The random-functions tag has no wiki summary.
5
votes
0answers
57 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
135 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 ...
0
votes
0answers
25 views
Moments in the Quantile Process
Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F:
$$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$
In this case, $F^{-1}$ is the (generalized) inverse of ...
2
votes
2answers
103 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
vote
0answers
66 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
votes
0answers
39 views
Isotropic correlation function for a vector valued random field
I'm having trouble with some of the implications of the following theorem.
Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...
0
votes
0answers
87 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 ...
0
votes
0answers
76 views
Bounding Random Quadratic Gauss sums
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$ and $|\epsilon_k|=1$ for all $k=1,2,\ldots,n$. We have
\begin{align*}
...
3
votes
1answer
133 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 $m<n$, ...
0
votes
0answers
46 views
Tail Bounds for the minimum value of a function
Consider y to be the minimum value of an objective function over some subspace. More specifically
$y= \min_x \|e+Bx\|_\infty \quad s.t. \quad x\in \mathcal{S}$
where $e$ is a known vector, $B$ is a ...
1
vote
0answers
25 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 ...
0
votes
0answers
29 views
Convergence integral in probability
Let $X_1,\dots,X_n$ be i.i.d with distribution function $F$. Let $\hat F_n$ be their modified empirical distribution function, i.e.,
$$
\hat F_n(x)=\frac1{n+2}\left(1+\sum_{i=1}^n1_{\{X_i\le ...
2
votes
1answer
211 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
72 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 ...
0
votes
3answers
337 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
199 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
votes
0answers
91 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
votes
3answers
528 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
273 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
398 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 ...
