An Asymptotic Property of Gaussian Processes. I
Hisao Watanabe · Transactions of the American Mathematical Society · 1970
The study of the asymptotic properties of stochastic processes has a long history.Such researches were developed from those of the asymptotic behavior of the first n of a sequence of independent random variables, namely, the so-called law of the iterated logarithm.A. Kolmogoroff proposed the final form of the very theory which is stated, without proof, in P. Levy's book [4].W. Feller [3] gave its complete proof.The final result on the Brownian motion corresponding to Feller's for the case of the partial sums of independent random variables were led by T. Sirao and T. Nisida [9].Their result is stated below.We introduce the following notation.Let M¿ ={; is a positive, nondecreasing, real function on [a, oo)}.Let {/?(;); 0 (0 = £(£(ß(02))1,2 = f1/2.Theorem A.Pithere is a t0io>) such that |£(r)| S vit) it) for all t > t0iw)) = 1 or 0, according as, for some a > 0, F il/t) eMa\ Ja converges or diverges, respectively.It will be possible to generalize in several ways Theorem A which is true for the Brownian motion.In this paper, by use of the method of T. Sirao [8], we will give some results on a generalization of Theorem A.2. Results.Let{x(/), -oo < / < oo} be a real, separable, measurable Gaussian process defined on a probability measure space (Q, êS, P).Without loss of generality, we may assume that £(x(?)) = 0. We put rit, í) = £(x(í)x(í)) and Eix\t)) = v2it).In the following, we will assume that r(r, s) is continuous with respect to / and s and vit) is positive.And we put pit, ,s) = r(r, s)/ivit)vis)).In the following, we, will obtain some results on the asymptotic behavior of the process x(/) as / tends to infinity.To state the results, we introduce the following conditions.