Questions tagged [random-functions]

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3answers
112 views

Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem. I am curious if someone developed kind of &...
1
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0answers
26 views

Get covariance from log-density function

Problem Given a following log-density function $$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$ where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters. Find $\Bbb Cov( g(Y),...
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0answers
39 views

Supremum of Lipschitz Gaussian process on sphere

Let $T:=\mathbb S_{m-1}$ be the unit sphere in $\mathbb R^m$ and consider a real-valued centered Gaussian process $(x_t)_{t \in \mathbb T}$ such that $\mathbb E X_t^2 = t$ for all $t$. consider the ...
1
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1answer
144 views

Approximate expectation of a random variable that is the logarithm of a function of a binomial

I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series: \begin{...
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0answers
26 views

(random fields / gaussian process): On rewritting a certain expectation as a kernel function

Let $v = (v_1,\ldots,v_n)$ and $(w_{1,1},\ldots,w_{1,n},\ldots, w_{n,m})$ be random vectors with iid coordinates, and also $v$ is independent of $w$, with $w_{i,j} \sim N(0,1/m)$ and $v_j \sim N(0,1/n)...
3
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1answer
106 views

Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field

Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow $$ \dot{x}(t) = -\nabla g(x(t)), \;...
0
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1answer
69 views

Central limit theorem for chi-squared random field on $\mathbb R^p$

Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
1
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1answer
75 views

Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
2
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0answers
50 views

Concentration inequalities for gradient flows induced by random fields

Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...
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1answer
43 views

Emergence of non-power-law behaviour under infinite summing

Suppose $X_1,X_2,...$ is a sequence of random vectors in $\mathbb{R}^n$ s.t for all $k \in \mathbb{Z}^+$ and $u \in \mathbb{R}^n$ we have that $E [ \langle u, X_i \rangle ^k]$ is finite. (The $X_i$s ...
5
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1answer
343 views

Forcing, constructibility, and random functions

This question is in some ways an offshoot of my recent question about trying to explain forcing to someone (such as Scott Aaronson, whose questions have prompted my questions) encountering it for the ...
3
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0answers
128 views

Two kinds of generating functions

Sorry for a possibly off-the-topic question, but I am afraid to gain the necessary overview to give an answer (supposed the question is not ill-posed) is beyond my capabilities. In the course of ...
3
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1answer
77 views

Reference for Function-Valued Random Variables?

Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following: Let $X$ be a topological space, $Y$ be a random variable with ...
1
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0answers
42 views

Negative moments of Steinhaus random variables

Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$. 1) Find $E\left(\sum_{i=1}^nf_i a_i\...
2
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1answer
85 views

Proving anti-concentration for the operator norm of a random matrix

If $X$ is a random matrix then I would like to find $\theta >0$ and $\delta \in (0,1)$ s.t I can say, $$\mathbb{P} \Bigg [ \Big \vert \Vert X \Vert - \mathbb{E} [ \Vert X \Vert ] \Big \vert > \...
3
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1answer
63 views

About concentration of eigenvalues values of a random symmetric matrix in a specific interval

Given a random symmetric matrix $M$ and two numbers $\lambda_\min$ and $\lambda_\max$ how do we calculate the expected or high probability value of the fraction of its eigenvalues which lie in the ...
2
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0answers
55 views

Less regular version of the Gaussian free field

One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
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0answers
106 views

Canonical embedding of Hilbert space in random $L^2$

This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...
0
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1answer
104 views

The distribution of the power of the sum of inner products of two independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$. What is the distribution of the following ...
0
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1answer
114 views

The distribution of the sum of inner products of two independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$. What is the distribution of the following ...
2
votes
1answer
119 views

Smooth transformation of a curve with fixed ends and length [duplicate]

I am simulating polymers of fixed length and fixed ends. I would like to search the phase space of all possible conformations quickly. Is there anyway I can generate efficiently a lot of (rather) ...
1
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1answer
78 views

Expected roots of polynomials with randomness in coefficients

I need some results on expected roots of polynomial with random coefficients. I searched for $\textit{roots of polynomial with random coefficients}$ online and found some papers, but they are dealing ...
3
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0answers
55 views

Largest component and number of components of random mappings with bounded in-degree

