Questions tagged [random-functions]
The random-functions tag has no usage guidance.
67
questions
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Expected value of number of collisions for a matrix valued random function
Consider the function
$$f(x_1, x_2, \ldots, x_k) = S_1^{x_1} S_2^{x_2} \cdots S_k^{x_k}.$$
Each $x_1, x_2, \ldots, x_k \in \{0, 1\}$ and each $S_i \in \mathbb{F}_q^{n \times n}$ is a randomly chosen $...
1
vote
1
answer
74
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On the growth of sample paths of Gaussian random fields
Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ...
- 361
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0
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57
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Distribution of multivariate polynomial evaluation
Let $R:\mathbb{F}^n \rightarrow \mathbb{F}^m$ be a multivariate quadratic map. Here $\mathbb{F}$ denotes the finite field of order $q$.
I am curious to know whether the distribution of $R(x)$ for ...
5
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0
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261
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Fastest sine of a large power of 2
What is the fastest known way to calculate $\sin(2^{n})$ for large integer $n$?
I only need the highest few bits to be correct. I suspect that the compute time required
scales with $n$ (and actually ...
- 1,445
1
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1
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66
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Partial derivative of expectation and Stein's lemma
Currently, I am reading a paper about the Gaussian Process in Neural Network [1]. In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...
- 95
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1
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193
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Construct a random vector as a function of another random vector
ASSUMPTION 1: there exists a continuous random vector $(X,Y,Z)$ such that
$$
\begin{cases}
p_1=\Pr(X\geq 0, Z\geq 0)\\
p_2=\Pr(Y\geq 0, Z< 0)\\
p_3=\Pr(X< 0, Y<0)\\
\end{cases}
$$
where $(p_1,...
user447159
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20
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Maxima/minima of a random curve
Let $\left\{ {{\xi _k}} \right\}$ be independent random variables with some known distribution function ${F_\xi }$ and $f(x)$ be a "good" function that is bounded and decrease to zero with ...
- 21
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0
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65
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Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 4,997
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3
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Practical pseudorandom generators
It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem.
I am curious if someone developed kind of &...
- 741
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0
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31
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Get covariance from log-density function
Problem
Given a following log-density function
$$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$
where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters.
Find $\Bbb Cov( g(Y),...
- 111
1
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1
answer
289
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Approximate expectation of a random variable that is the logarithm of a function of a binomial
I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series:
\begin{...
- 91
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votes
1
answer
147
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Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...
- 5,728
1
vote
1
answer
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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)) = \...
- 5,728
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1
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157
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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 \...
- 5,728
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57
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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}...
- 5,728
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1
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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 ...
- 2,016
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1
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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 ...
- 64.8k
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143
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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 ...
- 9,100
3
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1
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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 ...
- 1,606
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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\...
- 29
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1
answer
113
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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 > \...
- 2,016
3
votes
1
answer
83
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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 ...
- 2,016
2
votes
0
answers
57
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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$) ...
- 8,680
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127
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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 ...
- 1,259
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1
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154
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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 ...
- 105
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1
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304
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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 ...
- 105
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votes
1
answer
167
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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
vote
1
answer
89
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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 ...
- 103
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0
answers
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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 ...
- 131
3
votes
1
answer
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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 ...
- 263
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votes
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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 $\...
- 5,728
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vote
1
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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
votes
2
answers
196
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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} $$
be a $...
- 153
1
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0
answers
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Random solute transport equation
After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
- 141
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1
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571
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Can this criterion to indicate the randomness some numbers? [closed]
John Derbyshire in his book PRIME OBSESSION says on page 366:
CHAPTER 3
10.
"Here is an example of e turning up unexpectedly. Select a random number
between 0 and 1. Now select another and add ...
- 1,183
5
votes
1
answer
382
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A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...
- 2,016
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votes
1
answer
184
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2D wave equation with gaussian boundary condition
Given the 2D wave equation in polar coordinates:
$$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(...
- 399
2
votes
3
answers
361
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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 ...
- 161
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4
answers
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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?
...
- 11.1k
7
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2
answers
302
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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<...
- 17.3k
2
votes
0
answers
93
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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}, \...
- 323
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votes
1
answer
525
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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)...
- 243
3
votes
1
answer
280
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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
votes
0
answers
406
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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=...
- 189
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votes
1
answer
155
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 $P_k$, ...
- 341
1
vote
0
answers
221
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)\...
- 859
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votes
0
answers
306
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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{...
- 859
2
votes
2
answers
262
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 ...
- 309
1
vote
1
answer
153
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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 ...
- 11
1
vote
1
answer
257
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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 $n$?
- 401