2013/07/18 by Z. Tan, Zhongquan Tan, Enkelejd Hashorva +1
Economics, Econometrics and Finance · Environmental Science · Mathematics · #Combinatorics #Delta method #Discretization #Financial Risk and Volatility Modeling #Function (biology) #Gaussian #Gaussian process #Hydrology and Drought Analysis #Limit (mathematics) #Mathematical analysis #Mathematics #Physics #Quantum mechanics #Statistics #Stochastic processes and financial applications #Zero (linguistics) #math.PR
paper · pdf · doi:10.1016/j.jmaa.2013.07.022
published as 2014, Journal of Mathematical Analysis and Applications, 409,1, 299-314
openalex publication_date 2013/07/18 · arxiv created 2014/05/10 · arxiv updated 2014/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let \X(t), t≥0\ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004), which we refer to as Piterbarg's max-discretisation theorem gives the joint asymptotic behaviour (T→ ∞) of the continuous time maximum M(T)=maxt∈ [0,T] X(t), and the maximum Mδ(T)=max_t∈ \mathfrakR(δ)X(t), with \mathfrakR(δ) ⊂ [0,T] a uniform grid of points of distance δ=δ(T). Under some asymptotic restrictions on the correlation function Piterbarg's max-discretisation theorem shows that for the limit result it is important to know the speed δ(T) approaches 0 as T→ ∞. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.