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On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema

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Abstract

This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.

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References

  1. M. A. Lifshits Ya. Yu. Nikitin, V. V. Petrov A. Yu. Zaitsev, and A. A. Zinger “Toward the history of the Saint Petersburg School of Probability and Statistics. I. Limit theorems for sums of independent random variables,” Vestn. St. Petersburg Univ.: Math. 51, 144–163 (2018). https://doi.org/.doi10.3103/S1063454118020115

    Article  MathSciNet  MATH  Google Scholar 

  2. D. N. Zaporozhets D. N. Ibragimov M. A. Lifshits and A. I. Nazarov “Toward the history of the Saint Petersburg School of Probability and Statistics. II. Random processes and dependent variables,” Vestn. St. Petersburg Univ.: Math. 51, 213–236 (2018).

    Article  Google Scholar 

  3. A. N. Kolmogorov “On the analytic methods of probability theory,” Usp. Mat. Nauk 5, 5–41 (1938).

    Google Scholar 

  4. M. Kac “On distribution of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. N. Borodin and I. A. Ibragimov “Limit theorems for functionals of random walks,” Proc. Steklov Inst. Math. 195, 1–259 (1995).

    MathSciNet  MATH  Google Scholar 

  6. A. N. Borodin Stochastic Processes (Lan’, St. Petersburg, 2013; Springer-Verlag, Cham, 2017).

    Book  MATH  Google Scholar 

  7. A. N. Borodin and P. Salminen Handbook of Brownian Motion — Facts and Formulae, 2nd ed. (Lan’, St. Petersburg, 2016; Springer-Verlag, Basel, 2015).

    MATH  Google Scholar 

  8. A. V. Skorokhod and N. P. Slobodenyuk Limit Theorems for Random Walks (Naukova Dumka, Kiev, 1970) [in Russian].

    MATH  Google Scholar 

  9. Yu. A. Davydov “Convergence of distributions generated by stationary stochastic processes,” Theory Probab. Appl. 13, 691–696 (1968).

    Article  MATH  Google Scholar 

  10. Yu. A. Davydov “The invariance principle for stationary processes,” Theory Probab. Appl. 15, 487–498 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  11. Yu. A. Davydov “On strong mixing condition for countable Markov chains,” Dokl. Akad. Nauk SSSR 187, 252–254 (1969).

    MathSciNet  Google Scholar 

  12. Yu. A. Davydov “Mixing conditions for Markov chains,” Theory Probab. Appl. 18, 312–328 (1974).

    Article  MATH  Google Scholar 

  13. A. N. Kolmogorov “On the statistical theory of metal crystallization,” Izv. Akad. Nauk SSSR, Ser. Mat. 1, 355–359 (1937).

    Google Scholar 

  14. J. Møller “Generation of Johnson–Mehl crystals and comparative analysis of models for random nucleation,” Adv. Appl. Probab. 27, 367–383 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  15. Yu. Davydov and A. Illig “Mixing properties of crystallization processes,” North-West. Eur. J. Math. 1, 169–191 (2015).

    Google Scholar 

  16. Yu. A. Davydov “On absolute continuity for distributions of stochastic functionals,” Theory Probab. Appl. 23, 218–219 (1978).

    Article  Google Scholar 

  17. Yu. A. Davydov and M. A. Lifshits “Fibering method in some probabilistic problems,” J. Sov. Math. 31, 2796–2858 (1985).

    Article  MATH  Google Scholar 

  18. Yu. A. Davydov “A remark on the absolute continuity of distributions of Gaussian functionals,” Theory Probab. Appl. 33, 158–161 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu. A. Davydov “On distribution of the norm for Gaussian functionals,” Theory Probab. Appl. 29, 836–853 (1984).

    Google Scholar 

  20. Yu. A. Davydov “On distributions of multiple Wiener–Itô integrals,” Theory Probab. Appl. 35, 27–37 (1991).

    Article  MATH  Google Scholar 

  21. Yu. A. Davydov M. A. Lifshits and N. V. Smorodina Local Properties of Distributions of Stochastic Functionals (Am. Math. Soc., New York, 1998; Nauka, Moscow, 1995).

