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.

“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.

Reviewer: Symon Serbenyuk (Kyiv)

##### Keywords:

limit theorems for sums of dependent variables; Gaussian processes; small deviation probabilities; approximation of processes of growing dimension; functional law of iterated logarithm; sticky particle systems
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\textit{I. A. Ibragimov} et al., Vestn. St. Petersbg. Univ., Math. 51, No. 3, 213--236 (2018; Zbl 1433.01023); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 5(63), No. 3, 367--401 (2018)

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##### References:

[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]. |

[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]. |

[3] | S. N. Bernstein, Collected Works (Nauka, Moscow, 1964), Vol. 4 [in Russian]. · Zbl 0198.50902 |

[4] | N. A. Sapogov, “On singular Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 58, 193-196 (1947). |

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

[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]. |

[7] | N. A. Sapogov, “On multidimensional inhomogeneous Markov chains,” Dokl. Akad. Nauk SSSR (N. S.) 69, 133-135 (1949). |

[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). |

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

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

[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). |

[12] | B. A. Lifshits, “On the central limit theorem for Markov chains,” Theory Probab. Appl. 23, 279-296 (1979). · Zbl 0422.60052 |

[13] | M. Rosenblatt, “A Some limit theorems for stochastic processes stationary in the strict sense,” Dokl. Akad. Nauk SSSR 125, 711-714 (1959). · Zbl 0087.13302 |

[14] | Ibragimov, I. A., Some limit theorems for stochastic processes stationary in the strict sense (1959) · Zbl 0087.13302 |

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

[16] | I. A. Ibragimov, “Some limit theorems for stationary processes,” Theory Probab. Appl. 7, 349-382 (1962). · Zbl 0119.14204 |

[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). · Zbl 0217.49803 |

[18] | I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Nauka, Moscow, 1965; Walters-Noordhoff, Groningen, 1971). · Zbl 0154.42201 |

[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). |

[20] | Yu. A. Davydov, “Mixing conditions for Markov chains,” Theory Probab. Appl. 18, 312-328 (1973). · Zbl 0297.60031 |

[21] | R. C. Bradley, Introduction to Strong Mixing Conditions (Kendrick Press, Heber City, 2007), Vols. 1-3. |

[22] | I. A. Ibragimov, “A note on the central limit theorem for dependent random variables,” Theory Probab. Appl. 20, 135-141 (1975). · Zbl 0335.60023 |

[23] | Yu. A. Davydov, “Convergence of distributions generated by stationary stochastic processes,” Theory Probab. Appl. 13, 691-696 (1968). · Zbl 0181.44101 |

[24] | Yu. A. Davydov, “The invariance principle for stationary processes,” Theory Probab. Appl. 15, 487-498 (1970). · Zbl 0219.60030 |

[25] | V. V. Gorodetskii, “The invariance principle for stationary random fields with a strong mixing condition,” Theory Probab. Appl. 27, 380-385 (1983). · Zbl 0505.60065 |

[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. |

[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). · Zbl 0346.60016 |

[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). · Zbl 0471.60030 |

[29] | V. V. Gorodetskii, “On the convergence rate in the invariance principle for strongly mixing sequences,” Theory Probab. Appl. 28, 816-821 (1984). · Zbl 0544.60038 |

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

[31] | M. Kh. Reznik, “The law of the iterated logarithm for some classes of stationary processes,” Theory Probab. Appl. 13, 606-621 (1968). · Zbl 0281.60022 |

[32] | I. A. Ibragimov, “A central limit theorem for a class of dependent random variables,” Theory Probab. Appl. 8, 83-89 (1963). · Zbl 0123.36103 |

[33] | M. I. Gordin, “On the Central Limit Theorem for stationary processes,” Sov. Math. Dokl. 10, 1174-1176 (1969). · Zbl 0212.50005 |

[34] | Gordin, M. I., Central limit theorem without assumption on the variance finiteness, 173-174 (1973), Vilnius |

[35] | P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application (Academic, New York, 1980). · Zbl 0462.60045 |

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

[37] | M. I. Gordin and B. A. Lifshits, “Central limit theorem for Markov stationary processes,” Dokl. Akad. Nauk SSSR 239, 766-767 (1978). · Zbl 0395.60057 |

[38] | Gordin, M. I.; Lifshits, B. A., The central limit theorem for Markov processes with normal transition operator and applications to random walks on compact Abelian groups (1994), Moscow |

[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. · Zbl 1274.60066 |

[40] | M. Gordin and H. Holzmann, “The central limit theorem for stationary Markov chains under invariant splittings,” Stoch. Dyn. 4 (1), 15-30 (2004). · Zbl 1077.60023 |

[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). · Zbl 1306.60007 |

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

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

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

[45] | M. A. Lifshits, Gaussian Random Functions (TViMS, Kyiv, 1995; Kluwer, Dordrecht, 1995). · Zbl 0832.60002 |

[46] | Dudley, R. M., V. N. Sudakov’s work on expected suprema of Gaussian processes, 37-43 (2016), Cham, Switzerland · Zbl 1355.60047 |

