NDLI: Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications

Content Provider	Springer Nature Link
Author	Rama Murthy, Kavi Rajput, Balram S.
Copyright Year	2007
Abstract	Let ( $$\mathbb{B}$$ , $$\\|\cdot\\|$$ ) be a separable Banach space and let $$\mathcal{M}$$ be a class of probability measures on $$\mathbb{B}$$ , and let $$\bar{\mu}$$ denote the symmetrization of $$\mu\in\mathcal{M}$$ . We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of $$\\|\cdot\\|$$ relative to μ and $$\bar{\mu}$$ , $$\mu\in\mathcal{M}$$ ) for the “uniform comparison” of the μ and $$\bar{\mu}$$ measure of the complements of the closed balls of $$\mathbb{B}$$ centered at zero, for every $$\mu\in\mathcal{M}$$ . As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities, the Kwapień-Contraction Inequality, and a part of the Itô–Nisio Theorem) that are valid for the symmetric (but not for the general non-symmetric) independent $$\mathbb{B}$$ -valued random vectors do indeed hold for the independent random vectors whose laws belong to any $$\mathcal{M}$$ which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe form a valuable part of the paper.
Ending Page	105
Page Count	19
Starting Page	87
File Format	PDF
ISSN	08949840
e-ISSN	15729230
Journal	Journal of Theoretical Probability
Issue Number	1
Volume Number	20
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2007-02-06
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	semistable Tail probability stable Stationary processes Prediction theory Probability Theory and Stochastic Processes Infinitely divisible distributions; stable distributions log concave probability measures Prediction; filtering Statistics
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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