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

4.3. General remarks on §§ 4.1 and 4.2. A sequence irregularaccording to §§ 4.1 or 4.2 presents a simplest example of an intuitiveand rigorous mathematical model of trials which can be calledindependent and identical (identical since the distributions of theprobabilities for all the formed sequences coincide). The idea of a poorpredictability of one initial sequence is here indeed reduced rathernaturally to demanding statistical independence of the ensemble ofsequences. As a result, independence of trials is treated in such amanner that provides a sufficiently clear rule for its quantitativeempirical verification.Thus, after being clearly formulated, independence of trialsobviously becomes a concept derived from the notion of statisticalstability, cf. our assumption in § 2.10. It follows that the postulate of §3.6.3 even in its most simple clear form is evidently more complexthan the postulates of §§ 3.6.1 and 3.6.2. It can not be the assumptionfrom which, at least according to the pattern of one series, statisticalstability is deduced.The verification of any propositions of mathematical statistics willbe therefore aimed at verifying the postulate of § 3.6.3 rather than atmeasuring the sought parameters of the distributions of the initialmagnitudes. This measurement, for which, as it seems, mathematicalstatistics is indeed created, will only constitute a small and so to saypreliminary part of the work to be done.The formulations of the idea of independence of trials consideredabove are obviously only applicable when the n trials are actuallycarried out many times. The alternative to the method of mathematicalstatistics therefore means that the postulate of § 3.6.3 should beintroduced only after the sought parameters or the initial distributionitself were reliably measured.4.4. Specification of the traditional formulations of the limittheorems on the basis of the concept of an irregular collective. Theauthor interprets the Bernoulli theorem by applying the notion of irregularity ofcollectives. One of the conditions of his pertinent theorem is the existence of a limitof the sequence of trials, the probability according to Mises.He notes that his (and therefore the Bernoulli) theorem does not claim to justifythe statistical stability of the frequency which is now one of his preconditions. Heconcludes that the limit theorems (in general!) are not actually fundamentalpropositions as it was thought in the initial period of the development of the theoryof probability.4.5. An example from classical statistical physics. [Concerning thework of an oscillator being in thermal equilibrium with a thermostat.]5. ConclusionAn alternative to the method of mathematical statistics can bedescribed in a few words in the following way. In applied research,and more precisely beyond fundamental physics, we should as far aspossible abstain from introducing stochastic magnitudes not measuredin real experiments in our initial assumptions. The so-called numericalexperiments compare a computer and a paper model but not modeland reality.The objects of study in economics, sociology and even moderntechnology are most often too complicated and unstable forconstructing their useful models by issuing from general principles131