Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables

Abstract

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as important relevant problems formulated in the second half of the twentieth century. The latter include the approximation of the distributions of sums of independent variables by infinitely divisible distributions, estimation of the accuracy of strong Gaussian approximation of such sums, and the limit theorems on the weak almost sure convergence of empirical measures generated by sequences of sums of independent random variables and vectors.

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References

  1. 1.

    Yu. V. Linnik, “Probability theory and mathematical statistics,” in Mathematics in St. Petersburg–Leningrad University, Ed. by V. I. Smirnov (Leningr. Gos. Univ., Leningrad, 1970), pp. 243–255 [in Russian].

    Google Scholar 

  2. 2.

    V. I. Bunjakovskii, Foundations of Mathematical Probability Theory (St. Petersburg, 1846) [in Russian].

    Google Scholar 

  3. 3.

    B. V. Gnedenko, Essay on the History of Probability Theory (URSS, Moscow, 2001) [in Russian].

    Google Scholar 

  4. 4.

    K. A. Andreev, “Victor Yakovlevich Bunjakovskij. Obituary essay,” in Reports and Minutes of Meetings of Mathematical Society at Imperial Khar’kov University (Kharkov, 1891), Vol. 2, pp. 149–161 [in Russian].

    Google Scholar 

  5. 5.

    P. L. Chebyshev, Probability Theory, Lectures Read in 1879–80, by Records of A. M. Lyapunov (A. N. Krylov, Akad. Nauk SSSR, Moscow, 1936) [in Russian].

  6. 6.

    A. Markov, The Calculus of Probabilities, 3rd ed. (Imp. Akad. Nauk., St. Petersburg, 1913) [in Russian].

    Google Scholar 

  7. 7.

    S. N. Bernstein, Probability Theory, 4th ed. (Gostehteorizdat, Moscow, 1946) [in Russian].

    Google Scholar 

  8. 8.

    I. A. Ibragimov, “On S. N. Bernstein’s work in probability,” Am. Math. Soc. Transl., Ser. 2 205, 83–104 (2002).

    MATH  Google Scholar 

  9. 9.

    E. Seneta, “Sergei Natanovich Bernstein,” in Statisticians of the Century, ed. by C. C. Heyde and E. Seneta (Springer-Verlag, 2001), pp. 339–345.

    Google Scholar 

  10. 10.

    Yu. Linnik, Selected Works. Probability Theory (Nauka, Leningrad, 1981) [in Russian].

    Google Scholar 

  11. 11.

    Yu. Linnik, Selected Works. Mathematical Statistics (Nauka, Leningrad, 1982) [in Russian].

    Google Scholar 

  12. 12.

    Ya. Yu. Nikitin and I. V. Romanovsky, “To the 100th anniversary of the birth of Yuri Vladimirovich Linnik,” Vestn. of S.-Peterb. Univ, Ser. 1: Mat., Mekh., Astron. 60, 487–492 (2015).

    Google Scholar 

  13. 13.

    Yu. V. Linnik, “The accuracy of the approximation to the Gauss distribution of sums of independent random variables,” Izv. Akad. Nauk SSSR, Ser. Math. 11, 111–138 (1947).

    MATH  Google Scholar 

  14. 14.

    V. V. Petrov, “An estimate of the deviation of the distribution of a sum of independent random variables from the normal law,” Sov. Math. Dokl. 6, 242–244 (1965).

    MATH  Google Scholar 

  15. 15.

    I. A. Ibragimov, “On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables,” Theory Probab. Appl. 11, 632–655 (1966).

    Article  Google Scholar 

  16. 16.

    L. V. Osipov, “Refinement of Lindeberg’s theorem,” Theory Probab. Appl. 11, 299–302 (1966).

    MATH  Article  Google Scholar 

  17. 17.

    L. V. Osipov and V. V. Petrov, “On an estimate of the remainder term in the central limit theorem,” Theory Probab. Appl. 12, 281–286 (1967).

    MATH  Article  Google Scholar 

  18. 18.

