On the renewal measure for Gaussian sequences

Abstract

A form for U(t), the expected number of times a Gaussian sequence falls below a level of t, is given in terms of the mean M(x) and the variance V2(x) functions. It is shown that under general conditions U(t) ~ M(-1)(t), t --> [infinity]. Moreover, if M and V are regularly varying at infinity functions, then U(t) - M(-1)(t) is also regularly varying at infinity. A renewal theorem for stationary Gaussian sequences is given, where it is shown that the asymptotic behavior of U(t) - t/[mu] is determined by the asymptotic behavior of V2(t)/t.