    MATH  Google Scholar 

  22. Yu. A. Davydov “Local limit theorems for functionals of Gaussian processes,” Theory Probab. Appl. 26, 859–860 (1982).

    Article  Google Scholar 

  23. Yu. A. Davydov “On strong convergence of distributions of functionals of random processes. I,” Theory Probab. Appl. 25, 772–789 (1981).

    Article  MATH  Google Scholar 

  24. Yu. A. Davydov “On the rate of strong convergence for convolutions,” J. Math. Sci. (N. Y.) 83, 393–396 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  25. Yu. A. Davydov “A variant of an infinite-dimensional local limit theorem,” J. Sov. Math. 61, 1853–1856 (1992).

    Article  Google Scholar 

  26. J.-Ch. Breton and Yu. A. Davydov “Local invariance principle for independent and identically distributed random variables,” Theory Probab. Appl. 51, 256–278 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu. Davydov and A. Vershik “Réarrangements convexes des marches aléatoires,” Ann. Inst. Henri Poincaré 34, 73–95 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  28. Yu. Davydov, V. Paulauskas and A. Rackauskas “More on p-stable convex random sets in Banach spaces,” J. Theor. Probab. 13, 39–64 (2000).

    Article  MATH  Google Scholar 

  29. Yu. Davydov, I. Molchanov and S. Zuev “On strictly stable laws on convex cones,” Electron. J. Probab. 13, 259–321 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  30. Yu. Davydov, “On convex hulls of Gaussian samples,” Lith. Math.J. 51 171–179 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  31. Yu. Davydov and V. Paulauskas “On the asymptotic form of convex hulls of Gaussian fields,” Cent. Eur. J. Math. 12, 711–720 (2014).

    MathSciNet  MATH  Google Scholar 

  32. Yu. Davydov and C. Dombry “Convex hulls of regularly varying processes,” J. Math. Sci. (N. Y.) 199, 150–161 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. V. B. Nevzorov and V. V. Petrov “On the distribution of the maximum of the sequential sums of independent random variables,” Theory Probab. Appl. 14, 682–687 (1969).

    Article  Google Scholar 

  34. V. B. Nevzorov “On the convergence of the distribution of the maximum of the sequential sums of independent symmetrical random variables to the limit law,” Teor. Veroyatn. Mat. Stat., No. 3, 105–116 (1970).

    Google Scholar 

  35. V. B. Nevzorov “On the speed of convergence of the distribution of the maximum cumulative sum of independent random variables to the normal law,” Theory Prob. Appl. 16, 378–384 (1971).

    Article  MATH  Google Scholar 

  36. V. B. Nevzorov “On the convergence of the distribution of the maximum of the sequential sums of independent random variables to the normal law,” Vestn. Leningr. Univ., No. 1, 34–44 (1972).

    Google Scholar 

  37. V. B. Nevzorov “On the distribution of the maximal sum of independent summands,” Dokl. Akad. Nauk. SSSR 208, 43–45 (1973).

    MathSciNet  Google Scholar 

  38. V. B. Nevzorov “On the maximum of sequential sums for Cauchy distribution,” J. Sov. Math. 20, 2221–2224 (1979).

    Article  MATH  Google Scholar 

  39. T. V. Arak and V. B. Nevzorov “Some estimates for the maximum cumulative sum of independent random variables,” Theory Probab. Appl. 18, 384–387 (1973).

    Article  MATH  Google Scholar 

  40. I. S. Akhundov “On the distribution of the maximum of the normed sums,” Vestn. Leningr. Univ., No. 2, 99–102 (1987).

    MathSciNet  MATH  Google Scholar 

  41. I. S. Akhundov “On the estimate of the rate of convergence in the central limit theorem for elements of the variational series, based on sums of random variables,” Vestn. Leningr. Univ., No. 2, 120–121 (1987).

