Related papers: On convex hull of Gaussian samples

Let $\{X_{n}(t), t\in[0,\infty)\}, n\in\mathbb{N}$ be a sequence of centered dependent stationary Gaussian processes. The limit distribution of $\sup_{t\in[0,T(n)]}|X_{n}(t)|$ is established as $r_{n}(t)$, the correlation function of…

We study the convex hull of the graph of a quadratic function $f(\mathbf{x})=\sum_{ij\in E}x_ix_j$, where the sum is over the edge set of a graph $G$ with vertex set $\{1,\dots,n\}$. Using an approach proposed by Gupte et al. (Discrete…

The distribution of the hypervolume $V$ and surface $\partial V$ of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than $P = 10^{-1000}$ to estimate…

The asymptotic behavior, as $T\to\infty$, of some functionals of the form $I_T(t)=F_T(\xi_T(t))+\int_0^tg_T(\xi_T(s))\,dW_T(s)$, $t\ge0$ is studied. Here $\xi_T(t)$ is the solution to the time-inhomogeneous It\^{o} stochastic differential…

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes $\{X(Y(t)) : t \in [0, \infty)\}$, where $\{X(t) : t \in \mathbb{R} \}$ is a centered Gaussian process and $\{Y(t): t…

This paper extends Edgeworth-Cornish-Fisher expansions for the distribution and quantiles of nonparametric estimates in two ways. Firstly it allows observations to have different distributions. Secondly it allows the observations to be…

Let $S_k$ be a random walk in $R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does not…

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian…

Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and $\gamma_n$ the (standard) Gaussian measure on ${\bf R}^n$ with density $(2 \pi )^{-n/2} e^{- \| x\|^2 /2}$. Let $\vartheta$ ($ \simeq 1.3489795$) be defined by $\gamma_1 ([ -…

We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,\beta} (x) =…

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional…

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$…

Let $T$ be the Student one- or two-sample $t$-, $F$-, or Welch statistic. Now release the underlying assumptions of normality, independence and identical distribution and consider a more general case where one only assumes that the vector…

We study the set of directions asymptotically explored by a spatially homogeneous random walk in $d$-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some…

Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root…

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at…

This paper studies several properties of channel codes that approach the fundamental limits of a given (discrete or Gaussian) memoryless channel with a non-vanishing probability of error. The output distribution induced by an…

Nonparametric data envelopment analysis (DEA) estimators have been widely applied in analysis of productive efficiency. Typically they are defined in terms of convex-hulls of the observed combinations of…

We show that the expected value of the mean width of a random polytope generated by $N$ random vectors ($n\leq N\leq e^{\sqrt n}$) uniformly distributed in an isotropic convex body in $\R^n$ is of the order $\sqrt{\log N} L_K$. This…