kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

formulas for particular cases are in [27]. Later Slutsky examined the possibility of applyingthe 2 test and its distribution to connected series as well as of determining the requiredmagnitudes through the Fourier coefficients [25; 26].By issuing from his theory of connected series, and allowing for the course of randomprocesses, Slutsky was able to provide a methodology of forecasting them, includingsufficiently long-term forecasting, with given boundaries of error [29].We ought to dwell especially on his method of models (of statistical experimentation) fordiscovering connections between phenomena. His idea was as follows. When studying manyproblems not yet completely solved by theory, it is possible to arrange a statistical“experiment” and thus to decide whether the statistical correspondence between phenomenais random or not. For example, when selecting a number of best and worst harvests in Russiafrom among the series collected by Mikhailovsky for 115 years, we can compare them withthe series of maximums and minimums of the number of sunspots for more than 300 years. Ifsuch comparisons are{if the correspondence is} only possible after shifting one of the serieswith respect to the other one, then, obviously, the coincidences will be random. However,since the sum of the squares of the discrepancies 14 is minimal when those series arecompared without such shifting, we may be sufficiently convinced in that the coincidencesare not random [28].I am unable to appraise Slutsky’s purely mathematical studies and am therefore quotingmost eminent Soviet mathematicians. Smirnov (1948, pp. 418 – 419), after mentioningSlutsky’s investigation [13], wrote:The next stage in the same direction was his works on the theory of continuous stochasticprocesses or random functions. One of Slutsky’s very important and effective findings herewas the proof that any random stochastically continuous function on a segment isstochastically equivalent to a measurable function of an order not higher than the secondBaire class. He also derived simple sufficient conditions for a stochastic equivalence of arandom function and a continuous function on a segment, conditions for the differentiabilityof the latter, etc. These works undoubtedly occupy an honorable place among theinvestigations connected with the development of one of the most topical issues of thecontemporary theory of probability, that{issue or theory?}owes its origin to Slutsky’sscientific initiative.The next cycle of Slutsky’s works (1926 – 1927) was devoted to the examination of randomstationary series, and they served as a point of departure for numerous and fruitfulinvestigations in this important field. Issuing from a simplest model of a series obtained by amultiple moving summation of an unconnected series, he got a class of stationary serieshaving pseudo-periodic properties imitating, over intervals of any large length, seriesobtained by superposing periodic functions. His finding was a sensation of sorts; itdemanded a critical revision of the various attempts of statistical justification of periodicregularities in geophysics, meteorology, etc. It occurred that the hypothesis of superpositionof a finite number of regularly periodic oscillations was statistically undistinguishable fromthat of a random function with a very large zone of connectedness.His remarkable work on stationary processes with a discrete spectrum was a still deeperpenetration into the structure of random functions. In this case, the correlation function willbe almost periodical. Slutsky’s main result consisted here in that a random function was alsoalmost periodic, belonged to a certain type and was almost everywhere determined by itsFourier series.These surprisingly new and fearlessly intended investigations, far from exhausting a verydifficult and profound problem, nevertheless represent a prominent finding of our science.With respect to methodology and style, they closely adjoin the probability-theoretic conceptsof the Moscow school (Kolmogorov, Khinchin), that, historically speaking, originated on a