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On Analytical Methods In Probability Theory

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Selected Works of A. N. Kolmogorov

Part of the book series: Mathematics and Its Applications (Soviet Series) ((MASS,volume 26))

Abstract

A physical process (a change of a certain physical system) is called stochastically determined if, knowing a state X 0 of the system at a certain moment of time t0 we also know the probability distribution for all the states X of this system at the moments t > t 0.

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Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), 415-458.

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Notes

  1. A well-known example of this method is to introduce, in addition to positions of points, the components of their velocities when describing a state of a certain mechanical system.

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  2. I. ‘Théorie de la spéculation’, Ann. École Norm. Supér. 17 (1900), 21; II. ‘Les probabilités à plusieurs variables’, Ann. École Norm. Super. 27 (1910), 339; III. Calcul des probabilités, Paris, 1912.

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  3. Concerning these notions, as well as additive sets of systems, etc., see, for example, M. Fréchet,’ sur l’intégrale d’une fonctionnelle étendu à un ensemble abstrait’, Bull. Soc. Math. France 43 (1915), 248.

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  4. See the first of the papers cited in footnote 2.

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  5. C. R. Acad. Sci. Paris186 (1928), 59; 189; 275.

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  6. See Footnote 5.

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  7. Compare with the functions F(s,x,t,y) considered in Chapter 4, which necessarily have points of discontinuity at t =s.

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  8. We could equally well have taken the opposite approach: to assume a priori that the conditions (47a) and (50) hold and to derive from this the continuity and differentiability of the function P ij (s, t) with respect to t.

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  9. More details on this question can be found in R. von Mises Wahrscheinlichkeitsrechnung, Berlin, 1931, especially the chapter on “local” limit theorems. (Remark by Russian editor.)

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  10. H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1928.

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  11. See P. Levy, Calcul des probabilités, Paris, 1927, p.187.

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  12. Math. Z.15 (1922), 211.

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  13. See papers Nos. 1 and 3 in footnote 2.

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  14. See item II in footnote 2.

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Authors
  1. A. N. Shiryayev
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A. N. Shiryayev

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© 1992 Springer Science+Business Media Dordrecht

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Shiryayev, A.N. (1992). On Analytical Methods In Probability Theory. In: Shiryayev, A.N. (eds) Selected Works of A. N. Kolmogorov. Mathematics and Its Applications (Soviet Series), vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2260-3_9

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  • DOI: https://doi.org/10.1007/978-94-011-2260-3_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5003-6

  • Online ISBN: 978-94-011-2260-3

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