\cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots$ as $x \rightarrow \infty$
It is known that
$$
\cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1})
$$
Is it true that
$$
f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\...
- 1,608
1
vote
1
answer
251
views
Using Hoeffding inequality for risk / loss function
I've got a question to the Hoeffding Inequality which states, that for data points $X_1, \dots, X_n \in X$, which are i.i.d. according to a probability measure $P$ on $X$, we find an upper bound for:
$...
- 13
2
votes
1
answer
415
views
Self normalized sum of products of i.i.d. random variables
Let $p\in (0,1)$ and $X_1, X_2, ...X_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X_i$ is $1$ equals $p$.
Fix $a,b>0$ different from $1$ that satisfy ...
- 23
1
vote
1
answer
122
views
Expectation value of random GUE matrix
Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors.
I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value
$$\...
- 73
1
vote
0
answers
50
views
Characterizing set of IID average of symmetric positive semidefinite matrices matrices
Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices.
Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...
- 420
1
vote
1
answer
92
views
Compressing covariance matrix from 3X3 to 2X2
thanks for answering my question. My question is,
Let $v_3=[a,b,c]^T$ is a probabilistic 3-D vector variable which is distributed by zero mean Gaussian of dense covariance matrix $\Sigma_{3\times3}$.
...
- 11
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
1
answer
403
views
Law of large numbers for triangular arrays whose moments "look independent"
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...
- 1,124
3
votes
0
answers
142
views
$\epsilon$-net for the set of distributions
Let $S$ be a space of all distributions over a finite set $X$. What is the size of its $\epsilon$-net w.r.t. the total variation distance defined as $d(P,Q) = \frac12 \sum_{x \in X} |\Pr[P=x] - \Pr[Q=...
- 231
1
vote
1
answer
299
views
Bounding $2$-Wasserstein distance and the $L^1$ distance
My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
- 730
1
vote
2
answers
127
views
Existence of a strictly proper scoring rule on a $\sigma$-algebra that is not countably generated
Given a $\sigma$-algebra $\scr F$ on $\Omega$, say that an accuracy scoring rule for $\scr F$ is a function $s$ from the set of all (countably additive) probabilities on $\scr F$ to the $\scr F$-...
- 2,555
8
votes
2
answers
4k
views
Expected value of min of variables - what informations do I need?
I encountered a problem where I need to compute:
$$\mathbb{E}(U) = \mathbb{E}(\min(X_1, .. , X_6))$$
The problem is that I have little information on the $X_i$. Basically I know $\mathbb{E}(X_i)$ and $...
- 267
1
vote
1
answer
135
views
Tight upper-bound on dependent events
Given $r$ r.v.s $x_k$: $x_k=1$ with probability $p_k$ and $0$ otherwise. Let $s_i=\sum_{j=1}^r c_{ij}p_j$ where $c_{ij}\in[0,1]$, $1\le i\le n$. Let $E_i$ denote the event $\sum_{j=1}^r c_{ij}x_j>(...
- 367
1
vote
0
answers
142
views
What are the Lévy processes with specific increments?
It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$.
I wonder what are the Lévy processes with increments from ...
- 11
-1
votes
2
answers
158
views
Cumulants of a sequence of variables with zero mean and variance
Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
- 13
0
votes
1
answer
119
views
Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?
Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
- 11
2
votes
0
answers
69
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
3
votes
3
answers
2k
views
How close are two Gaussian random variables?
Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
- 159
0
votes
0
answers
36
views
How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?
I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...
0
votes
0
answers
63
views
The moment problem for $m_n=1/n$
What is the p.d.f. for the moments $m_n=1/n$ ?
(They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)
- 85
5
votes
2
answers
139
views
Number of resampling until obtaining a uniform list
Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
- 1,608