    B. A. Lifshits, “On the accuracy of approximation in the central limit theorem,” Theory Probab. Appl. 21, 108–124 (1976).

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    C. C. Heyde, “On the uniform metric in the context of convergence to normality,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 25, 83–95 (1973).

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    L. V. Osipov, “Accuracy in approximation of distribution of a sum of independent random variables to normal distribution,” Dokl. Akad. Nauk SSSR 178, 1013–1016 (1968).

    MathSciNet  Google Scholar 

  21. 21.

    V. A. Egorov, “On the rate of convergence to normal law which is equivalent to the existence of the second moment,” Theory Probab. Appl. 18, 175–180 (1973).

    MATH  Article  Google Scholar 

  22. 22.

    L. V. Rozovsky, “On the precision of an estimate of the remainder term in the central limit theorem,” Theory Probab. Appl. 23, 712–730 (1978).

    Article  Google Scholar 

  23. 23.

    I. A. Ibragimov, “On the Chebyshev–Cramér asymptotic expansions,” Theory Probab. Appl. 12, 455–469 (1967).

    MATH  Article  Google Scholar 

  24. 24.

    L. V. Osipov, “Asymptotic expansions of the distribution function of a sum of random variables with non-uniform estimates for the remainder term,” Vestn. Leningr. Univ., Ser. 1: Mat., Mekh., Astron., No. 1, 51–59 (1972).

    MathSciNet  Google Scholar 

  25. 25.

    V. V. Petrov, “On some polynomials encountered in probability theory,” Vestn. Leningr. Univ., Ser. 1: Mat., Mekh., Astron., No. 19, 150–153 (1962).

    Google Scholar 

  26. 26.

    V. V. Petrov, “Asymptotic expansions for the derivatives of the distribution function of a sum of independent terms,” Vestn. Leningr. Univ., Ser. 1: Mat., Mekh., Astron., No. 19, 9–18 (1960).

    MathSciNet  Google Scholar 

  27. 27.

    V. V. Petrov, “A local theorem for densities of sums of independent random variables,” Theory Probab. Appl. 1, 316–322 (1956).

    MathSciNet  Article  Google Scholar 

  28. 28.

    V. V. Petrov, “On local limit theorems for sums of independent random variables,” Theory Probab. Appl. 9, 312–320 (1964).

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    H. Cramér, “Sur un nouveau théorème-limite de la théorie des probabilités,” Actual. Sci. Ind. 736, 5–23 (Paris, 1938).

    MATH  Google Scholar 

  30. 30.

    V. V. Petrov, “A generalization of Cramér’s limit theorem,” Usp. Mat. Nauk 9, 195–202 (1954).

    Google Scholar 

  31. 31.

    W. Feller, “Generalization of a probability limit theorem of Cramér,” Trans. Am. Math. Soc. 54, 361–372 (1943).

    MATH  Google Scholar 

  32. 32.

    W. Richter, “Local limit theorems for large deviations,” Theory Probab. Appl. 2, 206–220 (1957).

    MATH  Article  Google Scholar 

  33. 33.

    Yu. V. Linnik, “On probability of large deviations for the sums of independent variables,” in Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability, Berkeley, CA, June 20–July 30,1960, 289–306 (Univ. California Press, Berkeley, CA, 1969), Vol. 2.

    Google Scholar 

  34. 34.

    Yu. V. Linnik, “ Limit theorems for sums of independent variables taking into account large deviations. I,” Theory Probab. Appl. 6, 131–148

  35. 34.

    Yu. V. Linnik, “Limit theorems for sums of independent variables taking into account large deviations. II,” Theory Probab. Appl. 6, 345–360 (1961)

    MATH  Article  Google Scholar 

  36. 34.

    Yu. V. Linnik, “Limit theorems for sums of independent variables taking into account large deviations. III,” Theory Probab. Appl. 7, 115–129 (1962).

    MATH  Article  Google Scholar 

  37. 35.

    V. V. Petrov, “Limit theorems for large deviations under violation of Cramér condition. I,” Vestn. Leningr. Univ., Ser. 1: Mat., Mekh., Astron., No. 19, 49–68 (1963)

    Google Scholar 

  38. 35.

