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

absolutely precisely if the pertinent path during any however shortinterval of time is known.You can encounter a viewpoint stating that a practical estimate ofthe coefficient of diffusion does not therefore present any difficulties.This opinion has been established to some extent in the literature onthe statistics of stationary processes, but it is completely wrong. Twocircumstances prevent its application to real Brownian motion. First,the mathematical Brownian motion, i. e., the Wiener process, does notdescribe the real process over short intervals of time whereas exactlythe change of the position of the particle during infinitely shortintervals enters the estimation of the coefficient of diffusion. Second,the idea of knowing exactly the path of some stochastic process duringsome interval of time is absolutely unrealistic; we do not at all knowhow to define precisely a non-regularly changing function which is notdescribable by an analytic expression. I am unable to dwell here inmore detail on the theory of stochastic processes and am returning toprobability P. For intervals, it coincides with their length.However, it is possible to construct very complicated sets ofintervals and mathematical correctness demands that it be possible todefine additionally that probability for all such sets while retaining themain property of countable additivity (2.5). The French mathematicianLebesgue provided a construction (the Lebesgue measure) allowing toascertain the possibility of such an additional definition. It iscomplicated and we will not discuss it here. However, it can be appliedfor spaces Ω of a very general kind, consisting for example offunctions which is important for the theory of stochastic processes.Until now, we have discussed the complications necessarilydemanded by the Kolmogorov axiomatics; on the other hand, it ishowever connected with most important simplifications. Theintroduction of a measure having the property of countable additivityallows to apply the concept of Lebesgue integral; as a concept, it isincomparably simpler and more general than the Riemann integral. Inthe general case, all the main notions of the theory of random variablesoccur not more complicated than those described above for the discretecase. Thus, a remarkable simplicity, generality and order is originatedin the main notions of the theory of probability. However, theLebesgue integral is not more than a concept. No one calculatesintegrals by applying the Lebesgue extension of measure, the Riemannintegral is preferred.It is necessary to mention here a certain difficulty that takes placewhen teaching mathematical analysis, both at home and abroad. Ingeneral, nothing negative can be said about its part dealing withfunctions of one variable, although it is somewhat tedious; the horrorbegins with the transition to functions of several variables. Thetreatment of the differential, and especially integral calculus is herenowadays absolutely unsatisfactory. Take for example the set of theGreen, Stokes and Ostrogradsky formulas introduced without anyconnection between them. Indeed, there exists now a united viewpointabout all of them and it even includes the Newton – Leibniz formula. Itis not treated in textbooks, but can be read in Arnold’s lectures (1968)on theoretical mechanics.23