7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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In the homogeneous case H s t = H t–s and the operators H form a semi-group: H + = H H . This makes it natural to assume the existence of an infinitesimal operator U by whosemeans the operators H are expressed in accordance with the formula H = e u (Kolmogorov[46]). It is natural to suppose that, also among processes non-homogeneous in time, animportant (in applications, the main) part should be played by processes for which theinfinitesimal operatorU(t) = lim+∆H tt− E, 0,∆exists and the operators H s t are expressed through U(t) by means of the multiplicativeintegration according to Volterra 3H s t = ts[1 + U( )]d .The concrete realization of this program (Iosida, Feller, Dynkin) revealed the expediencyof considering both the operators adjoining to H s t , f s = T s t f t , which transform the martingales,i.e., the functions of the state f t (x) obeying the relationE{f t [(t)]|(s) = x} = f s (x),and the adjoining infinitesimal operatorsA t = limtTt −∆− E, 0.∆For non-homogeneous cases it is inevitable to postulate the existence of infinitesimaloperators. Otherwise, we may only reckon on proving under wide conditions the existence offamilies of operators V(s; t), B(s; t) additively depending on the interval (s; t) of the time axiswhich will allow to express H s t and T s t as multiplicative Stieltjes integralsH s t = ts[1 + V( ; + d )], T s t = ts[1 + B( ; + d )]. (*)When, however, U(t) and A(t) exist, the operators V(s; t) and B(s; t) themselves areexpressed through these by usual integralstV(s; t) =stU( ) d , B(s; t) =sA( ) d .Dobrushin [4] fulfilled this program for a finite number of states. If the infinitesimaloperators are given, the reconstruction of the process by issuing from them is possible by theIto method of stochastic differential equations. In the Soviet literature the work of Martynov[1] is devoted to this issue.For the homogeneous case, a finite number of states and stochastic independence of theprocess, the existence of infinitesimal operators uniquely determining the process is simple toprove and was known long ago. Only Dynkin [37; 42] obtained a similar result for acontinuous manifold of states (a straight line, an n-dimensional Euclidean space or an
arbitrary metric space). He thus completed a number of investigations made by other authors(Iosida, Feller).3.1. Strictly Markov Processes. When studying Markov processes by direct stochasticmethods it is often assumed that independence of the future course of a process given ( )from its course at t < exists not only for a constant but also for a random determined, ina certain sense, by the previous course without looking ahead. Hunt, in 1956, indicated theneed to justify this assumption. Dynkin [43] and he & Youshkevich [7] defined theappropriate concept of a strict Markov process in a general way. They showed that not all theMarkov processes were strictly Markovian, and, at the same time, that the class of the latterprocesses was sufficiently wide. In future, exactly the theory of strictly Markov processeswill apparently be considered as the theory of real Markovian processes.3.2. The Nature of the Paths of the Process. For a finite or a countable set of states theprocesses, where, during finite intervals of time, there only occur (with probability 1) a finitenumber of transitions from one state to another one, are of an unquestionably real interest.Such processes are here called regular. For homogeneous processes with a countable numberof states, and for general (non-homogeneous in time) processes with a finite number of statesDobrushin [2; 5] discovered the necessary and sufficient conditions of regularity. Processescontinuous on the right constitute an important class of processes with a finite or countablenumber of states. In such cases, issuing from any given state, the process will pass discretelyover a sequence of states until the sequence of the moments of transition condenses near thelimit point. Thus, for example, behave explosive processes of propagation. When consideringsuch processes only on a random interval of time, up to the first condensation of the momentsof transition, it is natural to call them semi-regular. Youshkevich [3] studied the conditionsfor continuity on the right.Problems on reaching the boundaries play a similar part for continuous processes ofwandering of the diffuse type. In particular, non-attainment of the boundaries herecorresponds to regularity. Already Bernstein (for example, [114]), although applying anotherterminology, studied the attainment or non-attainment of the boundaries. In a modernformulation the problem is easily solved in a definitive way for one-dimensional diffuseprocesses (Khasminsky [1]).Dynkin [21] determined the conditions of continuity of the path and existence of onlyjumps of the first kind in the path for general Markov processes with a set of statesconstituting an arbitrary metric space. Later L.V. Seregin derived a sufficient condition forthe continuity of the paths which is also necessary for a wide class of processes.3.3. The Concrete Form of the Infinitesimal Operators. Infinitesimal operators for afinite or countable number of