On the rate of convergence for extremes of mean square differentiable stationary normal processes

Let ${xi(t); tgeq 0}$ be a normalized continuous mean square differentiable stationary normal process with covariance function $r(t)$. Further, let $$ rho(t)=frac{(1-r(t))^2}{1-r(t)^2+r'(t)|r'(t)|} $$ and set $$ delta=frac{1}{2}wedge inf_{tgeq 0} rho(t). $$ We give bounds which are roughly of the order $T^{-delta}$ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by $xi(t)$ in the interval $[0,T]$. The results assume that $r(t)$ and $r'(t)$ decay polynomially at infinity and that $r''(t)$ is suitably bounded. For the number of upcrossings it is in addition assumed that $r(t)$ is non-negative. Some applications are developed.

KRATZ, M. and ROOTZÉN, H. (1997). On the rate of convergence for extremes of mean square differentiable stationary normal processes. Journal of Applied Probability, 34(4), pp. 908-923.

Keywords : #AMS-classification, #60G70, #60G15, #60F05rate-of-convergence, #extremes, #normal-processes, #Poisson-convergence