On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables

Abstract

This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).

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References

  1. 1.

    A. A. Markov, “Extension of the law of large numbers to dependent quantities,” Izv. Fiz.-Mat. O-va. Kazan. Univ., Ser. 2 15 (4), 135–156 (1906); A. A. Markov, Selected Works (Akad. Nauk SSSR, Leningrad, 1951), pp. 339–361 [in Russian].

    Google Scholar 

  2. 2.

    A. A. Markov, “Investigation of the general case of trials associated into a chain,” Zap. Akad. Nauk (S.-Peterb.) Fiz.-Matem. Otd., Ser. 8 25 (3), 1–33 (1910); A. A. Markov, Selected Works (AN SSSR, Leningrad, (Akad. Nauk SSSR, Leningrad, 1951), pp. 465–507 [in Russian].

    Google Scholar 

  3. 3.

    S. N. Bernstein, Collected Works (Nauka, Moscow, 1964), Vol. 4 [in Russian].

    Google Scholar 

  4. 4.

    N. A. Sapogov, “On singular Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 58, 193–196 (1947).

    MathSciNet  Google Scholar 

  5. 5.

    Yu. V. Linnik, “On the theory of nonuniform Markov chains,” Izv. Akad. Nauk SSSR. Ser. Mat. 13, 65–94 (1949).

    MathSciNet  Google Scholar 

  6. 6.

    Yu. V. Linnik and N. A. Sapogov, “Multivariate integral and local laws for inhomogeneous Markov chains,” Izv. Akad. Nauk SSSR. Ser. Mat. 13, 533–566 (1949) [in Russian].

    Google Scholar 

  7. 7.

    N. A. Sapogov, “On multidimensional inhomogeneous Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 69, 133–135 (1949).

    MathSciNet  Google Scholar 

  8. 8.

    R. L. Dobrushin, “Central limit theorem for nonstationary Markov chains. I,” Theory Probab. Appl. 1, 65–80 (1956); R. L. Dobrushin, “Central limit theorem for nonstationary Markov chains. II,” Theory Probab. Appl. 1, 329–383 (1956).

    MathSciNet  Article  Google Scholar 

  9. 9.

    V. A. Statulyavičus, “On a local limit theorem for inhomogeneous Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 107, 516–519 (1956).

    MathSciNet  Google Scholar 

  10. 10.

    V. A. Statulyavičus, “Asymptotic expansion for inhomogeneous Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 112, 206 (1957).

    MathSciNet  Google Scholar 

  11. 11.

    B. A. Lifshits, “The central limit theorem for sums of random variables connected in a Markov chain,” Dokl. Akad. Nauk SSSR 219, 797–799 (1974).

    MathSciNet  Google Scholar 

  12. 12.

    B. A. Lifshits, “On the central limit theorem for Markov chains,” Theory Probab. Appl. 23, 279–296 (1979).

    MATH  Article  Google Scholar 

  13. 13.

    M. Rosenblatt, “A Some limit theorems for stochastic processes stationary in the strict sense,” Dokl. Akad. Nauk SSSR 125, 711–714 (1959).

    MathSciNet  Google Scholar 

  14. 14.

    I. A. Ibragimov, “Some limit theorems for stochastic processes stationary in the strict sense,” Dokl. Akad. Nauk SSSR 125, 711–714 (1959).

    Google Scholar 

  15. 15.

    I. A. Ibragimov, “Spectral functions of certain classes of stationary Gaussian processes,” Dokl. Akad. Nauk SSSR 137, 1046–1048 (1961).

    MathSciNet  Google Scholar 

  16. 16.

    I. A. Ibragimov, “Some limit theorems for stationary processes,” Theory Probab. Appl. 7, 349–382 (1962).

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    I. A. Ibragimov, “The central limit theorem for sums of functions of independent variables and sums of the form Σf(2kt),” Theory Probab. Appl. 12, 596–607 (1967).

    MATH  Article  Google Scholar 

  18. 18.

    I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Nauka, Moscow, 1965; Walters–Noordhoff, Groningen, 1971).

    Google Scholar 

  19. 19.

    Yu. A. Davydov, “The strong mixing property for Markov chains with a countable number of states,” Dokl. Akad. Nauk SSSR 187, 252–254 (1969).

