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On the history of St. Petersburg school of probability and mathematical statistics. II: Random processes and dependent variables. (English. Russian original) Zbl 1433.01023
Vestn. St. Petersbg. Univ., Math. 51, No. 3, 213-236 (2018); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 5(63), No. 3, 367-401 (2018).
The authors present the following:
“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).”
The first section of this survey is devoted to limit theorems for development random variables. The authors consider the following items:
– Inhomogeneous Markov chains
– Bernstein’s method (this technique is a general method for studying limit distributions for sums of dependent random variables, generally speaking, unrelated to Markov dependence)
– Processes generated by processes with mixing
– The invariance principle (functional limit theorems)
– Rate of convergence
– The law of the iterated logarithm
– The martingale-approximation method (the Gordin method).
The second section deals with Gaussian random processes. The authors present the following items of this topic noting a fact that several fundamental results of the theory of Gaussian processes were obtained in Leningrad:
– Classical results
– Connection with intrinsic volumes.
The third section of this paper is on the functional law of the iterated logarithm (FLIL).
The fourth section “Probabilities of small deviations” consists of the following items:
– Sums of independent variables
– Riemann-Liouville processes
– Gaussian stationary processes
– Green’s Gaussian processes
– Other results related to small deviations.
The fifth and sixth sections are devoted to approximation of random fields of growing dimension and to stochastic systems of sticky particles, respectively.
MSC:
01A72 Schools of mathematics
60-03 History of probability theory
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References:
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[78] M. A. Lifshits and Th. Simon, “Small deviations for fractional stable processes,” Ann. Inst. H. Poincaré 41, 725-752 (2005). · Zbl 1070.60042
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[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). · Zbl 0999.47034
[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). · Zbl 1068.60054
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[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). · Zbl 1288.60045
[89] Ya. Yu. Nikitin and E. Orsingher, “Exact small deviation asymptotics for the Slepian and Watson processes,” J. Math. Sci. 137, 4555-4560 (2006).
[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).
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[101] Rastegaev, N. V., On the spectrum of the Sturm-Liouville problem with arithmetically selfsimilar weight (2017)
[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). · Zbl 1127.60030
[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). · Zbl 1090.60037
[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). · Zbl 1244.60040
[105] Rastegaev, N. V., On the spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics (2017) · Zbl 1435.47030
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[111] F. Gao and W. V. Li, “Logarithmic level comparison for small deviation probabilities,” J. Theor. Probab. 20, 1-23 (2007). · Zbl 1112.60028
[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). · Zbl 1077.60034
[113] E. Yu. Shmileva, “Small ball probabilities for a centered Poisson process of high intensity,” J. Math. Sci. 128, 2656-2668 (2005). · Zbl 1074.60059
[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). · Zbl 1104.60315
[115] A. N. Frolov, “Limit theorems for small deviation probabilities of some iterated stochastic processes,” J. Math. Sci. 188, 761-768 (2013). · Zbl 1282.60039
[116] F. Aurzada and M. A. Lifshits, “On the small deviation problem for some iterated processes,” Electron. J. Probab. 14 (68), 1992-2010 (2009). · Zbl 1190.60016
[117] A. N. Frolov, “Small deviations of iterated processes in the space of trajectories,” Cent. Eur. J. Math. 11, 2089-2098 (2013). · Zbl 1297.60017
[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). · Zbl 1154.60071
[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). · Zbl 1214.60019
[120] F. Aurzada and M. A. Lifshits, “Small deviations of sums of correlated stationary Gaussian sequences,” Theory Probab. Appl. 61, 540-568 (2017). · Zbl 1377.60048
[121] S. Dereich and M. A. Lifshits, “Probabilities of randomly centered small balls and quantization in Banach spaces,” Ann. Probab. 33, 1397-1421 (2005). · Zbl 1078.60029
[122] E. Novak, H. Wo’zniakowski, Tractability of Multivariate Problems, Vols. I-III (European Mathematical Society, Zürich, 2008, 2010, 2012). · Zbl 1156.65001
[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). · Zbl 1254.65016
[124] M. A. Lifshits, A. Papageorgiou, and H. Woźniakowski, “Average case tractability of nonhomogeneous tensor products problems,” J. Complexity 28, 539-561 (2012). · Zbl 1262.65212
[125] M. A. Lifshits and E. V. Tulyakova, “Curse of dimensionality in approximation of random fields,” Probab. Math. Stat. 26, 97-112 (2006). · Zbl 1113.41038
[126] N. A. Serdyukova, “Dependence on the dimension for complexity of approximation of random fields,” Theory Probab. Appl. 54, 272-284 (2010). · Zbl 1214.60021
[127] A. A. Khartov, “Average approximation of tensor product-type random fields of increasing dimension,” J. Math. Sci. 188, 769-782 (2013). · Zbl 1272.60029
[128] A. A. Khartov, “Approximation in probability of tensor product-type random fields of increasing parametric dimension,” J. Math. Sci. 204, 165-179 (2015). · Zbl 1359.60065
[129] A. A. Khartov, “Approximation complexity of tensor product-type random fields with heavy spectrum,” Vestn. St. Petersburg Univ.: Math. 46, 98-101 (2013). · Zbl 1309.60051
[130] A. A. Khartov, “Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements,” J. Complexity 31, 835-866 (2015). · Zbl 1333.60108
[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). · Zbl 1416.65582
[132] V. V. Vysotsky, “On energy and clusters in stochastic systems of sticky gravitating particles,” Theor. Probab. Appl. 50, 265-283 (2006). · Zbl 1100.82018
[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. · Zbl 1288.82051
[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).
[135] M. A. Lifshits and Z. Shi, “Aggregation rates in one-dimensional stochastic systems with adhesion and gravitation,” Ann. Probab. 33, 53-81 (2005). · Zbl 1096.60041
[136] M. A. Lifshits and Z. Shi, “Functional large deviations in Burgers particle systems,” Comm. Pure Appl. Math. 60, 41-66 (2007). · Zbl 1121.35146
[137] V. V. Vysotsky, “Clustering in a stochastic model of one-dimensional gas,” Ann. Appl. Probab. 18, 1026-1058 (2008). · Zbl 1141.60068
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