Estimates of the supremum of a class of stationary random processes
Let \(\xi(X)\) be a stationary process with \(E\xi(X)=0\), and \(E\xi(t+\tau)\xi(t)=\int_ 0^ \infty \cos \lambda\tau dF(\lambda)\). \(F(\lambda)\) is a spectral function. Let \(F(\lambda)\) be continuous, \(F(0)=0\), \(F(\infty)=1\), \(1-F(\lambda)>0\), \(\lambda<\infty\). The process \(\xi(t)\) has a representation in the form of stochastic integral \[ \xi(t)=\int_ 0^ \infty \cos \lambda\tau d\eta_ 1(\lambda)+\int_ 0^ \infty \sin \lambda\tau d\eta_ 2(\lambda), \] where \(\eta_ i(\lambda)\) are orthogonal processes with orthogonal increments, \(E\eta_ i(\lambda)=0\), \(E| d\eta_ i(\lambda)|^ 2=dF(\lambda)\), \(i=1,2\). It is assumed that \(\eta_ i(\lambda)\in\text{Sub}_ \varphi(\Omega)\). Estimations for the distribution of the supremum of \(\xi(t)\) are obtained. Sufficient conditions are given in terms of the spectral function \(F(\lambda)\).
orthogonal processes with orthogonal increments
scientific article; zbMATH DE number 58674