    V. V. Petrov, “Limit theorems for large deviations under violation of Cramér condition. II,” Vestn. Leningr. Univ., Ser. 1: Mat., Mekh., Astron., No. 1, 58–75 (1964).

    Google Scholar 

  39. 36.

    L. V. Osipov, “On probabilities of large deviations for sums of independent random variables,” Theory Probab. Appl. 17, 309–331 (1972).

    MathSciNet  MATH  Article  Google Scholar 

  40. 37.

    V. V. Petrov, “On probabilities of large deviations for sums of independent random variables,” Theory Probab. Appl. 10, 287–298 (1965).

    MathSciNet  MATH  Article  Google Scholar 

  41. 38.

    L. V. Rozovsky, “On the Cramér series coefficients,” Theory Probab. Appl. 43, 152–157 (1999).

    MathSciNet  Article  Google Scholar 

  42. 39.

    L. V. Rozovsky, “Asymptotic expansions for probabilities of large deviations,” Theory Probab. Appl. 31, 255–268 (1987).

    Article  Google Scholar 

  43. 40.

    I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, Groningen, 1971).

    Google Scholar 

  44. 41.

    V. V. Petrov, Sums of Independent Random Variables (Nauka, Moscow, 1972; Springer-Verlag, Berlin, 1975).

    Google Scholar 

  45. 42.

    V. V. Petrov, Limit Theorems of Probability Theory (Oxford Univ. Press, New York, 1995).

    Google Scholar 

  46. 43.

    A. I. Martikainen, “On necessary and sufficient conditions for the strong law of large numbers,” Theory Probab. Appl. 24, 813–820 (1979).

    MathSciNet  MATH  Article  Google Scholar 

  47. 44.

    L. V. Rozovsky, “On the relation of the rate of convergence in the weak and strong law of large numbers,” Lith. Math. J. 21, 155–167 (1981).

    MathSciNet  Google Scholar 

  48. 45.

    V. V. Petrov, “On the strong law of large numbers,” Theory Probab. Appl. 14, 183–192 (1969).

    MATH  Article  Google Scholar 

  49. 46.

    V. A. Egorov, “Some theorems on the strong law of large numbers and law of the iterated logarithm,” Theory Probab. Appl. 17, 86–100 (1972).

    MATH  Article  Google Scholar 

  50. 47.

    V. V. Petrov, “On the order of growth of sums of dependent variables,” Theory Probab. Appl. 18 (2), 348–350 (1974).

    MATH  Article  Google Scholar 

  51. 48.

    V. V. Petrov, “On absolute convergence of series of random variables almost surely,” J. Math. Sci. 214, 513–516 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  52. 49.

    V. V. Petrov, “On the strong law of large numbers for nonnegative random variables,” Theory Probab. Appl. 53, 346–349 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  53. 50.

    V. V. Petrov, “On the strong law of large numbers for sequences of dependent random variables,” J. Math. Sci. 199, 225–227 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  54. 51.

    V. V. Petrov, “On a relation between an estimate of the remainder in the central limit theorem and the law of the iterated logarithm,” Theory Probab. Appl. 11, 454–458 (1966).

    MathSciNet  Article  Google Scholar 

  55. 52.

    V. A. Egorov, “On the law of the iterated logarithm,” Theory Probab. Appl. 14, 693–699 (1969).

    MATH  Article  Google Scholar 

  56. 53.

    V. A. Egorov, “Generalization of Hartman–Wintner theorem on the law of the iterated logarithm,” Vestn. Leningrad Univ. Math., No. 4, 117–124 (1977).

    Google Scholar 

  57. 54.

    V. V. Petrov, “On the law of the iterated logarithm for a sequence of independent random variables,” Theory Probab. Appl. 46, 542–544 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  58. 55.

    V. V. Petrov, “Sequences of m-orthogonal random variables,” J. Sov. Math. 27, 3136–3139 (1984).

    MATH  Article  Google Scholar 

  59. 56.

