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

expected that we know them accurately enough and do not need anystatistical description. But what should be done with irregularities on asmall scale which can influence the estimation of the reserves as well?Take finally radio physics in which the concept of stationary processis recognized best of all. All kinds of interferences and noises are hereusually considered as stationary stochastic processes. However, thereis a special noise, the flicker noise or shimmering explained by chaoticvariations of the emissive capability of the cathode electronic tubes. Itis sufficiently clearly indicated, see for example Rytov (1966), that theshimmering can hardly be described by the model of stationarystochastic process.It follows that at present we begin to realize that a mathematicaldescription of the largest waves of wavy processes by methods ofmathematical statistics is in most cases impossible. We have to reckonon describing phenomena on a smaller scale but we certainly have toforfeit much. Thus, the theory of the microstructure of turbulence isuseless for predicting the weather because it does not describe themost essential phenomena occurring on a large scale. However, it isuseful in other fields, for example when calculating the passage oflight through the atmosphere which is important for astronomy (fortaking into account the corruption of images in telescopes).Kolmogorov introduced a universal concept of process withstationary increments which can hopefully replace the concept ofstationary stochastic process in all the cases considered above. Fordiscrete time it means that we turn from an observed process... ξ −1 , ξ 0 , ξ 1 , ..., ξ n , ...to differences... η −1 = ξ −1 − ξ −2 , η 0 = ξ 0 − ξ −1 , η 1 = ξ 1 − ξ 0 , ...and consider them a realization of a stationary stochastic process.For processes with continuous time we turn instead from ξ(t) to thederivativeη (t) = ξ′(t)and call it stationary stochastic process; the differentiation shouldsometimes be understood in a generalized sense.Let us explain in more detail what do we expect when turning todifferences or derivatives. Imagine that the observed process is a sumξ(t) = a(t) + ς(t)of some random or not component a(t) similar to large waves and theother component changing much more rapidly and can reasonably becalled a stationary stochastic process. We recognize our inability todescribe the changes of a(t) and wish to study the changes on a smallscale mostly determined by the other component. This is indeed76