Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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2answers
128 views

Random walk on $n$-dimensional cube

Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each ...
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+50

Maximum mutual information of random unitary transformation

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity ...
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1answer
50 views

Local limit theorems for circular/spherical distributions

Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$): $$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...
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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 \...
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0answers
102 views

A slight generalization of Skorokhod's representation theorem

Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
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1answer
111 views

Kullback–Leibler chains

The following question was asked and then deleted by the post author: Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...
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1answer
121 views
+50

Mutual Information after Applying Random Unitary Matrix

Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied: \begin{align} \mathbf{y}=\...
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0answers
58 views

Is Axiom of Choice for convex sets of distributions on naturals necessary?

Take any family $(S_i)_{i∈I}$ such that each $S_i$ is a convex set of functions $f : ℕ→[0,1]$ where $\sum_{k∈ℕ} f(k) = 1$. By "convex" we mean that for any $f,g∈S_i$ and any $a,b∈[0,1]$ such ...
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1answer
123 views

Scaling of double convolution

I am interested in the scaling of $$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$ In particular, I suspect that $$F(...
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20 views

Mean-preserving spreads and equality of noise in distribution

Let $X$, $Y$ be mean preserving spreads (MPS) of the same random variable $Q$ and assume that $X =_d Y$ in distribution. Then, by the definition of MPS, there exist variables $Z$ and $Z'$ such that $Q ...
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1answer
138 views

How to check positive-definiteness of this function?

Consider a real random vector $\vec X =(X_1, X_2)$ with characteristic function $\phi(\vec t) \equiv \mathbb{E} \big[ e^{i \vec t \cdot \vec X} \big]$ (where $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
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1answer
158 views

Anti-concentration of Gaussian when conditioning on event

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
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1answer
59 views

Convoluted Cantor-like measure which has a continuous component [duplicate]

Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable $$ \sum_{k\ge 1}3^{-k}X_k $$...
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48 views

Conditional moments estimate

Let $V\in C^\infty(\mathbb R^{d_1}\times \mathbb R^{d_2};\mathbb R)$, such that $Hess V(x,y)\geq \alpha\,I$ for some $\alpha>0$ (namely $V$ is uniformly convex). Thus $V$ as a unique minimum point $...
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44 views

Student's t-test for lognormally distributed samples

I want to understand whether the differences in the means of two independent samples (distributed lognormally) are statistically significant. In order to use Student's t-test, the data must be ...
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1answer
99 views

Large deviations: Growth of empirical average of iid non-negative random varialbes with infinite expectations?

Let $X_1,X_2,X_3,...$ be iid non-negative random variables with $E[X_i]=\infty$. I am looking for references on the growth in $n$ of the empirical average under assumptions on $X_1.,..,X_n$. A more ...
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1answer
112 views

Lottery in O(1) per participant

Goal: implement in $O(1)$ per participant a lottery where each participant has some large number of tickets, and the best (e.g. least) one wins, without actually burning electricity in proportion to ...
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1answer
55 views

Sufficient conditions for finite mean of a non-negative random variable

Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition: $$\lim_{x\rightarrow\infty} x(1-F(x)...
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1answer
57 views

Concentration on discrete probability estimator

Let $t>1$ and $X_1,..., X_t$ a set of real random variables from a discrete distribution, whose pmf is $p(x)$, supported on the points $1,...,k$. Let $N_t(x) = \sum_{i = 1}^t \mathbb{1}_{X_i =\, x}....
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1answer
47 views

Is integration against an indicator Wasserstein-Continuous

Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map: $$ \mathbb{P} \mapsto \...
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1answer
47 views

Positivity of exponentially bounded characteristic functions

I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all ...
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0answers
52 views

Riemann-Stieltjes integral of a distribution function

I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
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1answer
76 views

Existence of independent linear combinations of random variables

Let $X,Y$ be independent multivariant random variables on $\mathbb{R}^d$. Let $\alpha,\beta$ be two positive real values such that $\alpha^2+\beta^2=1$. Then, $S=\alpha X+\beta Y$ is a new random ...
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0answers
42 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 ...
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222 views

When does $A-A$ avoid $A$?

Chavez and Allawala recently used a statistical model to explain the empirical observation that the imaginary parts of the nontrivial zeros of the Riemann zeta function form a set $A\subseteq\mathbb{R}...
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1answer
224 views

Is this geometrically-defined minimum an algebraic number?