    V. V. Petrov, “On the law of the iterated logarithm for sequences of dependent random variables,” Vestn. St. Petersburg Univ.: Math. 50, 32–34 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  60. 57.

    A. N. Frolov, “Limit theorems for increments of sums of independent random variables,” Theory Probab. Appl. 48, 93–107 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  61. 58.

    A. N. Frolov, Limit Theorems of Probability Theory (St. Peterburg Gos. Univ., St. Petersburg, 2014) [in Russian].

    Google Scholar 

  62. 59.

    A. N. Kolmogorov, “Two uniform limit theorems for sums of independent random variables,” Theory Probab. Appl. 1, 384–394 (1956).

    Article  Google Scholar 

  63. 60.

    I. A. Ibragimov and E. L. Presman, “On the rate of approach of the distributions of sums of independent random variables to accompanying distributions,” Theory Probab. Appl. 18, 713–727 (1973).

    MATH  Article  Google Scholar 

  64. 61.

    T. V. Arak and A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math. 174, 1–122 (1988).

    MathSciNet  Google Scholar 

  65. 62.

    T. V. Arak, “On the convergence rate in Kolmogorov’s uniform limit theorem, I,” Theory Probab. Appl. 26, 219–239

  66. 62.

    T. V. Arak, “On the convergence rate in Kolmogorov’s uniform limit theorem, II,” Theory Probab. Appl. 26, 437–451 (1981).

    MathSciNet  MATH  Article  Google Scholar 

  67. 63.

    T. V. Arak, “An improvement of the lower bound for the rate of convergence in Kolmogorov’s uniform limit theorem,” Theory Probab. Appl. 27, 826–832 (1982).

    MathSciNet  MATH  Article  Google Scholar 

  68. 64.

    E. L. Presman, “On a multidimensional version of the Kolmogorov uniform theorem,” Theory Probab. Appl. 18, 378–384 (1973).

    MATH  Article  Google Scholar 

  69. 65.

    T. V. Arak, “On the approximation by the accompanying laws of n-fold convolutions of distributions with nonnegative characteristic functions,” Theory Probab. Appl. 25, 221–243 (1980).

    MathSciNet  MATH  Article  Google Scholar 

  70. 66.

    A. Yu. Zaitsev, “Approximation of convolutions by accompanying laws under the existence of moment of low orders,” J. Math. Sci. 93, 336–340 (1999).

    MathSciNet  Article  Google Scholar 

  71. 67.

    E. L. Presman, “Approximation in variation of the distribution of a sum of independent Bernoulli variables with a Poisson law,” Theory Probab. Appl. 30, 417–422 (1986).

    MATH  Article  Google Scholar 

  72. 68.

    V. Cekanavicius, “On compound Poisson approximations under moment restrictions,” Theory Probab. Appl. 44, 18–28 (2000).

    MathSciNet  MATH  Article  Google Scholar 

  73. 69.

    V. Cekanavicius, “Infinitely divisible approximations for discrete nonlattice variables,” Adv. Appl. Probab. 35, 982–1006 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  74. 70.

    V. Cekanavicius and Y. H. Wang, “Compound Poisson approximations for sums of discrete nonlattice variables,” Adv. Appl. Probab. 35, 228–250 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  75. 71.

    F. Götze, Yu. S. Eliseeva, and A. Yu. Zaitsev, “Arak inequality for functions of concentration and the Littlewood–Offord problem,” Theory Probab. Appl. 62 (2) (2018).

    Google Scholar 

  76. 72.

    J. E. Littlewood and A. C. Offord, “On the number of real roots of a random algebraic equation (III),” Rec. Math. [Mat. Sbornik] N.S. 12(54) (3), 277–286 (1943).

    MathSciNet  MATH  Google Scholar 

  77. 73.

    P. Erdös, “On a lemma of Littlewood and Offord,” Bull. Am. Math. Soc. 51, 898–902 (1945).

    MathSciNet  MATH  Article  Google Scholar 

  78. 74.

    T. Tao and V. Vu, “Inverse Littlewood–Offord theorems and the condition number of random discrete matrices,” Ann. Math. 169, 595–632 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  79. 75.

