Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables
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Lithuanian Mathematical Journal
Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables Maria Rosaria Formica a,1 , Yuriy Vasil’ovich Kozachenko b, Eugeny Ostrovsky c, and Leonid Sirota c a
Parthenope University of Naples, via Generale Parisi 13,Palazzo Pacanowsky, 80132, Napoli, Italy b Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine c Department of Mathematics and Statistics, Bar-Ilan University, 52900, Ramat Gan, Israel (e-mail: [email protected]; [email protected]; [email protected]; [email protected]) Received April 2, 2019; revised August 16, 2019
Abstract. We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm. Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions. MSC: 60B05,60G50,62E20,46E30 Keywords: array of random variables, tail function, law of iterated logarithm, law of ordinary logarithm, Cramer condition, Orlicz spaces, grand Lebesgue spaces, slowly and regularly varying functions
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Statement of the problem. Notations. Previous results
Let (Ω, B, P) be a nontrivial probability space. Let ξi , i = 1, 2, . . . , be a sequence of centered (i.e., with mean Eξi = 0) independent identically distributed (i.i.d.) random variables (r.v.s) with finite nonzero variance σ 2 := Var(ξi ) = Eξi2 . Denote Sn =
n
ξi
i=1
for n ∈ N. The classical law of iterated logarithm (LIL) due to Hartman and Wintner [20] states that lim √
n→∞ 1
Sn 2n ln ln n
=σ
and
Sn = −σ lim √ n→∞ 2n ln ln n
The author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project “sostegno alla Ricerca individuale”.
c 2020 Springer Science+Business Media, LLC 0363-1672/20/6002-0001
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M.R. Formica et al.
with probability one (a.s.). The more general case of sequences of independent nonidentically distributed r.v.s can be found, for example, in [6, 32, 34, 43, 50] and for martingales, in [19, 33]. For a random variable θ taking values of both signs, the tail function can be defined by Tθ (u) = P |θ| u , u 0, (1.1) or, by the so-called Bernstein version (see [5]), (B)
Tθ
(u) = max P(θ u), P(θ −u) ,
u 0.
For a nonnegative r.v. θ , (1.1) becomes Tθ (u) = P(θ u),
u 0.
For the r.v. def
θ = sup √ n3
Sn , 2n ln ln n
the exponential bound for the corresponding tail function of the form Tθ (u) exp −Cum lnr u , m = const > 0, r = const ∈ R, u e, was first obtained in [29, 33]; see also [34, Chap. 2, Sect. 2.6.]. The situation is quite different if we consider a triangular array instead of the sequenc
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