Questions tagged [random-functions]
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85 questions
11 votes
1 answer
396 views
Determine whether the random dynamical system $f(z)=1/(U-z)$ is bounded or not
I am looking at the random iteration in $\mathbb{C}$ $$z_{n+1}=\frac{1}{U_n-z_n},\qquad n\ge 0,$$ where $(U_n)$ is an random i.i.d. sequence taking the two complex values $4+\mathrm{i}$ and $4-\mathrm{...
- 311
3 votes
1 answer
125 views
Linear stability analysis of invariant distribution in a random dynamical system?
Consider the following discrete random dynamical system: $$ X_{n+1} = F_\theta(X_n), $$ where at each step the index $\theta \in \{1,2\}$ is chosen randomly and independently, and $X_n \in \mathbb{C}$....
- 311
0 votes
1 answer
154 views
Uniform concentration bound (function-valued random variable / continuous stochastic process)
I'm trying to consider a probability space $\Omega$ and $f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...
- 321
0 votes
1 answer
241 views
Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$
A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...
- 121
3 votes
1 answer
307 views
How to generate a random function with conditions?
The background is as follows: I consider the following differential equation $$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$ where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
- 65
1 vote
1 answer
345 views
What is convergence in distribution of random variables taking values in a non-metrizable product space?
Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...
- 1,201
0 votes
1 answer
156 views
Lower bounding the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise. How ...
- 405
0 votes
2 answers
163 views
the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
- 405
2 votes
1 answer
228 views
Bounding random process
Def $\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that $$|X_t-X_s|\leq Cd(t,s),\text{ for all }t,s\in T.$$ Lemma Suppose $\{X_t\}_{t\in T}$ is ...
- 405
2 votes
0 answers
133 views
Fourier expansion of random functions
Consider a random mapping $f:\{0,1\}^n \to \{0,1\}^n$, .i.e, a function such that for each $x \in \{0,1\}^n$, $f(x) \in \{0,1\}^n$ is chosen uniformly at random. My question is what would the fourier ...
- 121
0 votes
1 answer
105 views
Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows
Let $T:=[-1,1]^{n-1}\times (0,1]$. Let $$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$ where (i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables (ii) ...
- 149
0 votes
1 answer
157 views
Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
- 149
1 vote
1 answer
241 views
concentration of random field to its expectation function
Question Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}_N$. Is there result regarding the concentration of the random field? For example $\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\...
- 405
1 vote
0 answers
117 views
Expected Number of roots in $\mathbb D (0;r)$
In the literature about roots of random polynomials there are many results about the expected number of real roots of a complex polynomial building on Kac formula especially in the the asymptotic ...
- 828
0 votes
1 answer
69 views
Diameter of the range of composition of random maps on the circle
My questions are related to the paper https://hal.science/hal-03933493v1 (accepted with corrections in Ergodic Theory and Dynamical Systems). I fix an irrational number $\theta \in [0,1[$. I define ...
- 3,968
1 vote
1 answer
113 views
Does the following expectation-based inequality hold?
Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...
- 189
0 votes
1 answer
100 views
Distribution of zeros and angles of a function with additive coloured noise
Let us consider some real-variable function $$ f(t) = f_0(t) + \xi(t), $$ where $f_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f_0 = \...
- 335
1 vote
0 answers
114 views
Constructing k-wise independent variables over a general set
We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...
- 11
0 votes
0 answers
174 views
Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?
Consider two non-orthogonal random variables $$ (1) \quad E(XU)\neq 0, $$ where $X$ can be a vector. Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...
- 108
2 votes
0 answers
158 views
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
177 views
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 ...
- 617
1 vote
0 answers
91 views
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 votes
0 answers
332 views
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 ...
- 1,567
1 vote
1 answer
426 views
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 ...
- 87
1 vote
1 answer
310 views
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
1 vote
0 answers
97 views
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: $$ [0,1]^d\ni x \...
- 4,942
1 vote
3 answers
371 views
Practical pseudorandom generators
It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem. I am curious if someone developed kind of &...
- 801
1 vote
0 answers
55 views
Get covariance from log-density function
Problem Given a following log-density function $$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$ where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters. Find $\Bbb Cov( g(Y),...
- 111
1 vote
1 answer
717 views
Approximate expectation of a random variable that is the logarithm of a function of a binomial
I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series: \begin{...
- 73
3 votes
1 answer
277 views
Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow $$ \dot{x}(t) = -\nabla g(x(t)), \;...
- 7,033
1 vote
1 answer
188 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)) = \...
- 7,033
1 vote
1 answer
381 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 \...
- 7,033
3 votes
0 answers
127 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}...
- 7,033
0 votes
1 answer
69 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 ...
- 2,256
6 votes
1 answer
545 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 ...
- 87.7k
3 votes
0 answers
253 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 ...
- 9,934
3 votes
1 answer
475 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 ...
- 2,035
1 vote
0 answers
102 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\...
- 97
3 votes
1 answer
210 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 > \...
- 2,256
3 votes
1 answer
205 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 ...
- 2,256
2 votes
0 answers
94 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 $\Omega \subset \mathbb{R}^d$) ...
- 9,634
0 votes
0 answers
211 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 ...
- 1,349
0 votes
1 answer
242 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 ...
- 105
0 votes
1 answer
959 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 ...
- 105
2 votes
1 answer
348 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 vote
1 answer
158 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 ...
- 113
3 votes
0 answers
120 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 ...
- 131
3 votes
1 answer
423 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 ...
- 345
2 votes
0 answers
125 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 $\...
- 7,033
1 vote
1 answer
192 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 ...