    T. Tao and V. Vu, “From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices,” Bull. Am. Math. Soc. 46, 377–396 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  80. 76.

    T. Tao and V. Vu, “A sharp inverse Littlewood–Offord theorem,” Random Struct. Algorithms 37, 525–539 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  81. 77.

    H. Nguyen and V. Vu, “Optimal inverse Littlewood–Offord theorems,” Adv. Math. 226, 5298–5319 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  82. 78.

    H. Nguyen and V. Vu, “Small ball probabilities, inverse theorems and applications,” in Erdös Centennial, Ed. by L. Lovász, (Springer-Verlag, Berlin, 2013), pp. 409–463.

    Google Scholar 

  83. 79.

    A. Yu. Zaitsev and T. Arak, “On the rate of convergence in Kolmogorov’s second uniform limit theorem,” Theory Probab. Appl. 28, 351–374 (1984).

    MATH  Article  Google Scholar 

  84. 80.

    A. Yu. Zaitsev, “Multidimensional version of the second uniform theorem of Kolmogorov,” Theory Probab. Appl. 34, 108–128 (1989).

    MathSciNet  Article  Google Scholar 

  85. 81.

    A. Yu. Zaitsev, “On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions,” Probab. Theory Rel. Fields 74, 535–566 (1987).

    MathSciNet  MATH  Article  Google Scholar 

  86. 82.

    A. Yu. Zaitsev, “Estimates for the Lévy–Prokhorov distance in the multivariate central limit theorem for random vectors with finite exponential moments,” Theory Probab. Appl. 31, 203–220 (1986).

    Article  Google Scholar 

  87. 83.

    A. Yu. Zaitsev, “On the accuracy of approximation of distributions of sums of independent random variables— which are nonzero with a small probability—by means of accompanying laws,” Theory Probab. Appl. 28, 657–669 (1984).

    MATH  Article  Google Scholar 

  88. 84.

    A. Yu. Zaitsev, “Approximation of a sample by a Poisson point process,” J. Math. Sci. 128, 2556–2563 (2005).

    MathSciNet  Article  Google Scholar 

  89. 85.

    A. Yu. Zaitsev, “On the uniform approximation of distributions of sums of independent random variables,” Theory Probab. Appl. 32, 40–47 (1987).

    MathSciNet  Article  Google Scholar 

  90. 86.

    A. Yu. Zaitsev, “Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions,” J. Math. Sci. 75, 1922–1930 (1995).

    MathSciNet  Article  Google Scholar 

  91. 87.

    L. Le Cam, “On the distribution of sums of independent random variables,” in Bernoulli, Bayes, Laplace (Anniversary Volume) (Springer-Verlag, Berlin, 1965), pp. 179–202.

    Google Scholar 

  92. 88.

    F. Götze and A. Yu. Zaitsev, “Approximation of convolutions by accompanying laws without centering, J. Math. Sci. 137, 4510–4515 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  93. 89.

    A. Yu. Zaitsev, “Multidimensional generalized method of triangular functions,” J. Sov. Math. 43, 2797–2810 (1988).

    MATH  Article  Google Scholar 

  94. 90.

    A. Yu. Zaitsev, “Approximation of convolutions of multi-dimensional symmetric distributions by accompanying laws,” J. Sov. Math. 61, 1859–1872 (1992).

    Article  Google Scholar 

  95. 91.

    A. Yu. Zaitsev, “Certain class of nonuniform estimates in multidimensional limit theorems,” J. Math. Sci. 68, 459–468 (1994).

    MathSciNet  Article  Google Scholar 

  96. 92.

    A. Yu. Zaitsev, “Estimation of proximity of distributions of sequential sums of independent identically distributed random vectors,” J. Sov. Math. 24, 536–539 (1984).

    MATH  Article  Google Scholar 

  97. 93.

    A. Yu. Zaitsev, “Some properties of n-fold convolutions of distributions,” Theory Probab. Appl. 26, 148–152 (1981).

    MathSciNet  MATH  Article  Google Scholar 

  98. 94.