    Google Scholar 

  42. V. B. Nevzorov “Ordinal statistics based on sums of random variables,” Teor. Veroyatn. Primen. 30, 193–194 (1985).

    Google Scholar 

  43. V. A. Egorov and V. B. Nevzorov “Some estimates of the rate of convergence of sums of order statistics to the normal law,” Zap. Nauchn. Semin. LOMI 41, 105–128 (1974).

    MATH  Google Scholar 

  44. V. A. Egorov and V. B. Nevzorov “On the rate of convergence of linear combinations of absolute order statistics to the normal law,” Theory Probab. Appl. 20, 203–211 (1975).

    Article  MATH  Google Scholar 

  45. V. A. Egorov and V. B. Nevzorov “Asymptotical decompositions of the distribution functions of sums of the absolute order statistics,” Vestn. Leningr. Univ. 19, 18–25 (1975).

    MATH  Google Scholar 

  46. V. A. Egorov and V. B. Nevzorov “Some theorems for induced order statistics,” Theory Probab. Appl. 27, 633–639 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  47. V. B. Nevzorov “The rate of convergence of order statistics for nonidentically distributed random variables to the normal law,” Zap. Nauchn. Semin. LOMI 130, 137–146 (1984).

    MATH  Google Scholar 

  48. V. B. Nevzorov “Representation of order statistics based on exponential variables with different scale parameters,” Zap. Nauchn. Semin. LOMI 136, 162–164 (1983).

    Google Scholar 

  49. S. M. Ananjevskii and V. B. Nevzorov “On families of distributions characterized by certain properties of ordered random variables,” Vestn. St. Petersburg Univ.: Math. 49, 197–203 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Berred and V. B. Nevzorov “Samples without replacement: Discrete analogs of some continuous distributions,” J. Math. Sci. (N. Y.) 133, 1224–1230 (2006).

    Article  MathSciNet  Google Scholar 

  51. V. O. Zykov and V. B. Nevzorov “Some characterizations of families of distributions, including logistic and exponential ones, by properties of order statistics,” J. Math. Sci. (N. Y.) 176, 203 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  52. N. M. Marudova and V. B. Nevzorov “A class of distributions that includes Student’s t2-distribution,” J. Math. Sci. (N. Y.) 159, 312 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  53. V. B. Nevzorov “One property of Student’s distribution with two degrees of freedom,” J. Math. Sci. (N. Y.) 127, 1757–1762 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  54. V. B. Nevzorov “On characterizations of Student distribution with two degrees of freedom,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 4, 38–42 (2005).

    MathSciNet  Google Scholar 

  55. V. B. Nevzorov “Extremes and Student’s t2-distribution,” J. Math. Sci. (N. Y.) 152, 928–931 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  56. V. B. Nevzorov “On the average number of records for sequences of not identically distributed random variables,” Vestn. St. Petersburg Univ.: Math. 45, 164–167 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  57. V. B. Nevzorov “Record values with restrictions,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron, No. 3, 70–74 (2013).

    Google Scholar 

  58. V. B. Nevzorov and S. A. Tovmasjan “On the maximal value of the expectation of record numbers,” Vestn. St. Petersburg Univ.: Math. 47, 64–67 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  59. V. B. Nevzorov “On regression relations between sample means and order statistics,” Vestn. St. Petersburg Univ.: Math. 48, 164–167 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  60. V. K. Sagatelyan “Characterization of distributions by equalities of order statistics,” J. Math. Sci. (N. Y.) 147, 6946–6949 (2007).

    Article  MathSciNet  Google Scholar 

  61. V. B. Nevzorov Records. Mathematical Theory (Fazis, Moscow, 2000) [in Russian].

    Book  MATH  Google Scholar 

  62. V. A. Balabekyan and V. B. Nevzorov “On the number of records in a sequence of series of nonidentically distributed random variables,” in Rings and Modules. Limit Theorems of Probability Theory (Leningr. Gos. Univ., Leningrad, 1986), Vol. 1, pp. 147–153 [in Russian].

