On norms in some class of exponential type Orlicz spaces of random variables
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On norms in some class of exponential type Orlicz spaces of random variables Krzysztof Zajkowski1 Received: 26 February 2019 / Accepted: 12 December 2019 © Springer Nature Switzerland AG 2019
Abstract A new characterization of the exponential type Orlicz spaces generated by the functions exp(|x| p ) − 1 ( p ≥ 1) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with the Luxemburg norms. On the example of Hoeffding’s inequality we present some application of these norms in a probabilistic context. Keywords Orlicz spaces of exponential type · Luxemburg norms · Convex conjugates · Hoeffding inequality Mathematics Subject Classification 46E30 · 60E15
1 Introduction The most important class of exponential type Orlicz spaces form spaces generated by the functions ψ p (x) = exp(|x| p ) − 1 ( p ≥ 1). In probability theory, for p = 1, we have the space of sub-exponential random variables and, for p = 2, the space of sub-gaussian random variables. Spaces of this type also appear naturally in asymptotic geometric analysis (see the monograph of Arstein-Avidan et al. [1]), especially, in the context of concentration of measures (see [1, Par. 3.5.2]) and [6], for instance), convex geometry (see, e.g., [3,9,11]) and also in information-based complexity (see Hinrichs et al. [7]). Let us recall that in probability theory, for any p ≥ 1, the Orlicz space L ψ p consists of all random variables X on a probability space for which the Luxemburg norm X ψ p := inf K > 0 : E exp(|X /K | p ) ≤ 2
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Krzysztof Zajkowski [email protected] Institute of Mathematics, University of Bialystok, Ciolkowskiego 1M, 15-245 Białystok, Poland
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K. Zajkowski
is finite (see Luxemburg [13]). Let us note that L ψ p1 ⊂ L ψ p2 , if p1 ≥ p2 , and moreover L ∞ ⊂ L ψ p ⊂ L r ( p, r ≥ 1), where L ∞ , L r denote the classical Lebesgue spaces. In other words, the spaces L ψ p form an increasing family, by decreasing p, smaller than all of L p -spaces but larger than the space of bounded random variables L ∞ . Estimates of the ψα -norm play often an important role in solving problems of classical probability theory and asymptotic geometric analysis. This can be quite challenging and delicate task, and therefore any alternative form of this norm may be useful. It is known an equivalent expression of the ψ p -norms in terms of the classical Lebesgue norms as supr ≥ p r −1/ p X L r ; see [1, Lem. 3.5.5]. Some modification of the ψα -norm one can find in Dick et al. [5]. In this paper I would like to present an equivalent form of the ψ p -norms on spaces of centered random variables. Let us recall that centered sub-gaussian random variables have another important classical characterization (see Kahane [10]): a random variable X is sub-gaussian if there exists a positive constant K such that E exp(t X ) ≤ exp(K 2 t 2 /2) for all t ∈ R. In other words when the moment generating function of X is majorized by the moment generating function of a centered Gaussian variable g
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