    A. Yu. Zaitsev, “Estimates for the closeness of successive convolutions of multidimensional symmetric distributions,” Probab. Theory Rel. Fields 79, 175–200 (1988).

    MathSciNet  MATH  Article  Google Scholar 

  99. 95.

    V. Cekanavicius, Approximations Methods in Probability Theory (Springer-Verlag, Basel, 2016).

    Google Scholar 

  100. 96.

    A. Yu. Zaitsev, “An example of a distribution whose set of n-fold convolutions is uniformly separated from the set of infinitely divisible laws in distance in variation,” Theory Probab. Appl. 36, 419–425 (1991).

    MathSciNet  Article  Google Scholar 

  101. 97.

    J. Komlós, P. Major, and G. Tusnády, “An approximation of partial sums of independent RV’-s, and the sample DF. I,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111–131 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  102. 97.

    J. Komlós, P. Major, and G. Tusnády, “An approximation of partial sums of independent RV’-s, and the sample DF. II,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 34 (1), 34–58 (1976).

    MathSciNet  MATH  Article  Google Scholar 

  103. 98.

    A. I. Sakhanenko, “Convergence rate in the invariance principle for non-identically distributed variables with exponential moments,” in Advances in Probability Theory: Limit Theorems for Sums of Random Variables, Ed. by A. A. Borovkov (Springer-Verlag, New York, 1985, pp. 2–73).

    Google Scholar 

  104. 99.

    A. Yu. Zaitsev, “Multidimensional version of the results of Komlós, Major, and Tusnády for vectors with finite exponential moments,” ESAIM: Probab. Stat. 2, 41–108 (1998).

    MATH  Article  Google Scholar 

  105. 100.

    U. Einmahl, “Extensions of results of Komlós, Major and Tusnády to the multivariate case,” J. Multivar. Anal. 28, 20–68 (1989).

    MATH  Article  Google Scholar 

  106. 101.

    A. Yu. Zaitsev, “Multidimensional version of the results of Sakhanenko in the invariance principle for vectors with finite exponential moments, I,” Theory Probab. Appl. 45, 624–641 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  107. 101.

    A. Yu. Zaitsev, “ Multidimensional version of the results of Sakhanenko in the invariance principle for vectors with finite exponential moments, II,” Theory Probab. Appl. 46, 490–514

  108. 101.

    A. Yu. Zaitsev, “Multidimensional version of the results of Sakhanenko in the invariance principle for vectors with finite exponential moments, III,” Theory Probab. Appl. 46, 676–698 (2002).

    MathSciNet  Article  Google Scholar 

  109. 102.

    A. Yu. Zaitsev, “Estimates for the strong approximation in multidimensional Central Limit Theorem,” in Proc. Int. Congress of Mathematicians (ICM 2002), Bejing, Aug. 20–28, 2002, Ed. by Li Tatsien, et al. (World Sci., Bejing, 2002), Vol. 3: Invited Lectures, pp. 107–116.

    MathSciNet  MATH  Google Scholar 

  110. 103.

    A. Yu. Zaitsev, “Estimates for the rate of strong approximation in the multidimensional invariance principle,” J. Math. Sci. 145, 4856–4865 (2007).

    MathSciNet  Article  Google Scholar 

  111. 104.

    A. Yu. Zaitsev, “Estimates for the rate of strong Gaussian approximation for sums of i.i.d. multidimensional random vectors,” J. Math. Sci. 152, 875–884 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  112. 105.

    A. Yu. Zaitsev, “Rate of strong Gaussian approximation for sums of i.i.d. multidimensional random vectors,” J. Math. Sci. 163, 399–408 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  113. 106.

    A. I. Sakhanenko, “On estimates of the rate of convergence in the invariance principle,” in Advances in Probability Theory: Limit Theorems for Sums of Random Variables, ed. A. A. Borovkov (Springer-Verlag, New York, 1984), pp. 124–135.

    Google Scholar 

  114. 107.

    F. Götze and A. Yu. Zaitsev, “Bounds for the rate of strong approximation in the multidimensional invariance principle,” Theory Probab. Appl. 53, 59–80 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  115. 108.