    Google Scholar 

  63. V. B. Nevzorov “Asymptotic distributions of records in nonstationary schemes,” J. Stat. Plann. Inference 44, 261–273 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  64. V. B. Nevzorov “Record times in the case of nonidentically distributed random variables,” Theory Probab. Appl. 29, 836–853 (1984).

    Google Scholar 

  65. V. B. Nevzorov “Record and interrecord times for sequences of nonidentically distributed random variables,” J. Sov. Math. 36, 510–516 (1985).

    Article  MATH  Google Scholar 

  66. V. B. Nevzorov “Record times and their generalizations,” Theory Probab. Appl. 31, 551–559 (1986).

    MATH  Google Scholar 

  67. V. B. Nevzorov and M. Ranne “On record moments in sequences of nonidentically distributed discrete random variables,” J. Math. Sci. (N. Y.) 75, 1951–1957 (1992).

    Article  MATH  Google Scholar 

  68. M. Rannen “Records in sequences of series of nonidentically distributed random variables,” Vestn. S.-Peterb. Univ., Ser. 1: Math., Mekh., Astron., No. 1, 79–83 (1991).

    Google Scholar 

  69. P. Deheuvels and V. B. Nevzorov “Records in the Fα-scheme. I. Martingale properties,” J. Math. Sci. (N. Y.) 81, 2368–2378 (1993).

    Article  MathSciNet  Google Scholar 

  70. P. Deheuvels and V. B. Nevzorov “Records in the Fα-scheme. II. Limit laws,” J. Math. Sci. (N. Y.) 88, 29–35 (1994).

    Article  MATH  Google Scholar 

  71. V. B. Nevzorov “Analogue of the Fα scheme for discrete distributions,” Vestn. St. Petersburg Univ.: Math. 48, 18–22 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  72. P. Deheuvels and V. B. Nevzorov “Limit laws for K-record times,” J. Stat. Plann. Inference 38, 279–307 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  73. V. B. Nevzorov “On k-th record times and their generalizations,” Zap. Nauchn. Semin. LOMI 153, 115–121 (1986).

    MATH  Google Scholar 

  74. V. B. Nevzorov “Generating functions for kth record times — A martingale approach,” J. Math. Sci. (N. Y.) 68, 545–550 (1990).

    Article  MATH  Google Scholar 

  75. V. K. Sagatelyan “A new model for record values,” Vestn. St. Petersburg Univ.: Math. 41, 282–285 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  76. V. B. Nevzorov and A. V. Stepanov “Records with confirmation,” Stat. Probab. Lett. 95, 39–47 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  77. I. V. Belkov and V. B. Nevzorov “Record values in sequences of sample ranges,” Vestn. St. Petersburg Univ.: Math. 50, 325–328 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  78. A. V. Stepanov “On logarithmic moments for interrecord times,” Theory Probab. Appl. 32, 708–710 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  79. A. V. Stepanov “Characterizations of geometrical type of distributions,” Teor. Veroyatn. Mat. Stat. 41, 113–116 (1996).

    Google Scholar 

  80. V. B. Nevzorov and A. Stepanov “Records: Martingale approach to finding of moments,” in Rings and Modules. Limit Theorems of Probability Theory (Leningr. Gos. Univ., Leningrad, 1988), Vol. 2, pp. 171–181 [in Russian].

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, 199034, Russia

    A. N. Borodin, Yu. A. Davydov & V. B. Nevzorov

  2. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg Branch, St. Petersburg, 191023, Russia

    A. N. Borodin

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  1. A. N. Borodin
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  2. Yu. A. Davydov
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  3. V. B. Nevzorov
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Correspondence to A. N. Borodin.

Additional information

Original Russian Text © A.N. Borodin, Yu.A. Davydov, V.B. Nevzorov, 2018, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2018, Vol. 51, No. 4, pp. 572–596.

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Borodin, A.N., Davydov, Y.A. & Nevzorov, V.B. On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema. Vestnik St.Petersb. Univ.Math. 51, 343–359 (2018). https://doi.org/10.3103/S1063454118040052

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