    MathSciNet  Google Scholar 

  20. 20.

    Yu. A. Davydov, “Mixing conditions for Markov chains,” Theory Probab. Appl. 18, 312–328 (1973).

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    R. C. Bradley, Introduction to Strong Mixing Conditions (Kendrick Press, Heber City, 2007), Vols. 1–3.

  22. 22.

    I. A. Ibragimov, “A note on the central limit theorem for dependent random variables,” Theory Probab. Appl. 20, 135–141 (1975).

    MATH  Article  Google Scholar 

  23. 23.

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

    MATH  Article  Google Scholar 

  24. 24.

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

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    V. V. Gorodetskii, “The invariance principle for stationary random fields with a strong mixing condition,” Theory Probab. Appl. 27, 380–385 (1983).

    MATH  Article  Google Scholar 

  26. 26.

    C. Stein, “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables,” in Proc. Sixth Berkeley Symp. Probab. Stat., Davis, CA, June 21–July 18, 1970 (Univ. of California Press, 1972), Vol. 2, pp. 583–602.

    MathSciNet  Google Scholar 

  27. 27.

    A. N. Tikhomirov, “The rate of convergence in the central limit theorem for weakly dependent variables,” Vestn. Leningr. Univ., Ser. 1: Mat. Mekh. Astron. 7 (4), 158–159 (1976).

    MathSciNet  MATH  Google Scholar 

  28. 28.

    A. N. Tikhomirov, “On the rate of convergence in the central limit theorem for weakly dependent random variables,” Theory Probab. Appl. 25, 790–809 (1981).

    MATH  Article  Google Scholar 

  29. 29.

    V. V. Gorodetskii, “On the convergence rate in the invariance principle for strongly mixing sequences,” Theory Probab. Appl. 28, 816–821 (1984).

    Article  Google Scholar 

  30. 30.

    N. A. Sapogov, “Law of the iterated logarithm for sums of dependent quantities,” Uchen. Zap. Leningr. Univ. Ser. Mat. 137, 160–179 (1950).

    MathSciNet  Google Scholar 

  31. 31.

    M. Kh. Reznik, “The law of the iterated logarithm for some classes of stationary processes,” Theory Probab. Appl. 13, 606–621 (1968).

    MATH  Article  Google Scholar 

  32. 32.

    I. A. Ibragimov, “A central limit theorem for a class of dependent random variables,” Theory Probab. Appl. 8, 83–89 (1963).

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    M. I. Gordin, “On the Central Limit Theorem for stationary processes,” Sov. Math. Dokl. 10, 1174–1176 (1969).

    MATH  Google Scholar 

  34. 34.

    M. I. Gordin, “Central limit theorem without assumption on the variance finiteness,” in Proc. Int. Conf. on Probability Theory and Mathematical Statistics, Vilnius, June 25–30, 1973 (Akad. Nauk. Lit. SSR, Vilnius, 1973), Vol. 1, pp. 173–174.

    Google Scholar 

  35. 35.

    P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application (Academic, New York, 1980).

    Google Scholar 

  36. 36.

    M. I. Gordin and B. A. Lifshits, “Invariance principle for stationary Markov processes,” Theory Probab. Appl. 23, 829–840 (1978).

    Google Scholar 

  37. 37.

    M. I. Gordin and B. A. Lifshits, “Central limit theorem for Markov stationary processes,” Dokl. Akad. Nauk SSSR 239, 766–767 (1978).

    MathSciNet  MATH  Google Scholar 

  38. 38.

    M. I. Gordin and B. A. Lifshits, “The central limit theorem for Markov processes with normal transition operator and applications to random walks on compact Abelian groups,” in A. N. Borodin and I. A. Ibragimov, Limit Theorems for Functionals of Random Walks (Nauka, Moscow, 1994; AMS, Providence, RI, 1995), in Ser: Proceedings of the Steklov Institute of Mathematics.

    Google Scholar 

  39. 39.

    M. Gordin, CLT for Stationary Normal Markov Chains via Generalized Coboundaries. (Springer-Verlag, Berlin, 2015), in Ser.: Springer Proceedings in Mathematics and Statistics. Limit Theorems in Probability, Statistics and Number Theory, Vol. 42.

