WHEN IS A TRUNCATED COVARIANCE FUNCTION ON THE LINE A COVARIANCE FUNCTION ON THE CIRCLE

Let gamma denote the covariance function of a real stationary process on R. Define a new function on [-K, K] by lambda(K)(t)= gamma(t), t epsilon [-K,K]. Note that by identifying the end points of the interval, we may interpret lambda(K) as a function on the circle with circumference 2K. We address the following question: if gamma is a covariance function on the line, will lambda(K) be a covariance function on the circle? We identify one class of covariance functions for which the answer is ''yes'' for all K > 0, and a second class for which it is ''yes'' for all K sufficiently large. However, our most substantial result is a negative one, and the answer will frequently be ''no'' for all K > 0. A statistical consequence of the positive results is mentioned briefly.