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

of the mathematical model of the Brownian motion allowing todetermine precisely the coefficient of diffusion given observations of ahowever small interval of a realization of that motion. He who believesthat this is indeed true for a physical Brownian motion will be wrong.Here is another such example. Any broadcasting station istransmitting over a waveband of restricted width. If a radio signal isconsidered as a stochastic process, its spectrum will be contained inthat finite interval. And there exists a mathematical theorem statingthat with probability 1 a realization ξ i of such a process is an analyticalfunction of t. Consequently, after listening for any however shortinterval of time, we may unambiguously establish what was and whatwill be broadcast, an obviously absurd conclusion.It is certainly easy to indicate the mistake here. First, a broadcast isnot a stochastic process since it is not an element of some statisticalensemble; second, an analytical function can be reconstructed given itsvalues on any interval only if they are given absolutely preciselywhich is impossible for a function of a continuous variable. Even asingle number can not be written down precisely, much less a totalityof an infinitely many numbers. Third, a radio signal is not an analyticalfunction of time because in the 19 th century there were no broadcastingstations whereas an analytical function vanishing on some intervalvanishes everywhere.A digression about the concept of function in mathematics is inorder here 7 . At the emergence of mathematical analysis it was usuallyunderstood as a formula determining a dependence y = y(x). And allfunctions except at a few points were continuous and differentiable.The problem concerning the proof of differentiability did not evenexist. Later, however, in the 19 th century an idea was established that afunction is simply a relation between the sets of values of the argumentx and the function y = y(x). It is usually demanded that exactly onevalue of y corresponded to each value of x, but that the inverse was notnecessarily true. And there was no cause for an arbitrarycorrespondence y = y(x) to be continuous or differentiable.It is rather difficult but therefore interesting to provide an exampleof a continuous but nowhere differentiable function. The first suchexample was due to Weierstrass, later other and more simple exampleswere discovered. Such objects proved very interesting formathematicians and to them their attention had been to a large extentswung. For us, it is especially interesting that mathematicalconsiderations concerning the theory of stochastic processes lead tothe realization of many such processes which should be recognized ascontinuous but not differentiable functions (or functions only twice,say, differentiable with a continuous but not anymore differentiablesecond derivative).This was indeed joyful because it apparently proved that nondifferentiablefunctions indeed existed in nature. However, we wish tocast a shadow on that joy: it is absolutely absurd to believe that such afunction can be experimentally observed. Such a realization of astochastic process can not be given either by a formula, or a table, agraph, or an algorithm of calculation. When considering it indeed real,exactly known at all of its points, we will be able to come to absurd67