    Google Scholar 

  40. 40.

    M. Gordin and H. Holzmann, “The central limit theorem for stationary Markov chains under invariant splittings,” Stoch. Dyn. 4 (1), 15–30 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    M. Denker and M. Gordin, “Limit theorems for fon Mises statistics of a measure preserving transformation,” Probab. Theory Rel. Fields 160 (1–2), 1–40 (2014).

    MATH  Article  Google Scholar 

  42. 42.

    V. N. Sudakov, “Gauss and Cauchy measures and ε-entropy,” Dokl. Akad. Nauk SSSR 185, 51–53 (1969).

    MathSciNet  Google Scholar 

  43. 43.

    V. N. Sudakov, “Gaussian random processes, and measures of solid angles in Hilbert space,” Dokl. Akad. Nauk SSSR 197, 43–45 (1971).

    MathSciNet  Google Scholar 

  44. 44.

    V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions,” Proc. Steklov Inst. Math. 141, 1–178 (1979).

    MathSciNet  Google Scholar 

  45. 45.

    M. A. Lifshits, Gaussian Random Functions (TViMS, Kyiv, 1995; Kluwer, Dordrecht, 1995).

    Google Scholar 

  46. 46.

    R. M. Dudley, “V. N. Sudakov’s work on expected suprema of Gaussian processes,” in High Dimensional Probability VII (Birkhäuser, Cham, Switzerland, 2016), in Ser.: Progress in Probability, Vol. 71, pp. 37–43.

    Google Scholar 

  47. 47.

    V. N. Sudakov and B. S. Tsirel’son, “Extremal properties of half-spaces for spherically invariant measures,” J. Sov. Math. 9, 9–18 (1978).

    MATH  Article  Google Scholar 

  48. 48.

    B. S. Tsirel’son, “A natural modification of a random process, and its applications to series of random functions and to Gaussian measures,” Zap. Nauchn. Semin. LOMI 55, 35–63 (1976).

    MathSciNet  Google Scholar 

  49. 49.

    B. S. Tsirel’son, “Supplement to an article on the natural modification,” Zap. Nauchn. Semin. LOMI 72, 202–211 (1977).

    Google Scholar 

  50. 50.

    M. Lifshits, Lectures on Gaussian Processes (Springer-Verlag, 2012; Lan’, 2016).

    Google Scholar 

  51. 51.

    Z. Kabluchko and D. Zaporozhets, “Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls,” Trans. Am. Math. Soc. 368, 8873–8899 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinitedimensional Gaussian location. II,” Theory Probab. Appl. 30, 820–828 (1986).

    MATH  Article  Google Scholar 

  53. 53.

    Z. Kabluchko and D. Zaporozhets, “Expected volumes of Gaussian polytopes, external angles, and multiple order statistics,” Preprint (2017). https://doi.org/arxiv.org/abs/1706.08092. Accessed April 11, 2018.

    Google Scholar 

  54. 54.

    Z. Kabluchko and D. Zaporozhets, “Absorption probabilities for Gaussian polytopes, and regular spherical simplices,” Preprint (2017). https://doi.org/arxiv.org/abs/1704.04968. Accessed April 11, 2018.

    Google Scholar 

  55. 55.

    V. Strassen, “An invariance principle for the law of the iterated logarithm,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, 211–226 (1964).

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    K. Grill, “Exact rate of convergence in Strassen’s law of iterated logarithm,” J. Theor. Probab. 5, 197–205 (1992).

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    M. Talagrand, “On the rate of clustering in Strassen’s LIL for Brownian motion,” in Proc. 8th Int. Conf. Probability in Banach Spaces 8, Brunswick, ME, 1991 (Birkhäuser, Basel, 1992), pp. 339–347.

    Google Scholar 

  58. 58.

    P. Deheuvels and M. Lifshits, “Strassen-type functional laws for strong topologies,” Probab. Theory Rel. Fields 97, 151–167 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    P. Deheuvels and M. Lifshits, “Necessary and sufficient condition for the Strassen law of the iterated logarithm in non-uniform topologies,” Ann. Probab. 22, 1838–1856 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    A. V. Bulinskii and M. A. Lifshits, “The best rate of convergence in the Strassen law for random broken lines,” Moscow Univ. Math. Bull. 50 (5), 31–36 (1996).

