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2
votes
1answer
163 views

Central Limit Theorem for additive function of permutations of sequences

Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational. For each $n$ such ...
1
vote
1answer
330 views

Calculate channel capacity of general channel under constraint

Hi! Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this ...
3
votes
1answer
127 views

Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...
0
votes
2answers
193 views

Are all variables in a set of random variables independent if all pairs are independent?

If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized: $$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$ for all pairs $(X_i, X_j)$, does ...
1
vote
1answer
215 views

Distribution of the powers of a primitive element of a finite field

What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending ...
0
votes
1answer
545 views

Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?

Changing my question in light of Dan's answer. Thanks, Dan. Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
7
votes
2answers
426 views

Concentration bounds for sums of random variables of permutations

I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds. As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
7
votes
1answer
302 views

Probability density that minimizes the sample range

Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in ...
8
votes
2answers
478 views

What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia) For some sets, the ...
0
votes
1answer
77 views

Multinomial — how many trials in order to see all the values with prob 1-\alpha

Let suppose that I have a box with $k$ different balls, each one with a different color. At each time I have to extract a ball and observe the color. Then I put the ball back in the box. How many ...
3
votes
3answers
331 views

Support of an infinitely divisible measure.

Hello, if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of ...
1
vote
1answer
75 views

Numerical optimisation for multivariate Gaussians

Hi, I want to calculate $ f_{\mathbf x}(x_1,\ldots,x_k)\, = \frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf ...
1
vote
2answers
624 views

minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$. Is there any closed form expression or at least a good approximation for ...
2
votes
0answers
791 views

Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its satisfactory answer anywhere. My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...
4
votes
1answer
277 views

Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$. For example if $n = p^i$ for some prime $p$ then $D_n(x) ...
0
votes
0answers
147 views

softmax activation function with infinite support ?

Hi, How do we calculate the terms of a softmax activation function with an infinite support ? That is, finding the $\{p_i\}_i$ with $p_i = {{e^{q_i}} \over {\sum_{j=1}^\infty e^{q_j} }}$ (how to ...
1
vote
1answer
141 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
1
vote
1answer
89 views

Equivalence between choosing a subspace and choosing its orthogonal

Hi, We consider subspaces of $\mathbb{R}^N$. Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of ...
0
votes
1answer
289 views

Simple markov chain problem

I know this is an easy problem, but I can't figure it out. A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$. ...
6
votes
1answer
209 views

Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as “natural” / “induced”?

Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
0
votes
1answer
176 views

Limit of the stochastic process at time 0

This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic ...
0
votes
0answers
572 views

Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative

Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that : ...
0
votes
0answers
98 views

power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind: user1, timestamp1 user1, timestamp2 user1, timestamp3 user2, timestamp4 user1, timestamp5 ... These ...
0
votes
0answers
80 views

Marginal Distribution.

Consider a couple of real random variables $(X,Y)$ and $\mu_{X,Y}$ the induced probability distribution. Denote by $\mu_X$ and $\mu_Y$ the distributions of $X$ and $Y$. Is it true that ...
2
votes
2answers
391 views

Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N ...
2
votes
1answer
176 views

If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?

If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
4
votes
2answers
193 views

Sampling from a recursively defined distribution

I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult. As ...
1
vote
1answer
277 views

Product of probability densities of the form x^{-t} exp (-ax)

I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, ...
1
vote
1answer
162 views

Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...
0
votes
0answers
86 views

Proving that a property holds for random sequences with given marginal distribution by rearrangement

I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
1
vote
2answers
245 views

Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...
3
votes
2answers
1k views

Upper bound on expectation value of the product of two random variables [closed]

Hello, I am trying to find an upper bound on the expectation value of the product of two random variables. So suppose x, y are two non-independent random variables, given that I know the distribution ...
1
vote
2answers
118 views

limit of functionals on weak convergent random variables

Suppose real value random variables satisfy $X_{n} \Rightarrow X$ (convergence in distribution) as $n\to \infty$ in the same probability space $(\Omega, \mathcal F, \mathbb P)$. It is well known that ...
1
vote
0answers
267 views

Distribution of random vectors

Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$). A vector $u\in ...
1
vote
1answer
129 views

Ergodicity and convergence time in Probabilistic Cellular Automata

Has the following conjecture been prooved, or has any step in the direction of its proof been done? "ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff ...
3
votes
2answers
210 views

Ergodicity for a Probabilistic Cellular Automaton on a finite space

Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: ...
0
votes
2answers
1k views

Cumulative distribution function and convolution.

Hello, Given a probability distribution of a discrete variable p1(x) and a probability distribution of a discrete variable p2(y) defined by p2(y) = Sum_{x,x'} p1(x) p1(x') * KroneckerDelta((x+x')/2 = ...
3
votes
1answer
393 views

When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
1
vote
3answers
135 views

$Y|X \sim N(\mu X,X^2)$ and $X\sim N(\alpha, \beta)$. How is $Y$ distributed?

Dear all, I have recently been breaking my head over this question. The idea is that a certain variable $Y$ is normally distributed with a parameter $X$ in both mean and variance. $Y|X \sim N(\mu ...
1
vote
2answers
155 views

Probability of first collision with replacement

Suppose I have a bag of N balls, each labeled with a distinct integer. The experiment starts when I draw a ball from the bag, record its label, and then return the ...
0
votes
1answer
405 views

The Probability distribution of Random variable of Random variable

In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...
3
votes
3answers
393 views

What is the name for a non-normalized distribution?

For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
2
votes
1answer
214 views

Tracking down locality assumption in CHSH inequality

CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness. Now counterfactual definiteness tells us that given two different measurements on the ...
0
votes
0answers
100 views

Stationarity of an Integral Process

Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with ...
3
votes
1answer
268 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 ...
1
vote
2answers
281 views

Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$. It ...
1
vote
0answers
177 views

Why this two model have same probability distribution?

(1) Consider the following method of generating a random tree with $n$ nodes. First expand the root node into two branches. Then expand one of the two terminal nodes at random. At time $k$, ...
1
vote
2answers
657 views

Distribution of Maximum of a uniform multinomial distribution

Hello, I'm working with a data structure which uses a uniform distribution to bucket the inputs into $k$ buckets. The efficiency of the structure is bounded by the $\frac{k_{max}}n$, where $n$ is the ...
1
vote
1answer
315 views

Azuma's Inequality when the conditions hold with high probability?

In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
3
votes
0answers
907 views

E[ | X - Y | ] where X and Y are independent Poisson random variable

What is the expected value of the absolute difference of two independent Poisson variables? E[ |X - Y| ] Seems like an easy question but I haven't found an easy solution. I've split the double sum ...