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Limit theorems on large deviations. (Russian) Zbl 0810.60024

Gamkrelidze, R. V. (ed.) et al., Probability theory - 6. Limit theorems in probability theory. Moskva: Vsesoyuznyj Institut Nauchnoj i Tekhnicheskoj Informatsii, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya. 81, 219-312 (1991).
The article presents a survey of large deviations theorems obtained earlier mainly by the authors and their contributors. More precisely, an application of the method of cumulants for normal approximation is discussed. The first chapter is devoted to fundamental lemmas concerning normal approximation of distributions of random variables under suitable restrictions on their cumulants. Next, on the basis of these results, large deviations theorems for sums of independent random variables are given (Chapter 2). In Chapter 3, centered moments, cumulants of finite vectors and cumulants of sums of mixing sequences are estimated, and then large deviations for distributions of sums of stationary Markov chains under various mixing conditions are investigated. Chapter 4 concerns large deviations theorems for multinomial forms of i.i.d. r.v.’s, multinomial Pitman’s estimators, \(U\)-statistics, multiple Wiener and Poisson integrals, and estimators of the spectral density of stationary random sequences. Finally, in Chapter 5 the method of cumulants to the central limit theorem and asymptotic expansions for sums of general Markov chains are applied. For some results the main ideas of proofs are given, the other proofs are sketched or omitted.
More comprehensive and detailed information can be found in the research papers by R. Rudzkis and the authors [Lith. Math. J. 18, 226-238 (1978), translation from Litov. Mat. Sb. 18, No. 2, 99-116 (1978; Zbl 0405.60025) and ibid. 19, 118-125 (1979), resp. ibid. 19, No. 1, 169-179 (1979; Zbl 0401.60024)], the first author [Litov. Mat. Sb. 21, No. 2, 175-189 (1981; Zbl 0476.60025)], the second author [ibid. 9, 345-362, 635-672 (1969) and 10, 161-169 (1970; Zbl 0203.502)], A. Plikusas [Lith. Math. J. 20, 150-156 (1981), translation from Litov. Mat. Sb. 20, No. 2, 119-128 (1980; Zbl 0446.60017)], R. Bentkus and R. Rudzkis [Litov. Mat. Sb. 20, No. 1, 15-30 (1980; Zbl 0428.60027)] and A. A. Basalykas, A. E. Plikusas and the second author [in: Probability theory and applications, Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986, vol. 1, 629-639 (1987; Zbl 0679.60038)].
For the entire collection see [Zbl 0742.00020].

MSC:

60F10 Large deviations
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60E15 Inequalities; stochastic orderings
60G50 Sums of independent random variables; random walks
60G10 Stationary stochastic processes
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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