    F. Götze and A. Yu. Zaitsev, “Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments,” J. Math. Sci. 167, 495–500 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  116. 109.

    F. Götze and A. Yu. Zaitsev, “Estimates for the rate of strong approximation in Hilbert space,” Sib. Math. J. 52, 628–638 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  117. 110.

    A. Yu. Zaitsev, “Optimal estimates for the rate of strong Gaussian approximate in a Hilbert space,” J. Math. Sci. 188, 689–693 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  118. 111.

    A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors,” Russian Math. Surveys 68, 721–761 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  119. 112.

    A. Yu. Zaitsev, “Nonstability of the inversion of the Radon transform,” J. Math. Sci. 88, 53–58 (1998).

    Article  Google Scholar 

  120. 113.

    F. Götze and A. Yu. Zaitsev, “Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms,” J. Math. Sci. 176, 162–189 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  121. 114.

    F. Götze and A. Yu. Zaitsev, “Explicit rates of approximation in the CLT for quadratic forms,” Ann. Probab. 42, 354–397 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  122. 115.

    V. Bentkus and F. Götze, “Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces,” Probab. Theory Rel. Fields 109, 367–416 (1997).

    MathSciNet  MATH  Article  Google Scholar 

  123. 116.

    Yu. S. Eliseeva, “Multivariate estimates for the concentration functions of weighted sums of independent, identically distributed random variables,” J. Math. Sci. 204, 78–89 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  124. 117.

    Y. S. Eliseeva and A. Yu. Zaitsev, “Estimates of the concentration functions of weighted sums of independent random variables,” Theory Probab. Appl. 57, 670–678 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  125. 118.

    Yu. S. Eliseeva, F. Götze, and A. Yu. Zaitsev, “Estimates for the concentration functions in the Littlewood–Offord problem,” J. Math. Sci. 206, 146–158 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  126. 119.

    O. Friedland and S. Sodin, “Bounds on the concentration function in terms of Diophantine approximation,” C. R. Math. Acad. Sci. 345, 513–518 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  127. 120.

    M. Rudelson and R. Vershynin, “The Littlewood–Offord problem and invertibility of random matrices,” Adv. Math. 218, 600–633 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  128. 121.

    M. Rudelson and R. Vershynin, “Smallest singular value of a random rectangular matrix,” Comm. Pure Appl. Math. 62, 1707–1739 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  129. 122.

    R. Vershynin, “Invertibility of symmetric random matrices,” Random Struct. Algorithms 44, 135–182 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  130. 123.

    C. G. Esseen, “On the concentration function of a sum of independent random variables,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 9, 290–308 (1968).

    MathSciNet  MATH  Article  Google Scholar 

  131. 124.

    F. Götze and A. Yu. Zaitsev, “Estimates for the rapid decay of concentration functions of n-fold convolutions,” J. Theoret. Probab. 11, 715–731 (1998).

    MathSciNet  MATH  Article  Google Scholar 

  132. 125.

    F. Götze and A. Yu. Zaitsev, “A multiplicative inequality for concentration functions of n-fold convolutions,” in High Dimensional Probability. II, Ed. by. E. Giné, D. Mason, and J. A. Wellner (Birkhäuser, Boston, 2000), in Ser.: Progress in Probability, Vol. 47, pp. 39–47.

    MathSciNet  MATH  Article  Google Scholar 

  133. 126.

    A. Yu. Zaitsev, “On the rate of decay of concentration functions of n-fold convolutions of probability distributions,” Vestnik St. Petersburg. Univ.: Math. 44, 110–114 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  134. 127.

    B. V. Gnedenko, “On the role of the maximal summand in sums of independent random variables,” Ukr. Math. J. 5, 291–298 (1953).

    Google Scholar 

  135. 128.

    V. M. Zolotarev and V. S. Korolyuk, “On a conjecture proposed by B. V. Gnedenko,” Theory Probab. Appl. 6, 431–435 (1961).

    MATH  Article  Google Scholar 

  136. 129.