Let $S$ be a finite set of size $n$ and consider all functions $f$ from $S$ to itself, such that the preimage $f^{-1} (k)$ obeys $0 \leq |f^{-1} (k)| \leq m$. Let $F$ be chosen uniformly at random ...
3
votes
1answer
294 views

Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
2
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0answers
93 views

Average number of pieces of a random piecewise-linear function

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
1
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1answer
62 views

Proof of variance bounds for transformed random variables

Given I have a random variable $a$ that can be realised in the domain $D$ and has a finite variance $\sigma^2$. Furthermore I have a function $f$ which is differentiable(hence continuous) with an ...
2
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2answers
173 views

Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables. Furthermore, let $$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$ be a $...
1
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0answers
38 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
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1answer
570 views

Can this criterion to indicate the randomness some numbers? [closed]

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10. "Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add ...
5
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1answer
361 views

A question about the paper “The Condition Number of a Randomly Perturbed Matrix”

My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307 Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...
0
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1answer
148 views

2D wave equation with gaussian boundary condition

Given the 2D wave equation in polar coordinates: $$u_{\rho\rho}+\dfrac{1}{\rho}u_{\rho}+\dfrac{1}{\rho^2}u_{\theta\theta}=\dfrac{1}{a^2}u_{tt}$$ with $u=u(\rho,\theta,t),(\rho,\theta,t)\in [0,c]\times(...
2
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3answers
322 views

Is it possible to check if a sequence of numbers (1 to 10) have been generated randomly or by a human? [closed]

Let's suppose to have sequence like "10 1 2 4 5 3 2 1 2 3 10 1" and so on... It is possible to understand if it has been generated randomly (by a program) or by a human? Do they correspond to a kind ...
7
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4answers
2k views

Generating Random Curves with Fixed Length and Endpoint Distance

Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given? ...
7
votes
2answers
276 views

Sign-oscillations for power series with random coefficients

Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<...
2
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0answers
88 views

Stochastic approximation in two dimensions

I have a question about a stochastic approximation method presented in J. R. Blum, Ann. Math. Stat. 25(4): 737 (1954). Given a $k$-dimensional vector $\bf x$ and $k$ random variables $Y^1_{\bf x}, \...
5
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1answer
502 views

Zeroes of a not quite holomorphic (but random if helpful) function

I’m interested in the zeroes of the complex function $f(z,\bar{z}) = p(z) + \frac{1}{log(|z|)} q(z)$ where both $p$ and $q$ are polynomials of the complex variable $z$ (and are therefore holomorphic)...
3
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1answer
238 views

The best linear approximation of a random function

Let $\mathcal{F}_n$ be the set of all boolean functions of $n$ variables and let $\xi$ be a random variable with values in the set $\mathcal{F}_n$ with the uniform distribution. We define a new random ...
4
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0answers
342 views

Inequality for Conditional Variance

Let $X,Y$ be real random variables with joint density $p(x,y)$. Let the variance of $X, Y$ be $\operatorname{Var}(X) = \operatorname{Var} (Y)=\sigma^2$. The conditional variance of $X$ for a given $Y=...
2
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1answer
145 views

Expected number of local minima of random polynomial in high dimensions

Consider the inner product space $P_k$ of polynomials of degree $\le k$ on the unit sphere $\mathbb S^{n-1}\subset \mathbb R^n$. Let $p\sim N(0,\sigma^2 I)$ be a randomly chosen polynomial in $P_k$, ...
1
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0answers
202 views

One-sided Talagrand concentration inequality for empirical processes

Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that \begin{align*} \mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
4
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0answers
261 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
2
votes
2answers
192 views

Random processes with smooth paths

Is there any prototypical example of a Random process with smooth paths? I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path. It's easy to ...
1
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1answer
141 views

Is there analogs of perlin noise algorithm?

I want to create procedure generated map, but all resources that I found talks about using of "perlin noise" algorithm. Maybe better (higher perfomance, more realistic terrain generation) analogs ...
1
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1answer
222 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 $n$?
9
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1answer
1k 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, Y)...
2
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1answer
230 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 $< f,X>$ ...
7
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2answers
322 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
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3answers
553 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 $X$, and ...
5
votes
2answers
955 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
88 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 $\...