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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\...
YJ Kim's user avatar
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1 answer
186 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 ...
some1fromhell's user avatar
3 votes
1 answer
240 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 ...
miao zhengwu's user avatar
1 vote
1 answer
206 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 ...
SBK's user avatar
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1 answer
76 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 ...
tony's user avatar
  • 405
0 votes
2 answers
105 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 $...
tony's user avatar
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2 votes
1 answer
153 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 ...
tony's user avatar
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2 votes
0 answers
94 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 ...
David ross's user avatar
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1 answer
67 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) ...
happyle's user avatar
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1 answer
105 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),...
happyle's user avatar
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1 answer
179 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)\...
tony's user avatar
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1 vote
0 answers
92 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 ...
AgnostMystic's user avatar
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46 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 ...
Christophe Leuridan's user avatar
1 vote
1 answer
79 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 ...
Math_Y's user avatar
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1 answer
72 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 = \...
MightyPower's user avatar
1 vote
0 answers
81 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 ...
tamir's user avatar
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0 answers
116 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 ...
Star's user avatar
  • 76
2 votes
0 answers
137 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 $...
RandomMatrices's user avatar
1 vote
1 answer
106 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 ...
S.Z.'s user avatar
  • 555
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0 answers
78 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 ...
Vikas Srivastava's user avatar
5 votes
0 answers
290 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 ...
bobuhito's user avatar
  • 1,547
1 vote
1 answer
319 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 ...
Quicky2357's user avatar
1 vote
1 answer
274 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,...
user avatar
1 vote
0 answers
72 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 \...
ABIM's user avatar
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1 vote
3 answers
246 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 &...
Rubi Shnol's user avatar
1 vote
0 answers
39 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),...
student's user avatar
  • 111
1 vote
1 answer
500 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{...
qwert's user avatar
  • 89
3 votes
1 answer
215 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)), \;...
dohmatob's user avatar
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1 vote
1 answer
135 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)) = \...
dohmatob's user avatar
  • 6,824
1 vote
1 answer
321 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 \...
dohmatob's user avatar
  • 6,824
3 votes
0 answers
96 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}...
dohmatob's user avatar
  • 6,824
0 votes
1 answer
48 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 ...
gradstudent's user avatar
  • 2,186
6 votes
1 answer
441 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 ...
Timothy Chow's user avatar
  • 80.3k
3 votes
0 answers
193 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 ...
Hans-Peter Stricker's user avatar
3 votes
1 answer
341 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 ...
Daron's user avatar
  • 1,915
1 vote
0 answers
78 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\...
volond's user avatar
  • 97
3 votes
1 answer
155 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 > \...
gradstudent's user avatar
  • 2,186
3 votes
1 answer
137 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 ...
gradstudent's user avatar
  • 2,186
2 votes
0 answers
65 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 $\Omega \subset \mathbb{R}^d$) ...
truebaran's user avatar
  • 9,230
0 votes
0 answers
180 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 ...
pre-kidney's user avatar
  • 1,299
0 votes
1 answer
199 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 ...
William's user avatar
  • 105
0 votes
1 answer
750 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 ...
William's user avatar
  • 105
2 votes
1 answer
289 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) ...
Davood Norouzi's user avatar
1 vote
1 answer
115 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 ...
dave2d's user avatar
  • 103
3 votes
0 answers
94 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 ...
mam's user avatar
  • 131
3 votes
1 answer
362 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 ...
Matheus Manzatto's user avatar
2 votes
0 answers
109 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 $\...
dohmatob's user avatar
  • 6,824
1 vote
1 answer
155 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 ...
Peter Mølgaard Pallesen's user avatar
2 votes
2 answers
238 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 $...
Joker123's user avatar
  • 153
1 vote
0 answers
43 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/...
user39756's user avatar
  • 141