[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). · Zbl 0395.28007 |

[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). · Zbl 0408.60033 |

[49] | B. S. Tsirel’son, “Supplement to an article on the natural modification,” Zap. Nauchn. Semin. LOMI 72, 202-211 (1977). · Zbl 0409.60037 |

[50] | M. Lifshits, Lectures on Gaussian Processes (Springer-Verlag, 2012; Lan’, 2016). · Zbl 1248.60002 |

[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). · Zbl 1366.60025 |

[52] | B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinitedimensional Gaussian location. II,” Theory Probab. Appl. 30, 820-828 (1986). · Zbl 0604.62081 |

[53] | Kabluchko, Z.; Zaporozhets, D., Expected volumes of Gaussian polytopes, external angles, and multiple order statistics (2017) |

[54] | Kabluchko, Z.; Zaporozhets, D., Absorption probabilities for Gaussian polytopes, and regular spherical simplices (2017) |

[55] | V. Strassen, “An invariance principle for the law of the iterated logarithm,” Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, 211-226 (1964). · Zbl 0132.12903 |

[56] | K. Grill, “Exact rate of convergence in Strassen’s law of iterated logarithm,” J. Theor. Probab. 5, 197-205 (1992). · Zbl 0747.60028 |

[57] | Talagrand, M., On the rate of clustering in Strassen’s LIL for Brownian motion, 339-347 (1992), Basel · Zbl 0787.60040 |

[58] | P. Deheuvels and M. Lifshits, “Strassen-type functional laws for strong topologies,” Probab. Theory Rel. Fields 97, 151-167 (1993). · Zbl 0793.60037 |

[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). · Zbl 0840.60027 |

[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). |

[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). |

[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). · Zbl 1018.60030 |

[63] | N. Gorn and M. A. Lifshits, “Chung law and Csáki function,” J. Theor. Probab. 12, 399-420 (1999). · Zbl 0937.60013 |

[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). · Zbl 0489.60043 |

[65] | Lifshits, M. A., Asymptotic behavior of small ball probabilities, 453-468 (1999), Vilnius · Zbl 0994.60017 |

[66] | M. A. Lifshits, “On the lower tail probabilities of some random series,” Ann. Probab. 25, 424-442 (1997). · Zbl 0873.60012 |

[67] | L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables,” J. Math. Sci. 147, 6935-6945 (2007). · Zbl 1137.60013 |

[68] | Dunker, Th.; Lifshits, M. A.; Linde, W., Small deviations of sums of independent variables, 59-74 (1998), Basel · Zbl 0902.60039 |

[69] | L. V. Rozovsky, “On small deviations of series of weighted positive random variables,” J. Math. Sci. 176, 224-231 (2011). · Zbl 1290.60052 |

[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). · Zbl 1284.60088 |

[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). · Zbl 1359.60047 |

[72] | L. V. Rozovsky, “Small deviations of series of independent nonnegative random variables with smooth weights,” Theory Probab. Appl. 58, 121-137 (2014). · Zbl 1321.60069 |

[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). · Zbl 1341.60021 |

[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). · Zbl 1334.60050 |

[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). · Zbl 1388.60089 |

[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). |

[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). · Zbl 1336.60075 |

[78] | M. A. Lifshits and Th. Simon, “Small deviations for fractional stable processes,” Ann. Inst. H. Poincaré 41, 725-752 (2005). · Zbl 1070.60042 |

[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). · Zbl 1192.60055 |

[80] | Lifshits, M. A.; Nazarov, A. I., L2-small deviations for weighted stationary processes (2018) |

[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). · Zbl 1107.60014 |

[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 |

[84] | B. S. Tsirel’son, “Stationary Gaussian processes with a compactly supported correlation function,” J. Math. Sci. 68, 597-603 (1994). · Zbl 0836.60041 |

[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). |

[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). · Zbl 1051.60041 |

[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). · Zbl 1187.60025 |

[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). |

[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). · Zbl 1252.82019 |

[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). · Zbl 1215.60027 |

[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). · Zbl 1078.60028 |

[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). · Zbl 1322.60036 |

[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). · Zbl 1301.60045 |

[96] | Ya. Yu. Nikitin and R. S. Pusev, “Exact L2-small deviation asymptotics for some Brownian functionals,” Theory Probab. Appl. 57, 60-81 (2013). · Zbl 1278.60072 |

[97] | A. I. Nazarov, “On a set of transformations of Gaussian random functions,” Theory Probab. Appl. 54, 203-216 (2010). · Zbl 1214.60011 |

[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). · Zbl 1347.60038 |

[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). |

[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). · Zbl 1334.34186 |

[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 |

[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). · Zbl 1301.60044 |

[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). · Zbl 1136.60024 |

[108] | Rozovsky, L., Small ball probabilities for certain Gaussian fields (2017) |

[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). · Zbl 1052.60027 |

[110] | A. I. Nazarov, “Log-level comparison principle for small ball probabilities,” Stat. Probab. Lett. 79, 481-486 (2009). · Zbl 1166.60310 |

[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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