states in simple problems having real meaning are prescribedby densities of the transition probabilities a ij (t) from state i to state j (i 2 j). Kolmogorov[124] somewhat strengthened a finding achieved by Doob: he established that in ahomogeneous and stochastically continuous case these densities always exist and are finite.However, they only determine the process if it is regular (for semi-regularity, they determinethe course up to the first condensation of the jumps).Infinitesimal differential operators of the second order appeared already in the classicalworks of Fokker and Planck on continuous processes with sets of states being differentiablemanifolds. Kolmogorov, in his well-known work [27], ascertained their statistical meaningand indicated some arguments for considering the case when a process was determined bysuch a differential operator as being general in some sense. However, it was clear from thevery beginning that an exact sense could have only been attached to this assumption byadequately generalizing the concept of differential operators. Feller outlined the approaches
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[3] I bear in mind the well-known p
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successes of physical statistics. B
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classes of independent facts whose
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distribution is a corollary of the
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examine in the first place the curv
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12. According to Bortkiewicz’ ter
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generality, the similarities taking
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one on another, as well as the corr
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is inapplicable because the right s
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Instead, Slutsky introduced new not
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abandoned in August 1936, but it is
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last decades, mathematicians more o
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charged with making the leading ple
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motion and a number of others) are
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phenomena. It is self-evident that
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Such new demands were formulated in
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The addition of independent random
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automatic lathes, etc. Here, the ma
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11. Kolmogorov, A.N. Grundbegriffe
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period 1 and remained, until the ap
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of the analytical tool rather than
Page 46 and 47: with probability approaching unity,
Page 48 and 49: logic. The ensuing vagueness in his
Page 50 and 51: 2. Gnedenko, B.V. (1949), On Lobach
Page 52 and 53: will be sufficient, although not ne
Page 54 and 55: nlimk = 1P(| k (n) - m k (n) | > H
Page 56 and 57: favorite classical issue as the gam
Page 58 and 59: and some quite definite (not depend
Page 60 and 61: influenced by a construction that a
Page 62 and 63: P ij (1) = p ij (1) , P ij (t) =kP
Page 64 and 65: 4.2d. Bebutov [1; 2] as well as Kry
Page 66 and 67: are yet no limit theorems correspon
Page 68 and 69: described, from the viewpoint that
Page 70 and 71: conditional variance and determined
Page 72 and 73: Romanovsky [45] and Kolmogorov [46]
Page 74 and 75: Let S be the general population wit
Page 76 and 77: Part 1. Russian/Soviet AuthorsAmbar
Page 78 and 79: 2. On necessary and sufficient cond
Page 80 and 81: Gnedenko, B.V., Groshev, A.V. 1. On
Page 82 and 83: 52. ( (Mathematical Principl
Page 84 and 85: Kozuliaev, P.A. 1. Sur la répartit
Page 86 and 87: Obukhov, A.M. 1. Normal correlation
Page 88 and 89: 30. Généralisations d’un théor
Page 90 and 91: 22. Alcune applicazioni dei coeffic
Page 92 and 93: 10. A.N. Kolmogorov. The Theory of
Page 94 and 95: Kuznetsov, Stratonovich & Tikhonov
Page 98 and 99: to such a generalization. He only s
Page 100 and 101: In the particular case of a charact
Page 102 and 103: as it is usual for the modern theor
Page 104 and 105: 1. {The second reference to Pugache
Page 106 and 107: Smirnov, Romanovsky and others made
Page 108 and 109: determined the precise asymptotic c
Page 110 and 111: for finite values of N, M and R 2 .
Page 112 and 113: Mikhalevich’s findings by far exc
Page 114 and 115: Uch. Zap. = Uchenye ZapiskiUkr = Uk
Page 116 and 117: Khinchin, A.Ya. 43. Math. Ann. 101,
Page 118 and 119: 7. DAN 115, 1957, 49 - 52.Pinsker,
Page 120 and 121: Anderson, T.W., Darling, D.A. (1952
Page 122 and 123: Statistical problems in radio engin
Page 124 and 125: observations for its power with reg
Page 126 and 127: securing against mistakes (A.N. Kry
Page 128 and 129: of the others, then its distributio
Page 130 and 131: In Kiev, in the 1930s, N.M. Krylov
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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