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0answers
19 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$ [migrated]

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
0
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0answers
33 views

Supremum distribution of band-limited functions with random spectrum

Consider the properties of band-limited functions $f_N:[-\pi,\pi]\to\mathbb{R}$ defined through their Fourier series $f_N(x)=\sum_{n=-N}^N c_n e^{inx}$ where $c_n=a_n+i b_n$ and both ...
1
vote
1answer
50 views

Probability of a random divisor of a given integer $n$ [closed]

Suppose we have a given integer $n$. We randomly pick an integer $k$, where $k\leq n$. Then What is the probability that $k$ divides $n$?
7
votes
1answer
295 views

Decompose dependent random variables into function of dependent and independent parts

Let $X$ and $Z$ be two (possibly dependent) random variables. Is it necessarily the case that there exists a Borel function f and a random variable $Y$ that is independent of $X$ such that $Z = f(X, ...
2
votes
1answer
136 views

Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...
6
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0answers
98 views

Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws

It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...
0
votes
3answers
255 views

An inequality based on expectation of continuous random variables

I am trying to prove the following statement: $$ E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)] $$ where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...
3
votes
2answers
271 views

PDF of the product of normal and Cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
1
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0answers
75 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
0
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0answers
109 views

Reference: Bochner Integral`

What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
3
votes
1answer
153 views

Strictly positive solutions of a random linear system

Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...
1
vote
0answers
34 views

the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...
2
votes
1answer
230 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 ...
2
votes
0answers
75 views

Link between presence of attracting random fixed points and synchronisation - is this an open question?

This is a question in the theory of random dynamical systems. Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an ...
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3answers
383 views

Random infinite sequence : Can machines generate truly random sequences. [closed]

Test : "A True Random Sequence Source and a computer producing a certain sequence of numbers are kept in separate rooms and judges try to tell them apart by conducting a series of tests on the ...
0
votes
1answer
231 views

Random infinite sequences

An Algorithm/Turing machine Produces a symbol from a finite alphabet, and continues doing so infinitely. Another algorithm gets a copy of this symbol, ...
0
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0answers
121 views

Random variables related through nonlinear system of equations

I asked this question on http://math.stackexchange.com/questions/377140/random-variables-related-through-nonlinear-system-of-equations, however I received no answer for a while so I'm posting it here: ...
5
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3answers
592 views

How to test if two sets of random numbers might be from the same random number generator?

I have a sequence of sets of random numbers, with each set generated by an unknown random number generator. I am assuming that in the sequence, the random number generator is the same one for a ...
2
votes
2answers
294 views

Scale random variables in a way they have equal probabilities of being minimal

I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...
5
votes
1answer
542 views

Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions

Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...