Central limit theorem and exponential tail estimations in hybrid Lebesgue-continuous spaces

Abstract

We study the Central Limit Theorem (CLT) in the so-called hybrid Lebesguecontinuous spaces and tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces.

Exact exponential tail estimations in the Law of Iterated Logarithm for Bochner's mixed Lebesgue spaces

“…C. The proposition of theorem 4.1. follows from the main result of article G.Pisier [40]; see also [20], [63], [64].…”

“…The method presented here may be used by investigation of the Law of Iterated Logarithm as well as Central Limit Theorem in the so-called mixed Sobolev's spaces W A p , see e.g. [61], [63].…”

Banach spaces characterization of random vectors with exponential decreasing tails of distribution

“…Vector notations: we obtain some generalization of the known one-dimensional Grand Lebesgue Space (GLS) norm, see [14], [7]- [9], [11]- [12], [16], chapter 1. These multivariate generalization of a form ||ξ||Gψ Φ based in turn on the theory of the so-called mixed (anisotropic) Lebesgue-Riesz spaces [2], chapters 1,2; appears at first perhaps in the authors preprints [21]- [22].…”

Stochastic fields with paths in arbitrary rearrangement invariant spaces

“…Many sufficient conditions for the equality P(ξ(•) ∈ B) = 1 for different separable Banach spaces B are obtained in [26], [19], [31], [5], [6], [29], [30], [32], [9] - [12]. The case of rearrangement invariant spaces, especially ones exponential type Orlicz's spaces, is considered in the articles [13], [14], [15].…”

“…CLT in another separable Banach spaces is investigated, e.g. in [17], [21], [23], [19], [31], [29], [30], [32], [33]. The article [14] is devoted to the CLT in the exponential Orlicz space, more exactly, to the CLT in some separable subspace of the exponential type Orlicz space.…”