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

the structural function of the contents of the useful component is of thetypeD(r) = αlnr + βwhere r is the distance between sample points and α and β, parametersdetermined by observation. However, the book has a number ofinconsistencies. Thus, the logarithmic dependence is continued into theinterval of small values of r which is impossible because D(r) is a nonnegativemagnitude. Then, in some cases the subject concerns thecontent, in other instances, its logarithm. In addition, no statisticalchecks are made. But still, the factual material impresses so stronglythat careful reliable studies in the same direction become desirable.In radio physics, the scientific level is high and similarinconsistencies just can not occur. However, as far as we know, noreports about successful applying the model of process with stationaryincrements are in existence. Rytov (1966) only formulated ahypothesis that the phenomenon of flicker should be thus described.In concluding, I deal in more detail with the statistical theory ofturbulence and the problem of forecasting.3.6. Statistical theory of turbulence. This theory provides abrilliant success of a purely statistical description of a phenomenon, ofa highly developed and very complicated turbulence with a largenumber of vortical movements on differing scales. Kolmogorov andObukhov founded the basis of the theory before 1941. Experimentalconfirmation of their theoretical conclusions demanded perfectmeasuring instruments and up to 25 years. Application of that theoryto problems in propagation of electromagnetic and acousticoscillations in the atmosphere is also being developed.A precise knowledge of the field of velocities in a turbulent currentis understandably both impossible and useless. Indeed, had we somemethod of calculating all the velocities at all points, their registrationwith sufficient precision would have alone demanded an unimaginableamount of paper or magnetic tape and work with so much informationis absolutely impossible. The situation should be resolved by someversion of a statistical description.It occurred that the main suitable notions can be borrowed from thecorrelation theory; however, in their initial form they were insufficient.There is a scientific law stating that ex nihilo nihil fit which means thatan application of established theories does not cover anything new.Without going into mathematical detail, I will attempt to showexactly how does this law work in case of turbulence and what newconsiderations it was necessary to draw for getting the things moving.Imagine a turbulent current. Its mean velocity depends on concreteconditions (what and where is the current set into motion [...]) and it issenseless to describe it by statistical methods. However, thedifferences of velocity in various points of the current and in differingmoments of time less depend on initial conditions and to a largerextent are determined by the properties of the liquid or gas itself. So,let us study the differences78