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Moment inequalities for nonnegative random variables

Castañeda-Leyva Netzahualcóyotl, Rodríguez-Narciso Silvia, Hernández-Quintero Angélika, Pérez Aroldo

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

41--61

We give reciprocal versions of the Sclove et al. and Feller inequalities for moments of nonnegative random variables. Our results apply to any nonnegative random variable. The strongest assumption is that the moments involved must be finite. Thus, the results obtained also hold for any empirical distribution with nonnegative data. These facts allow potential applications in numerical analysis, probability, and statistical inference, among other disciplines. Moreover, the proposed methodology offers an alternative approach to obtain new inequalities and even to improve some known inequalities. For instance, we give new inequalities for the ratio of gamma functions. In this context, we also improve an inequality by Bustoz and Ismail and some cases of inequalities due to Gurland and Dragomir et al. Additionally, we present a new inequality for finite sums of nonnegative or nonpositive numbers. For some cases, this relation improves even the Cauchy-Bunyakovsky-Schwarz inequality.

An Empirical Functional Central Limit Theorem for weakly dependent sequences

Prieur C.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

259--287

In this paper we obtain a Functional Central Limit Theorem for the empirical process of a stationary sequence under a new weak dependence condition introduced by Doukhan and Louhichi [5]. This result improves on the Empirical Functional Central Limit Theorem in Doukhan and Louhichi [5]. Our proof relies on new moment inequalities and on a Central Limit Theorem. Techniques of proofs come from Louhichi [12] and Rio [16], respectively. We also deduce a rate of convergence in a Marcinkiewicz-Zygmund Strong Law.