Etude stochastique de l'erreur dans un calcul numérique approché

On the estimation of the time mean-value of a stationary process

“…A second approach to the problem is found in some fairly recently published papers on numerical methods (see Blanc [1], BlancLiniger [2] and Uhlmann [11]). These authors consider the classical situation when one has to compute a definite integral by numerical integration, but depart from the classical case in assuming that the function to be integrated is a realization of a stochastic process of the kind considered here.…”

“…F (Je) has all its mass in the points 2 nj, j = ... , -1,0, 1, .... To get a~, we have to put tv=v/n III (6) and minimize w.r. to w o '" W n . As x (1) …”

“…EX(t)2< 00, and Ex(t+s) x(t) =e (s) is independent oft. We will also assume that the covariance function e (s) is continuous, so that it has a spectral representation (1) where the spectral distribution function F (A) is non-decreasing, continuous to the right and bounded, F ( -00) = 0, F (00) = e(0) =…”