# 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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### 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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105 views

+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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vote

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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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59 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 \...

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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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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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**1**answer

121 views

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### 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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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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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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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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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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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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**1**answer

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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**1**answer

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...

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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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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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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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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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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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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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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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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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**3**answers

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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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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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 ...

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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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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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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 ...

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**1**answer

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 ...

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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 ...

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**2**answers

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\...