Projections of determinantal point processes

Let x = {x((1)), ..., x((n))} be a space filling-design of n points defined in [0, 1](d). In computer experiments, an important property seek for x is a nice coverage of [0, 1](d). This property could be desirable as well as for any projection of x onto [0, 1](iota) for iota < d . Thus we expect that x(I) = {x(I)((1)), ..., x(I)((n))}, which represents the design x with coordinates associated to any index set I subset of {1, ..., d}, remains regular in [0, 1](iota) where iota is the cardinality of I. This paper examines the conservation of nice coverage by projection using spatial point processes, and more specifically using the class of determinantal point processes. We provide necessary conditions on the kernel defining these processes, ensuring that the projected point process X-I is repulsive, in the sense that its pair correlation function is uniformly bounded by 1, for all I subset of {1, ..., d}. We present a few examples, compare them using a new normalized version of Ripley's function. Finally, we illustrate the interest of this research for Monte-Carlo integration. (C) 2020 Elsevier B.V. All rights reserved.