1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

achieved by differentiating because the large component a(t) likelychanges slowly, so that its derivation is small. We haveη(t) = ξ′(t) = a′(t) + ς′(t) ≈ ς′(t)which means that ξ′(t) practically does not include any componentconnected with a(t). The same is achieved by taking the differences incase of discrete time.For continuous time, rather than differentiating, we certainly canalso study differences∆ τ ξ(t) = ξ(t + τ) − ξ(t) ≈t+τ∫tη( s)dswhere η(s) is a stationary process. The second equality is needed forconstructing a correlation and spectral theory of processes withstationary increments being integrals of a stationary process.Instead of a correlation function a structural function introduced byKolmogorov is being used:D2(τ) = E[ ∆τξ( t)] ,that is, the variance of the increment of the process during time τ.Practical application of processes with stationary increments can bestudied by means of Monin & Jaglom (1967, pt. 2/1975).The situation that emerged nowadays in science can be thereforedescribed in the following way. We do not expect that generalstatistical methods can characterize wavy processes as a whole, i. e.,including large waves. In general, the notion of stationary stochasticprocess is compromised. For applications, it is the turn of the conceptof stochastic process with stationary increments that does not claim tocover a phenomenon as a whole but can cover it in the sphere of thesmall scale. Its possibilities are not yet sufficiently investigated. Thesituation concerning the examples with which we have dealt is this.In economics, there exist works of the American school founded byBox, for example Box, Jenkins & Bacon (1967); Box & Jenkins(1970). However, the quality of statistical approach is there doubtful:no statistical checks are made, attention is concentrated on forecastingwhereas the exclusion of the large-scale component compels us tothink that it would have been better to abandon altogether predictionssince they depend in the first place on the excluded component. Ingeneral, the situation is doubtful. We will consider it later.For meteorology, processes with stationary increments are of nospecial significance. The Kolmogorov – Obukhov theory of turbulencerather belongs to aero-hydrodynamics. Brilliant success is achievedthere: conclusions made by the creators of the theory wereexperimentally confirmed. That success remains, however, the onlyone attained.In geology, we have the book of Matheron (1962). The factualmaterial included there supports in some measure the hypothesis that77