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-1
votes
0answers
28 views

Maximum chi-square distance between norm vectors [on hold]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below. $d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$
1
vote
2answers
49 views

Approximating Probability Distribution by Sampling

Consider a discrete probability distribution over $n$ events. Assume that the probabilistic kernel is a black box, that is, we can only sample from it without knowing anything about the type or ...
0
votes
1answer
235 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
0
votes
1answer
60 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 ...
0
votes
1answer
76 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 ...
26
votes
1answer
2k views

Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes. Let $\mathrm{Li}(x)$ be the offset logarithmic ...
0
votes
0answers
42 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 ...
3
votes
2answers
171 views

Gradient descent-like optimization on a convex landscape with noisy sampling

This is a rewrite of the original positing (below), and is crossposted to ...
0
votes
0answers
36 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
3answers
106 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
0answers
32 views

Concentration bound for $f(w) = w \times \sin wz$

I need to find an exponential bound for $P(|S_n - \mu| > \lambda)$ where $S_n = \frac{1}{D} \sum_{i=1}^D w_i \sin w_iz$ for a constant $z$, $E(S_n) = \mu$ and $w_i$ are drawn from the normal ...
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 ...
3
votes
1answer
51 views

Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...
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
1answer
210 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 ...
0
votes
1answer
96 views

Is any derivative of $f_1^x f_0^{1-x}$ w.r.t. $x$ integrable?

For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then ...
17
votes
1answer
953 views

When should we expect Tracy-Widom ?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on 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 ...
3
votes
2answers
377 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 ...
0
votes
1answer
47 views

one divided by (constant plus complex Gaussian) [closed]

Let $X$ be a circular symmetric complex Gaussian random variable with zero mean and unit variance. Define $Y=\frac{1}{A+x}$ for some real-valued constant A. What is the distribution of $Y$? When is ...
3
votes
1answer
60 views

concentration of random matrices involving normal random variables

Define the random variable \begin{align*} A=|a_1|^2\mathbf{a}\mathbf{a}^* \end{align*} where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as ...
7
votes
3answers
231 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 ...
1
vote
0answers
33 views

Angular distribution for Gaussian vector with non-zero mean

The angular central Gaussian distribution (ACG) is the distribution of $\frac{\mathbf{x}}{\|\mathbf{x}\|}$, when $\mathbf{x}\sim\mathcal{N}\left(\boldsymbol{0},\mathbf{A}\right)$, where $\mathbf{x}$ ...
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
377 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, ...
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 ...
0
votes
0answers
28 views

Probability distribution of a function of a random variable $P(y(x))$ [migrated]

Do you have an idea about how to prove the following? (references will be useful) Let $x$ a random variable with probability distribution $ρ(x)$, and $y=f(x)$ another random variable, then the ...
1
vote
0answers
24 views

Conditional Distribution of Inverse Wishart

Suppose $\begin{bmatrix} K_{11} K_{12}\\K_{12}^T K_{22} \end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix} \Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22} \end{bmatrix}\right)$. What is 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?
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 ...
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 ...
3
votes
2answers
84 views

expectation of log(x+a) when X follows a beta distribution

Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?
5
votes
0answers
205 views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by ...
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
90 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
51 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 ...
1
vote
1answer
389 views

What can be said about an infinite linear chain of conjugate prior distributions?

We can sample a discrete value from the multinomial distribution. We can also sample the parameters of the multinomial distribution from its conjugate prior the dirichlet distribution. Since the ...
1
vote
1answer
54 views

Is there a simple closed form solution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution?

I have an exponential distribution with rate $\lambda$, where $\lambda$ is drawn from a Gamma distribution with shape and scale parameters $(k,\theta)$. I'd like to calculate an exact PDF for values, ...
1
vote
0answers
36 views

Does this kind of integral equations have unique solution?

Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on ...
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 ...
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
49 views

Distribution of the local time for reflected Brownian motion on the quadrant

If $W$ and $B$ are two independent Brownian motions, $X_t=W_t-\sup_{u\le t} B_u + 1$. I want to find the distribution of $S$ the first hitting time of $0$ by $X$. If someone could give me a direction ...
5
votes
1answer
172 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
71 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, ...
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 ...
5
votes
1answer
171 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
202 views

First colour to be drawn $n$ times from a hypergeometric distribution

Consider a set up of drawing balls without replacement from a population of $R$ red balls and $B$ black balls. Is anyone aware of an efficient way to calculate the probability, $p$, that I draw $n$ ...
0
votes
0answers
70 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. ...