    MathSciNet  Google Scholar 

  61. 61.

    A. V. Bulinskii and M. A. Lifshits, “Estimates for the rate of convergence in the Strassen law for random broken lines,” J. Math. Sci. 93, 287–293 (1999).

    MathSciNet  Article  Google Scholar 

  62. 62.

    Ph. Berthet and M. A. Lifshits, “Some exact rates in the functional law of the iterated logarithm,” Ann. Inst. H. Poincaré 38, 811–824 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    N. Gorn and M. A. Lifshits, “Chung law and Csáki function,” J. Theor. Probab. 12, 399–420 (1999).

    MATH  Article  Google Scholar 

  64. 64.

    I. A. Ibragimov, “On the probability that a Gaussian vector with values in a Hilbert space hits a sphere of small radius,” J. Sov. Math. 20, 2164–2174 (1982).

    MATH  Article  Google Scholar 

  65. 65.

    M. A. Lifshits, “Asymptotic behavior of small ball probabilities,” in Proc. VII International Vilnius Conf. on Probability Theory and Mathematical Statistics, Vilnius, 1998 (VSP/TEV, Vilnius, 1999), pp. 453–468.

    Google Scholar 

  66. 66.

    M. A. Lifshits, “On the lower tail probabilities of some random series,” Ann. Probab. 25, 424–442 (1997).

    MathSciNet  MATH  Article  Google Scholar 

  67. 67.

    L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables,” J. Math. Sci. 147, 6935–6945 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  68. 68.

    Th. Dunker, M. A. Lifshits, and W. Linde, “Small deviations of sums of independent variables,” in Proc. Conf. High Dimensional Probability, Oberwolfach, Germany, August 1996 (Birkhäuser, Basel, 1998), in Ser.: Progress in Probability, Vol. 43, pp. 59–74.

    Google Scholar 

  69. 69.

    L. V. Rozovsky, “On small deviations of series of weighted positive random variables,” J. Math. Sci. 176, 224–231 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables whose density has a power decay at zero,” J. Math. Sci. 188, 748–752 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  71. 71.

    L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables with a distribution that slowly varies at zero,” J. Math. Sci. 204, 155–164 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    L. V. Rozovsky, “Small deviations of series of independent nonnegative random variables with smooth weights,” Theory Probab. Appl. 58, 121–137 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  73. 73.

    L. V. Rozovsky, “Probabilities of small deviations of a weighted sum of independent random variables with a common distribution that decreases at zero not faster than a power,” J. Math. Sci. 214, 540–545 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  74. 74.

    L. V. Rozovsky, “Small deviations of probabilities for weighted sum of independent positive random variables with a common distribution that decreases at zero not faster than a power,” Theory Probab. Appl. 60, 142–150 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    L. V. Rozovsky, “Small deviation probabilities for a sum of independent positive random variables whose general distribution decreases at zero no faster than a power,” J. Math. Sci. 229, 767–771 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  76. 76.

    L. V. Rozovsky, “Small deviation probabilities of a weighted sum of independent random variables with a common distribution having a power decrease in zero under minimal moment assumptions,” Teor. Veroyatn. Ee Primen. 62, 610–616 (2017).

    Article  Google Scholar 

  77. 77.

    S. Y. Hong, M. A. Lifshits, and A. I. Nazarov, “Small deviations in L2-norm for Gaussian dependent sequences,” Electronic Comm. Probab. 21 (41), 1–9 (2016).

    MATH  Google Scholar 

  78. 78.

    M. A. Lifshits and Th. Simon, “Small deviations for fractional stable processes,” Ann. Inst. H. Poincaré 41, 725–752 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  79. 79.

    F. Aurzada, I. A. Ibragimov, M. A. Lifshits, and H. van Zanten, “Small deviations of smooth stationary Gaussian processes,” Theory Probab. Appl. 53, 697–707 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  80. 80.

