# Questions tagged [random-functions]

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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 $... 1 vote 1 answer 74 views ### On the growth of sample paths of Gaussian random fields Consider a centered Gaussian random field on$\mathbb{R}^n$with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ... 1 vote 0 answers 57 views ### Distribution of multivariate polynomial evaluation Let$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)$for ... 5 votes 0 answers 261 views ### Fastest sine of a large power of 2 What is the fastest known way to calculate$\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$(and actually ... 1 vote 1 answer 66 views ### Partial derivative of expectation and Stein's lemma Currently, I am reading a paper about the Gaussian Process in Neural Network . In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ... 1 vote 1 answer 193 views ### Construct a random vector as a function of another random vector ASSUMPTION 1: there exists a continuous random vector$(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,... 20 views

### 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 ...
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### 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 $P_k$, ...
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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)\...