I'm trying to find the maximum value $c$ so that there is a probability distribution with support in $R_c:=[-2,2]\times[-2,2]\cap\{x+y\geq c\}$ so that $32$ expectational equations hold. In ...
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2answers
263 views

Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
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0answers
114 views

Relation satisfied by a Gaussian random variable

I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ...
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1answer
141 views

Quantization of normal distribution

For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points. Question: Is it known which element in $\mathcal{Q}_n$ is ...
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48 views

A closed form or a good approximation of an infinite series related to the negative binomial distribution

Does anyone know a closed form for this expression: $$\sum_{r =1}^{\infty}{{\alpha + 2r - 1}\choose{ r - 1}}(1 - p)^{\alpha + r}p^{r},$$ where $\alpha \geq 1$ and $0<p<1$. A good approximation ...
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1answer
40 views

Covariance in the limit of random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ are two sequences of random variables and we know that $X_n \overset{L^2}{\to} X$ and $Y_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean ...
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1answer
47 views

Expectation of the sum of the squares of the cardinal of an inverse function

I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as: $$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$ where $\oplus$ is the bitwise XOR. ...
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0answers
192 views

Maximum mutual information of a matrix representation

Let $\mathbf{Z}$ be a $m\times n$ matrix with zero-mean unit-variance i.i.d complex Gaussian entries. What is the maximum value of mutual information $I(\mathbf{X}_1,\mathbf{X}_2;\mathbf{Y})$ such ...
1
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1answer
60 views

KL-divergence and sub-$\sigma$-algebras

I am trying to understand if the following claim is true: Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\...
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0answers
21 views

Assigning negative integer moments to random variables with Hadamard regularization

Let $X\sim F_X$ denote a continuous random variable that admits a density $f_X$ with support $\mathcal S=\operatorname{supp}(X)\ni 0$ and assume $f_X(0)>0$. I am interested in defining a ...
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0answers
41 views

Bootstrap-$t$ confidence intervals

I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need ...
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2answers
80 views

Greater contribution in a sum of independent random variables

In section 2.4 (Summation of strictly stable random variables), page 54, of the book "chance and stability", Zolotarev, Uchaikin there is the following consideration : The general relation ...
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1answer
47 views

Random sampling from modified Erlang distribution

I am tasked with randomly sampling from the following probability density function, which is a modified Erlang Function: $$f(k,q,\nu)=\frac{(k q)^{k-1}}{[(k-1) !]^{v}} \quad \text { with } \quad q \...
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1answer
35 views

Simplification on the estimation on error of the ratio of 2 random variables

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables. Distribution of $Z=\dfrac{X}{Y}$ Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
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0answers
36 views

Distribution of total offspring of Poisson multitype branching process

The question I have is related to the question asked here: Total offspring of Poisson multitype branching process Fix $d\in\mathbb{N}$ and let $Z_n\in\mathbb{N}^d$ be a multitype branching process, ...
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1answer
75 views

Using $\delta$-method to “estimate” undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is. Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
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3answers
512 views

Convergence speed of a random dyadic rational generator

We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$ two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
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0answers
30 views

Log-concave probability measure with slowest decay

Let $X$ be a real valued random variable with log-concave distribution $\mu$. For each $x \in {\mathbb R}$, let $$ \phi_\mu(x)=\min\limits_{c\in{\mathbb R}}E[e^{c(X-x)}] $$ be the minimal value of the ...
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1answer
102 views

Is the topology generated by the convergence of finite-dimensional distributions metrizable?

Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
6
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0answers
69 views

Distribution of iid hypergeometric random variables conditioned on the sum

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$....
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1answer
114 views

What is the expected value of the sum of the k (out of a set of n) smallest normal random variables?

Given $n$ independent normally distributed random variables $X_1,X_2,...,X_n \sim N(\mu,\sigma)$. For any $k\leq n$, let $X_{(k)}$ be the k-th order statistics (i.e., the k-th smallest value). What is ...
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0answers
68 views

sub-exponential type upper bound on the Poisson probability

I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received. Question: For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
1
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1answer
62 views

Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables

Consider a set of i.i.d. (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance. In particular, if $P_{X_i}(x)$ is the P.D.F. of ...
3
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3answers
71 views

Testing uniformity for continuous probability distributions

Suppose I can sample from a random variable $X$ which is distributed on a compact interval, say, $[0,1]$. Fix a distance measure between distributions, say total variation. Let $\epsilon\in(0,1)$. How ...
3
votes
2answers
130 views

Distribution of a certain functional of iid $N(0,1)$ random variables

Suppose that $X_1,\ldots,X_n$ are iid $N(0,1)$ random variables. Consider the random variable given by $$ \xi_n =\Bigl|\frac1{\sqrt{n}}\sum_{t=1}^nX_t\Bigr|^2-\frac1n\sum_{t=1}^nX_t^2 =\frac1n\sum_{s\...

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