    M. A. Lifshits and A. I. Nazarov, “L2-small deviations for weighted stationary processes,” Preprint (2018). https://doi.org/arxiv.org/abs/1705.00422. Accessed April 11, 2018.

    Google Scholar 

  81. 81.

    A. I. Nazarov and Ya. Yu. Nikitin, “Logarithmic L2-small ball asymptotics for some fractional Gaussian processes,” Theory Probab. Appl. 49, 645–658 (2004).

    MATH  Article  Google Scholar 

  82. 82.

    M. A. Lifshits and W. Linde, “Approximation and entropy numbers of Volterra operators with application to Brownian motion,” Memoirs Am. Math. Soc. 157 (745), 1–87 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  83. 83.

    M. A. Lifshits and W. Linde, “Small deviations of weighted fractional processes and average nonlinear approximation,” Trans. Am. Math. Soc. 357, 2059–2079 (2005).

    MATH  Article  Google Scholar 

  84. 84.

    B. S. Tsirel’son, “Stationary Gaussian processes with a compactly supported correlation function,” J. Math. Sci. 68, 597–603 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  85. 85.

    A. I. Nazarov, “On the sharp constant in the small ball asymptotics of some Gaussian processes under L2-norm,” J. Math. Sci. 117, 4185–4210 (2003).

    MathSciNet  Article  Google Scholar 

  86. 86.

    A. I. Nazarov and Ya. Yu. Nikitin, “Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems,” Probab. Theory Relat. Fields 129, 469–494 (2004).

    MATH  Article  Google Scholar 

  87. 87.

    A. I. Nazarov, “Exact L2-small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems,” J. Theoret. Probab. 22, 640–665 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  88. 88.

    A. I. Nazarov and R. S. Pusev, “Exact small deviation asymptotics in L2-norm for some weighted Gaussian processes,” J. Math. Sci. 163, 409–429 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  89. 89.

    Ya. Yu. Nikitin and E. Orsingher, “Exact small deviation asymptotics for the Slepian and Watson processes,” J. Math. Sci. 137, 4555–4560 (2006).

    MathSciNet  Article  Google Scholar 

  90. 90.

    P. A. Kharinski and Ya. Yu. Nikitin, “Sharp small deviation asymptotics in L2-norm for a class of Gaussian processes,” J. Math. Sci. 133, 1328–1332 (2006).

    MathSciNet  Article  Google Scholar 

  91. 91.

    R. S. Pusev, “Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm,” Theor. Math. Phys. 165, 1348–1357 (2010).

    MATH  Article  Google Scholar 

  92. 92.

    R. S. Pusev, “Asymptotics of small deviations of Matérn processes with respect to a weighted quadratic norm,” Theory Probab. Appl. 55, 164–172 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  93. 93.

    L. Beghin, Ya. Yu. Nikitin, and E. Orsingher, “Exact small ball constants for some Gaussian processes under L2-norm,” J. Math. Sci. 128, 2493–2502 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  94. 94.

    A. A. Kirichenko and Ya. Yu. Nikitin, “Precise small deviations in L2 of some Gaussian processes appearing in the regression context,” Cent. Eur. J. Math. 12, 1674–1686 (2014).

    MathSciNet  MATH  Google Scholar 

  95. 95.

    A. I. Nazarov and R. S. Pusev, “Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted L2-norms,” St. Petersburg Math. J. 25, 455–466 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  96. 96.

    Ya. Yu. Nikitin and R. S. Pusev, “Exact L2-small deviation asymptotics for some Brownian functionals,” Theory Probab. Appl. 57, 60–81 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  97. 97.

    A. I. Nazarov, “On a set of transformations of Gaussian random functions,” Theory Probab. Appl. 54, 203–216 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  98. 98.

    A. I. Nazarov and Yu. P. Petrova, “The small ball asymptotics in Hilbert norm for the Kac–Kiefer–Wolfowitz processes,” Theory Probab. Appl. 60, 460–480 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  99. 99.

    A. I. Nazarov, “Logarithmic L2-small ball asymptotics with respect to self-similar measure for some Gaussian random processes,” J. Math. Sci. 133, 1314–1327 (2006).

    MathSciNet  Article  Google Scholar 

  100. 100.