    A. A. Zinger, “On a problem of B. V. Gnedenko,” Dokl. Akad. Nauk SSSR 162, 1238–1240 (1965).

    MathSciNet  Google Scholar 

  137. 130.

    A. A. Zinger, “On a class of limit distributions for normalized sums of independent random variables,” Theory Probab. Appl. 10, 607–626 (1965).

    MathSciNet  MATH  Article  Google Scholar 

  138. 131.

    Y. V. Linnik, “Linear forms and statistical criteria,” Ukr. Math. J. 5, 247–290 (1953).

    MathSciNet  MATH  Google Scholar 

  139. 132.

    I. Berkes, “Results and problems related to the pointwise central limit theorem,” in Asymptotic Methods in Probability and Statistics (Elsevier, 1998), pp. 59–96.

    Google Scholar 

  140. 133.

    I. A. Ibragimov, “On almost sure versions of limit theorems,” Dokl. Ross. Akad. Nauk 350, 301–303 (1996).

    MathSciNet  Google Scholar 

  141. 134.

    I. A. Ibragimov and M. A. Lifshits, “On almost sure limit theorems,” Theory Probab. Appl. 44, 254–272 (2000).

    MathSciNet  MATH  Article  Google Scholar 

  142. 135.

    M. A. Lifshits, “The almost sure limit theorem for sums of random vectors,” J. Math. Sci. 109, 2166–2178 (2002).

    Article  Google Scholar 

  143. 136.

    I. A. Ibragimov and M. A. Lifshits, “On the convergence of generalized moments in almost sure limit theorems,” Stat. Probab. Lett. 40, 343–351 (1998).

    MathSciNet  MATH  Article  Google Scholar 

  144. 137.

    M. A. Lifshits and E. S. Stankevich, “On the large deviation principle for the almost sure CLT,” Stat. Probab. Lett. 51, 263–267 (2001).

    MathSciNet  MATH  Article  Google Scholar 

  145. 138.

    M. A. Lifshits, “Almost sure limit theorem for martingales,” in Limit Theorems in Probability and Statistics, Balatonlelle, 1999, Ed. by I. Berkes, E. Csáki, M. Csörgo (János Bolyai Math. Soc., Budapest,2002), Vol. 2, pp. 367–390.

    MathSciNet  MATH  Google Scholar 

  146. 139.

    I. Berkes and E. Csáki, “A universal result in almost sure central limit theory,” Stochastic Processes Appl. 94, 105–134 (2001).

    MathSciNet  MATH  Article  Google Scholar 

  147. 140.

    I. Berkes and H. Dehling, “Some limit theorems in log density,” Ann. Probab. 21, 1640–1670 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  148. 141.

    M. K. Heck, “The principle of large deviations for almost everywhere central limit theorem,” Stochastic Processes Appl. 76, 61–75 (1998).

    MathSciNet  MATH  Article  Google Scholar 

  149. 142.

    P. March and T. Seppäläinen, “Large deviations from the almost sure central limit theorem,” J. Theor. Probab. 10, 935–967 (1997).

    MATH  Article  Google Scholar 

  150. 143.

    A. I. Martikainen, “Almost sure central limit theorem without logarithmic sums,” J. Math. Sci. 137, 4549–4554 (2004).

    MATH  Article  Google Scholar 

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Correspondence to M. A. Lifshits.

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Original Russian Text © M.A. Lifshits, Ya.Yu. Nikitin, V.V. Petrov, A.Yu. Zaitsev, A.A. Zinger, 2018, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2018, Vol. 63, No. 2, pp. 231–264.

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Lifshits, M.A., Nikitin, Y.Y., Petrov, V.V. et al. Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables. Vestnik St.Petersb. Univ.Math. 51, 144–163 (2018). https://doi.org/10.3103/S1063454118020115

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Keywords

  • sums of independent random variables
  • central limit theorem
  • law of large numbers
  • law of the iterated logarithm
  • infinitely divisible distributions
  • concentration functions
  • Littlewood–Offord problem
  • empirical measure
  • almost sure limit theorem