The Law of the Iterated Logarithm and Probabilities of Moderate Deviations of Sums of Dependent Random Variables
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THE LAW OF THE ITERATED LOGARITHM AND PROBABILITIES OF MODERATE DEVIATIONS OF SUMS OF DEPENDENT RANDOM VARIABLES V. V. Petrov∗
UDC 519.2
Sufficient conditions are obtained for the applicability of one of the classical forms of the law of iterated logarithm to a sequence of random variables without conditions of independence and the existence of any moments. Bibliography: 5 titles.
Let {Xn ; n = 1, 2, . . . } be a sequence of random variables on a certain probability space and {an ; n = 1, 2, . . . } a nondecreasing sequence of positive numbers such that an → ∞ as n → ∞. n Xk . The following condition (the condition D) was introduced in [1]: for any Put Sn = k=1
positive ε and ε0 < ε there exists a positive number γ such that P max Sk > (1 + ε)an ≤ γ P Sn > (1 + ε0 )an 1≤k≤n
(1)
for n large enough. Inequality (1) goes back to the classical Kolmogorov inequality P( max Sk ≥ x) ≤ 2P(Sn ≥ 1≤k≤n
x − (2Bn )1/2 ) for all x, where Sn is the sum of n independent random variables with zero n EXk2 . Sufficient conditions for the inequality expectations and finite variances and Bn = k=1
lim sup Sn /an ≤ 1 a.s.,
(2)
including the condition D, were found in [1]. Moreover, in [1], as well as in the present paper, there are no assumptions of independence of random variables from the original sequence {Xn } or existence of any moments of these random variables. We are interested in conditions sufficient to satisfy the inequality lim sup Sn /Ln ≤ 1 a.s.,
(3)
Ln = (2Bn log log Bn )1/2
(4)
where with a nondecreasing sequence {Bn } of positive numbers, Bn → ∞ as n → ∞, i.e., the conditions sufficient for the classical form of the one-sided law of iterated logarithm. Theorem. Let {Bn ; n = 1, 2, . . . } be a nondecreasing sequence of positive numbers, Bn → ∞ as n → ∞, and let the condition D with an replaced by Ln defined by equality (4), be satisfied. Let 2 (5) P(Sn ≥ xLn ) ≤ M (log(Bn ))−x for every x belonging to a nondegenerate interval (1, 1 + β) and all sufficiently large n, where M is a constant. Then inequality (3) holds. We note that inequality (5) can be written in the following form: P(Zn ≥ xtn ) ≤ M exp{−x2 t2n /2}, ∗
(6)
St. Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 195–198. Original article submitted October 11, 2017. 128 1072-3374/20/2511-0128 ©2020 Springer Science+Business Media, LLC
1/2
where Zn = Sn /Bn and tn = (2 log log Bn )1/2 . Thus, condition (5) gives an estimate of the probabilities of moderate deviations of the sums of dependent random variables. In [2], sufficient conditions are found for inequality (3) with Ln as in (4) to hold for a sequence of independent random variables with finite variances and expectations equal to zero. Moreover, it is assumed there that there is some estimate of the probabilities of moderate 1/2 deviations of the normalized sum Sn /Bn , where Bn = EX12 + · · · + EXn2 . Verification of the fulfillment of estimates of type (6) for the tail probabilities of sums of ra
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