    N. V. Rastegaev, “On spectral asymptotics of the Neumann problem for the Sturm–Liouville equation with self-similar weight of generalized Cantor type,” J. Math. Sci. 210, 814–821 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  101. 101.

    N. V. Rastegaev, “On the spectrum of the Sturm–Liouville problem with arithmetically selfsimilar weight,” Preprint of St. Petersburg Math. Soc. (2017). https://doi.org/www.mathsoc.spb.ru/preprint/2017/17-06.pdf. Accessed April 11, 2018.

    Google Scholar 

  102. 102.

    M. A. Lifshits, W. Linde, and Z. Shi, “Small deviations of Gaussian random fields in Lq-spaces,” Electron. J. Probab. 11 (46), 1204–1223 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  103. 103.

    M. A. Lifshits, W. Linde, and Z. Shi, “Small deviations of Riemann–Liouville processes in Lq-norms with respect to fractal measures,” Proc. Lond. Math. Soc. 92 (1), 224–250 (2006).

    MATH  Article  Google Scholar 

  104. 104.

    A. I. Nazarov and I. A. Sheipak, “Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes,” Bull. Lond. Math. Soc. 44 (1), 12–24 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  105. 105.

    N. V. Rastegaev, “On the spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics,” Preprint of St. Petersburg Math. Soc. (2017); Algebra i Analiz 29 (6), 197–229 (2017). https://doi.org/www.mathsoc.spb.ru/preprint/2017/17-04.pdf. Accessed April 11, 2018.

    Google Scholar 

  106. 106.

    A. I. Karol’ and A. I. Nazarov, “Small ball probabilities for smooth Gaussian fields and tensor products of compact operators,” Math. Nachr. 287, 595–609 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  107. 107.

    A. I. Karol’, A. I. Nazarov, and Ya. Yu. Nikitin, “Small ball probabilities for Gaussian random fields and tensor products of compact operators,” Trans. Am. Math. Soc. 360, 1443–1474 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  108. 108.

    L. Rozovsky, “Small ball probabilities for certain Gaussian fields,” Preprint (2017). https://doi.org/arxiv.org/abs/1705.05001. Accessed April 11, 2018.

    Google Scholar 

  109. 109.

    J. A. Fill and F. Torcaso, “Asymptotic analysis via Mellin transforms for small deviations in L2-norm of integrated Brownian sheets,” Probab. Theory Relat. Fields 130, 259–288 (2003).

    MATH  Google Scholar 

  110. 110.

    A. I. Nazarov, “Log-level comparison principle for small ball probabilities,” Stat. Probab. Lett. 79, 481–486 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  111. 111.

    F. Gao and W. V. Li, “Logarithmic level comparison for small deviation probabilities,” J. Theor. Probab. 20, 1–23 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  112. 112.

    P. Deheuvels and M. A. Lifshits, “Probabilities of hitting shifted small balls by a centered Poisson process,” J. Math. Sci. 118, 5541–5554 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  113. 113.

    E. Yu. Shmileva, “Small ball probabilities for a centered Poisson process of high intensity,” J. Math. Sci. 128, 2656–2668 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  114. 114.

    E. Yu. Shmileva, “Small ball probabilities for jump Lévy processes from the Wiener domain of attraction,” Stat. Probab. Lett. 76 (17), 1873–1881 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  115. 115.

    A. N. Frolov, “Limit theorems for small deviation probabilities of some iterated stochastic processes,” J. Math. Sci. 188, 761–768 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  116. 116.

    F. Aurzada and M. A. Lifshits, “On the small deviation problem for some iterated processes,” Electron. J. Probab. 14 (68), 1992–2010 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  117. 117.

    A. N. Frolov, “Small deviations of iterated processes in the space of trajectories,” Cent. Eur. J. Math. 11, 2089–2098 (2013).

    MathSciNet  MATH  Google Scholar 

  118. 118.

    A. I. Martikainen, A. N. Frolov, and J. Steinebach, “On probabilities of small deviations for compound renewal processes,” Theory Probab. Appl. 52, 328–337 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  119. 119.

