Selected Works II: Probability Theory and Mathematical Statistics

ISBN-13: 9789401050036 / Angielski / Miękka / 2012 / 585 str.

A. N. Shiryayev

Opis

lEt moi *...* si j'avait so comment en revenir, One service mathematics has rendered the je n'y serais point alle:' human race. It has put common sense back Ju1e. Veme where it belongs, 01 the topmost shelf next to the dUlty canister labelled 'discarded non- Tbe series is divergent; therefore we may be sense'. able to do something with it. Erie T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple reweiting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'tftre of this series.

Spis treści

1. On convergence of series whose terms are determined by random events.- 2. On the law of large numbers.- 3. On a limit formula of A. Khinchin.- 4. On sums of independent random variables.- 5. On the law of the iterated logarithm.- 6. On the law of large numbers.- 7. General measure theory and probability calculus.- 8. On the strong law of large numbers.- 9. On analytical methods in probability theory.- 10. The waiting problem.- 11. The method of the median in the theory of errors.- 12. A generalization of the Laplace-Lyapunov Theorem.- 13. On the general form of a homogeneous stochastic process.- 14. On computing the mean Brownian area.- 15. On the empirical determination of a distribution law.- 16. On the limit theorems of probability theory.- 17. On the theory of continuous random processes.- 18. On the problem of the suitability of forecasting formulas found by statistical methods.- 19. Random motions.- 20. Deviations from Hardy’s formulas under partial isolation.- 21. On the theory of Markov chains.- 22. On the statistical theory of metal crystallization.- 23. Markov chains with a countable number of possible states.- 24. On the reversibility of the statistical laws of nature.- 25. Solution of a biological problem.- 26. On a new confirmation of Mendel’s laws.- 27. Stationary sequences in Hubert space.- 28. Interpolation and extrapolation of stationary random sequences...- 29. On the logarithmic normal distribution of particle sizes under grinding.- 30. Justification of the method of least squares.- 31. A formula of Gauss in the method of least squares.- 32. Branching random processes.- 33. Computation of final probabilities for branching random processes..- 34. Statistical theory of oscillations with continuous spectrum.- 35. On sums of a random number of random terms.- 36. A local limit theorem for classical Markov chains.- 37. Solution of a probabilistic problem relating to the mechanism of bed formation.- 38. Unbiased estimators.- 39. On differentiability of transition probabilities of time-homogeneous Markov processes with a countable number of states.- 40. A generalization of Poisson’ s formula for a sample from a finite set.- 41. Some recent work on limit theorems in probability theory.- 42. On A.V. Skorokhod’s convergence.- 43. Two uniform limit theorems for sums of independent terms.- 44. Random functions and limit theorems.- 45. On the properties of P. Levy’s concentration functions.- 46. Transition of branching processes to diffusion processes and related genetic problems.- 47. On the classes ?(n) of Fortet and Blanc-Lapierre.- 48. On conditions of strong mixing of a Gaussian stationary process.- 49. Random functions of several variables almost all realizations of which are periodic.- 50. An estimate of the parameters of a complex stationary Gaussian Markov process.- 51. On the approximation of distributions of sums of independent terms by infinitely divisible distributions.- 52. Estimators of spectral functions of random processes.- 53. On the logical foundations of probability theory.- Comments On the papers on probability theory and mathematical statistics.- Analytical methods in probability theory (No. 9).- Markov processes with a countable number of states (No. 10).- Homogeneous random processes (No. 13).- Homogeneous Markov processes (No. 39).- Branching processes (Nos. 25, 32, 33, 46).- Stationary sequences (No. 27).- Stationary processes (No. 48).- Statistics of processes (No. 50).- Spectral theory of stationary processes (No. 34).- Spectral representation of random processes (Nos. 47, 49).- Brownian motion (Nos. 14, 19, 24).- Markov chains with a countable number of states (No. 23).- Wald identities (No. 35).- S-Convergence (No. 42).- Uniform limit theorems (Nos. 43, 51).- Concentration functions (No. 45).- Empirical distributions (No. 15).- The method of least squares (Nos. 30, 31).- Unbiased estimators (No. 38).- Statistical prediction (No. 18).- On inter-bed washout (No. 37).

Blurb

The creative work of Andrei N. Kolmogorov is exceptionally wide-ranging. In his studies on trigonometric and orthogonal series, the theory of measure and inte­gral, mathematical logic, approximation theory, geometry, topology, functional analysis, classical mechanics, ergodic theory, superposition of functions, and in­ formation theory, he solved many conceptual and fundamental problems and posed new questions which gave rise to a great deal of further research. Kolmogorov is one of the founders of the Soviet school of probability theory, mathematical statistics, and the theory of turbulence. In these areas he obtained a number of central results, with many applications to mechanics, geophysics, linguistics and biology, among other subjects. This edition includes Kolmogorov's most important papers on mathematics and the natural sciences. It does not include his philosophical and ped­agogical studies, his articles written for the "Bolshaya Sovetskaya Entsiklopediya", his papers on prosody and applications of mathematics or his publications on general questions. The material of this edition was selected and compiled by Kolmogorov himself.

The first volume consists of papers on mathematics and also on turbulence and classical mechanics. The second volume is devoted to probability theory and mathematical statistics. The focus of the third volume is on information theory and the theory of algorithms.