Multivariate Normal Probabilities using Vecchia Approximation

Under a different representation of the multivariate normal (MVN) probability, we can use the Vecchia approximation to sample the integrand at a linear complexity with respect to n. Additionally, both the SOV algorithm from Genz (92) and the exponential-tilting method from Botev (2017) can be adapted to linear complexity. The reference for the method implemented in this package is Jian Cao and Matthias Katzfuss (2024) "Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities" . Two major references for the development of our method are Alan Genz (1992) "Numerical Computation of Multivariate Normal Probabilities" and Z. I. Botev (2017) "The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting" .