    F. Aurzada, M. A. Lifshits, and W. Linde, “Small deviations of stable processes and entropy of associated random operators,” Bernoulli 15, 1305–1334 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  120. 120.

    F. Aurzada and M. A. Lifshits, “Small deviations of sums of correlated stationary Gaussian sequences,” Theory Probab. Appl. 61, 540–568 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  121. 121.

    S. Dereich and M. A. Lifshits, “Probabilities of randomly centered small balls and quantization in Banach spaces,” Ann. Probab. 33, 1397–1421 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  122. 122.

    E. Novak, H. Wo’zniakowski, Tractability of Multivariate Problems, Vols. I–III (European Mathematical Society, Zürich, 2008, 2010, 2012).

    Google Scholar 

  123. 123.

    M. A. Lifshits, A. Papageorgiou, and H. Woźniakowski, “Tractability of multi-parametric Euler and Wiener integrated processes,” Probab. Math. Stat. 32, 131–165 (2012).

    MathSciNet  MATH  Google Scholar 

  124. 124.

    M. A. Lifshits, A. Papageorgiou, and H. Woźniakowski, “Average case tractability of nonhomogeneous tensor products problems,” J. Complexity 28, 539–561 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  125. 125.

    M. A. Lifshits and E. V. Tulyakova, “Curse of dimensionality in approximation of random fields,” Probab. Math. Stat. 26, 97–112 (2006).

    MathSciNet  MATH  Google Scholar 

  126. 126.

    N. A. Serdyukova, “Dependence on the dimension for complexity of approximation of random fields,” Theory Probab. Appl. 54, 272–284 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  127. 127.

    A. A. Khartov, “Average approximation of tensor product-type random fields of increasing dimension,” J. Math. Sci. 188, 769–782 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  128. 128.

    A. A. Khartov, “Approximation in probability of tensor product-type random fields of increasing parametric dimension,” J. Math. Sci. 204, 165–179 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  129. 129.

    A. A. Khartov, “Approximation complexity of tensor product-type random fields with heavy spectrum,” Vestn. St. Petersburg Univ.: Math. 46, 98–101 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  130. 130.

    A. A. Khartov, “Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements,” J. Complexity 31, 835–866 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  131. 131.

    A. A. Khartov, “A simplified criterion for quasi-polynomial tractability of approximation of random elements and its applications,” J. Complexity 34, 30–41 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  132. 132.

    V. V. Vysotsky, “On energy and clusters in stochastic systems of sticky gravitating particles,” Theor. Probab. Appl. 50, 265–283 (2006).

    MathSciNet  Article  Google Scholar 

  133. 133.

    V. F. Zakharova, “Aggregation rates in one-dimensional stochastic gas model with finite polynomial moments of particle speeds,” J. Math. Sci. 152, 885–896.

  134. 134.

    L. V. Kuoza and M. A. Lifshits, “Aggregation in one-dimensional gas model with stable initial data,” J. Math. Sci. 133, 1298–1307 (2006).

    MathSciNet  Article  Google Scholar 

  135. 135.

    M. A. Lifshits and Z. Shi, “Aggregation rates in one-dimensional stochastic systems with adhesion and gravitation,” Ann. Probab. 33, 53–81 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  136. 136.

    M. A. Lifshits and Z. Shi, “Functional large deviations in Burgers particle systems,” Comm. Pure Appl. Math. 60, 41–66 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  137. 137.

    V. V. Vysotsky, “Clustering in a stochastic model of one-dimensional gas,” Ann. Appl. Probab. 18, 1026–1058 (2008).

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to I. A. Ibragimov.

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Original Russian Text © I.A. Ibragimov, M.A. Lifshits, A.I. Nazarov, D.N. Zaporozhets, 2018, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2018, Vol. 63, No. 3, pp. 367–401.

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Ibragimov, I.A., Lifshits, M.A., Nazarov, A.I. et al. On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables. Vestnik St.Petersb. Univ.Math. 51, 213–236 (2018). https://doi.org/10.3103/S1063454118030123

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Keywords

  • limit theorem for sums of dependent variables
  • Gaussian processes
  • small deviation probabilities
  • approximation of processes of growing dimension
  • functional law of iterated logarithm
  • sticky particle systems