# Questions tagged [random-functions]

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### 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 vote
49 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 ...
1 vote
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1 vote
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### 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 ...
44 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 ...
1 vote
73 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 ...
69 views

1 vote
84 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 ...
1 vote
71 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 ...
277 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 vote
199 views

### Partial derivative of expectation and Stein's lemma

Currently, I am reading a paper about the Gaussian Process in Neural Network . In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...
1 vote
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1 vote
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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 ...
418 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 ...
164 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 ...
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 ...