# Ahmed Arafat's research while affiliated with Mansoura University and other places

## Publications (4)

Article
Full-text available
We consider the class Ψd of continuous functions that define isotropic covariance functions in the d-dimensional sphere Sd. We provide a new recurrence formula for the solution of Problem 1 in Gneiting (2013b), solved by Fiedler (2013). In addition, we have improved the current bounds for the curvature at the origin of locally supported covariances...
Preprint
We consider the class $\Psi_d$ of continuous functions $\psi \colon [0,\pi] \to \mathbb{R}$, with $\psi(0)=1$ such that the associated isotropic kernel $C(\xi,\eta)= \psi(\theta(\xi,\eta))$ ---with $\xi,\eta \in \mathbb{S}^d$ and $\theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $\mathbb{S}^d$. We...
Article
Full-text available
The equivalence of Gaussian measures is a fundamental tool to establish the asymptotic properties of both prediction and estimation of Gaussian fields under fixed domain asymptotics. The paper solves Problem 18 in the list of open problems proposed by Gneiting (2013). Specifically, necessary and sufficient conditions are given for the equivalence o...
Article
We propose and define a family of marked point processes in noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump-diffusion processes. Then we build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group.

## Citations

... The approach was then generalized to nonparametric spectral estimation (Castruccio and Genton, 2014), three-dimensional variables (Castruccio and Genton, 2016), different land/ocean behavior (Castruccio and Guinness, 2017) and also multivariate processes (Edwards et al., 2019). On the more theoretical side, substantial progress has been made in the determination of properties of high dimensional spheres for isotropic processes via basis decomposition see, e.g., Arafat et al. (2020);Porcu et al. (2020). We refer to Jeong et al. (2017); Porcu et al. (2018) for two recent reviews on the topic. ...
... The zonal kernel induced by any continuous function 1 1 possesses a Fourier-type expansion in spherical harmonics where 1 is a real orthonormal basis for the space of spherical harmonics of degree and the collection 1 0 forms a real orthonormal basis for 2 1 . Using Schoenberg's [30] pioneering work, it can be shown that if the expansion coefficients are positive for 0, then is a strictly positive definite kernel on 1 . ...