0
votes
1answer
71 views

Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...
0
votes
0answers
40 views

Quantiles moments and Convergence

QUESTION: Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
0
votes
0answers
35 views

Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian? In ...
1
vote
1answer
55 views

A calculation involving a uniform random variable quantile

THE PROBLEM: Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to ...
0
votes
1answer
53 views

Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent. 1): $X$ is $L^p$-integrable. 2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...
1
vote
3answers
105 views

Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
0
votes
1answer
57 views

Cramér-Wold device with limited angle and independence assumption

Let $X$ be a random vector taking values in $\mathbb R^2$ with probability density $p(x) = p_1(x_1)p_2(x_2)$, i.e. the components of $X$ are independent. Let $V$ be an open set in $\mathbb S^1$, the ...
4
votes
2answers
152 views

Joint probability distribution as functions

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...
7
votes
3answers
229 views

Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of $$\mathbb{E}\max_{1\le i\le n}|X_i|$$ and ...
2
votes
0answers
66 views

Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
3
votes
1answer
46 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 ...
0
votes
0answers
45 views

Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...
4
votes
1answer
80 views

General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
0
votes
3answers
193 views

Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
2
votes
1answer
123 views

Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
6
votes
1answer
380 views

Mean of i.i.d Random Variables With No Expected Value

Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
3
votes
1answer
89 views

Variance of maximum of mixture of gaussians

Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some ...
4
votes
0answers
50 views

Cramér-Wold theorem with independence assumption

Let $X = (X_1, X_2)$ be a random vector with joint probability density $p$. The celebrated Cramér-Wold theorem says that we can reconstruct $p$ from knowing the push-forward densities of $X$ under all ...
5
votes
0answers
204 views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by ...
5
votes
1answer
171 views

Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution. Specifically, from the general convergence rates stated in the Berry–Esseen ...
1
vote
1answer
70 views

explicit expressions of the distribution of sums of i.i.d. logistic random variables

Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4... The expressions given in "On the convolution of logistic random ...
1
vote
0answers
49 views

What is entropy of a variable described by Knightian uncertainty? [closed]

I have asked this question at Theoretical Computer Science and received no response. Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, ...
2
votes
3answers
152 views

Expected value of swaps

Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...
4
votes
3answers
356 views

Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$

Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...
0
votes
0answers
69 views

Fitting distribution to spatial data

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below. ...
2
votes
0answers
52 views

Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods?

Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ...
1
vote
0answers
100 views

approximation of probability distribution

I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying $$\int_{\mathbb{R}}|x|d\mu<+\infty$$ Set $$A_n=\Big\{\frac{i}{n}:~ ...
1
vote
0answers
55 views

Cramér-Wold like theorem for independent random variables

Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions \begin{align*} \left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
4
votes
1answer
105 views

Does second order stochastical domination with increasing likelihood ratio imply first order domination?

This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing ...
4
votes
2answers
222 views

Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that $$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$ and such that ...
1
vote
0answers
45 views

Characteristic function known on subsets

Let $X$ be a random variable in $\mathbb R^n$ with distribution $\mathbb P_X$. Given a (infinite) family of matrices $W_t \in \mathbb R^{n \times m}$ parameterized by $t \in \mathbb R$, suppose we ...
3
votes
1answer
72 views

Relaxing conditions for Cramer-Wold type theorem

Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions \begin{align*} \left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
4
votes
1answer
191 views

Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution

Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total ...
2
votes
1answer
84 views

Question about infinite-dimensional BM

Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by $$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$ where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...
5
votes
1answer
170 views

“Smallest” event such that probability greater than a given value

Very briefly, consider the probability space $(\mathbb R^n, \mathcal{B}(\mathbb R^n),P)$. During a problem I am studying, I came to a point where i need to compute \begin{equation*} \begin{aligned} ...
2
votes
1answer
115 views

Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. What is the ...
2
votes
1answer
81 views

Maximal component of a multivariate Gaussian distribution

Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...
0
votes
0answers
71 views

Obtaining the 'threshold' of a distribution

Context of Research Consider the expression: \begin{align} \widehat{\Theta}(\rho)_i = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) ...
0
votes
0answers
62 views

Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
3
votes
3answers
259 views

Repeated draws from multinomial distribution

(This is a cross-post from Math StackExchange http://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities) Let $\vec X = (X_1, \dots, X_k)$ be a draw from a ...
2
votes
2answers
272 views

What is the maximum entropy distribution on the natural numbers?

On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution. Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...
0
votes
0answers
43 views

Distinguishing two different matrix distributions in polynomial time

I have two distributions: $\{ (f^TA + e_1, f^T(As+e) \}$ and $\{ (f^TA, f^T(As+e) + s_i \}$ where $A$ is a randomly generated $m \times n$ binary matrix $A, A_{ij} \in \{0,1\}$, $f$ and $e$ are a ...
1
vote
1answer
151 views

Set of distributions that minimize KL divergence,

Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence, is there a ...
2
votes
1answer
302 views

Expectation of Maximum of Uniform Multinomial Distribution

Suppose we have a uniform multinomial distribution with $k$ buckets, i.e. we put $n$ items uniformly at random in $k$ buckets leading to $n_1, \dots, n_k$ items in each bucket respectively. Let $m = ...
0
votes
1answer
99 views

concentration of sums of fourth moment of normals

I was wondering what is the best tail bound for \begin{equation*} \mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ? \end{equation*} where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.
4
votes
1answer
167 views

Limit of pushforward measures of random variables is “represented” by a random variable

Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of real random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures ...
2
votes
2answers
118 views

Drawing random variates from a partially described probability distribution

I have a probability distribution over $\{0,1\}^n$ but instead of knowing the full joint distribution $p(x_1,\dots,x_n)$, I only know $p(x_i=x_j)$ for each $i,j$. How could I draw a random binary ...
2
votes
1answer
323 views

Central limit theorem for $P(x)\sim 1/x^3$ distribution

I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...
3
votes
2answers
376 views

Distribution of a product of two discrete i.i.d. variables

The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like ...
3
votes
0answers
107 views

Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...