Equivalence between tails, Grand Lebesgue Spaces and Orlicz norms for random variables without Kramer's condition
DOI:
https://doi.org/10.17721/1812-5409.2018/4.3Abstract
We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between the tail behavior, moments (Grand Lebesgue Spaces) norm and Orlicz’s norm for random variables (r.v.), which does not satisfy in general case the Kramer’s condition.
Key words: Random variable and random vector (r.v.), saddle-point method, tail of distribution, moments, Lebesgue-Riesz, Orlicz and Grand Lebesgue Spaces (GLS); slowly varying functions, Young-Fenchel transform, theorem and inequality of Fenchel-Moreau, Young-Orlicz function, Markov-Chernov’s estimate, non-asymptotical estimates, Kramer’s condition.
Pages of the article in the issue: 20-29
Language of the article: English
References
BENNET C. (1988) Interpolation of operators, Orlando, Academic Press Inc.
BULDYGIN, V., KOZACHENKO, YU. (2000) Metric characterization of random variables and random processes, Amer. Math. Soc., Providence, RI.
FIORENZA A. (2000) Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51, 131 - 148.
FIORENZA A AND KARADZHOV G.E. (2005) Grand and small Lebesgue spaces and their analogs. Consiglio Nationale Delle Ricerche, Instituto per le Applicazioni del Calcoto Mauro PiconeвЂ´к, Sezione di Napoli, Rapporto tecnico 272/03.
IWANIEC T. AND SBORDONE C. (1992) On the integrability of the Jacobian under minimal hypotheses. Arch. Rat.Mech. Anal., 119, 129-143.
KOZACHENKO YU. V., OSTROVSKY E.I. (1985) The Banach Spaces of random Variables of subgaussian Type. Theory of Probab. and Math. Stat. (in Russian). Kiev, 32, 43-57.
KOZACHENKO YU.V., OSTROVSKY E., SIROTA L. (2017) Relations between exponential tails, moments and moment generating functions for random variables and vectors. arXiv:1701.01901v1 [math.FA], 8 Jan 2017.
OSTROVSKY E.I. (1999). Exponential estimations for Random Fields and its applications, (in Russian), Moscow-Obninsk, OINPE.
OSTROVSKY E. AND SIROTA L. (2015) Vector rearrangement invariant Banach spaces of random variables with exponential decreasing tails of distributions. arXiv:1510.04182v1 [math.PR], 14 Oct 2015.
OSTROVSKY E. AND SIROTA L. (2015) Non-asymptotical sharp exponential estimates for maximum distribution of discontinuous random fields. arXiv:1510.08945v1 [math.PR], 30 Oct 2015.
OSTROVSKY E.I. (1980) About supports of probability measures in separable Banach spaces. Soviet Math., Doklady, V. 255, NO 6, p. 836-838, (in Russian).
OSTROVSKY E. AND SIROTA L. (2015) Criterion for convergence almost everywhere, with applications. arXiv:1507.04020v1 [math.FA], 14 Jul 2015.
OSTROVSKY E. AND SIROTA L. (2008) Schl¨omilch and Bell series for Bessels functions, with probabilistic applications. arXiv:0804.0089v1 [math.CV], 1 Apr 2008.
OSTROVSKY E. AND SIROTA L. (2014) Sharp moment estimates for polynomial martingales. arXiv:1410.0739v1 [math.PR], 3 Oct 2014.
OSTROVSKY E., ROGOVER E. (2008) Exact exponential bounds for the random field maximum distribution via the majorizing measures (generic chaining). arXiv:0802.0349v1 [math.PR], 4 Feb 2008.
OSTROVSKY E. AND SIROTA L. (2015) Entropy and Grand Lebesgue Spaces approach for the problem of Prokhorov - Skorokhod continuity of discontinuous random fields. arXiv:1512.01909v1 [math.Pr], 7 Dec 2015.
OSTROVSKY E. AND SIROTA L. (2015) Fundamental function for Grand Lebesgue Spaces. arXiv:1509.03644v1 [math.FA], 11 Sep 2015.
SENETA E. (1976) Regularly Varying Functions. Lectures Notes in Mathematics, 508.
VASYLYK O.I., KOZACHENKO YU.V., YAMNENKOR.E. (2008) ϕ − subgaussian random processes. Monograph, Kyiv, (in Ukrainian).
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
