7. Probability and Statistics Soviet Essays - Sheynin, Oscar

lim P(|(t+ h) – (t)| > ) = 0 as h ¤ ¥Doob) have developed this logical pattern. Independently of polishing its formal logical side, Slutskysystematically studied the most interesting problems naturally emerging when considering randomfunctions.An n-dimensional law of distribution Ft 1 , t 2 ,..., t n(x 1 ; x 2 ; …; x n ) of the random variables (t 1 ), (t 2 ),…, (t n ) corresponds to each finite group of values t 1 , t 2 , …, t n of the argument t. One of the mainproblems of the theory of random functions is, to determine the conditions to be imposed on thefunctions F so that they can correspond to a random function of some type E. In any case, thefunctions should be consistent in the elementary sense; additional conditions differ, however, fordiffering functional spaces. Slutsky and Kolmogorov (especially Slutsky [28]) offered sufficientlygeneral conditions of this type for the most important instances. These conditions are expressedthrough the laws of distribution of the differences (t) – (t), that is, through the two-dimensionaldistributions F t, t (x; x). According to Slutsky, stochastic continuity of (t), i.e., the condition that, forany > 0,for all (or almost for all) values of t, is sufficient for this space of measurable functions. For the samecase, Kolmogorov established a somewhat more involved necessary and sufficient condition(Ambrose 1940). For the space of continuous functions Kolmogorov’s sufficient condition is thatthere exist such m > 0 and > 1 thatE|(t+ h) – (t)| m = O|h| .The most essential from among the more special results concerning random functions of a realvariable were obtained in connection with the concepts of the theory of random processes (where theargument is treated as time), see the next section. For statistical mechanics of continuous mediumsboth scalar and vectorial random functions of several variables (points in space) are, however,essential. Findings in this direction are as yet scarce (Obukhov, Kolmogorov).4. The Theory of Random ProcessesThe direction of research now united under the heading General theory of random processesoriginated from two sources. One of these is Markov’s work on trials connected into a chain; theother one is Bachelier’s investigations of continuous probabilities which he began in accordance withPoincaré’s thoughts. The latter source only acquired a solid logical base after the set-theoretic systemof constructing the foundations of probability theory (§3) had been created.In its further development the theory of random processes is closely interwoven with the theory ofdynamic systems. Both conform to the ideas of the classical, pre-quantum physics. The fascination ofboth of them consists in that, issuing from the general notions of determinate process and randomprocess, they are able to arrive at sufficiently rich findings by delimiting, absolutely naturally freefrom the logical viewpoint, the various types of phase spaces (the sets of possible states) and ofregularities in the changes of the states (the absence, or the presence of aftereffect, stationarity, etc).A similar logical-mathematical treatment of the concepts of the modern quantum physics remains to aconsiderable extent a problem for the future.4.1. Markov Chains. Suppose that the system under study can be in one of the finite or countablestates E 1 , E 2 , …, E n , … and that its development is thought to occur in steps numbered by integers t(discrete time). Suppose in addition that the conditional probability of the transition P(E i ¤ E j ) = p ij(t)during step t does not depend on the earlier history of the system (absence of aftereffect). Suchrandom processes are called Markov chains.It is easy to see that the probability of the transition E i ¤ E j during t steps having numbers 1, 2, …,t which we shall denote by P ij (t) can be calculated by means of recurrent formulas