Asymptotic approximation of inverse moments of nonnegative random variables. (English) Zbl 1168.60340

Summary: Let \(\{Z_n,n\geq 1\}\) be a sequence of independent nonnegative r.v.’s (random variables) with finite second moments. It is shown that under a Lindeberg-type condition, the \(\alpha \)th inverse moment \(E\{a+X_n\}^{ - \alpha} \) can be asymptotically approximated by the inverse of the \(\alpha \)th moment \(\{a+EX_n\} ^{- \alpha} \) where \(a>0,\alpha>0\), and \(\{X_n\}\) are the naturally-scaled partial sums. Furthermore, it is shown that, when \(\{Z_n\}\) only possess finite \(r\)th moments, \(1\leq r<2\), the preceding asymptotic approximation can still be valid by using different norming constants which are the standard deviations of partial sums of suitably truncated \(\{Z_n\}\).


60F99 Limit theorems in probability theory
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI


[1] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[2] Fujioka, T., Asymptotic approximations of the inverse moment of the non-central chi-squared variable, J. Japan Statist. Soc., 31, 99-109 (2001) · Zbl 1031.62011
[3] Garcia, N. L.; Palacios, J. L., On inverse moments of nonnegative random variables, Statist. Probab. Lett., 53, 235-239 (2001) · Zbl 0991.60003
[4] Gupta, R. C.; Akman, O., Statistical inference based on the length-biased data for the inverse Gaussian distribution, Statistics, 31, 325-337 (1998) · Zbl 0930.62020
[5] Hu, S.-H.; Chen, G.-J.; Wang, X.-J.; Chen, E.-B., On inverse moments of nonnegative weakly convergent random variables, Acta Math. Appl. Sin., 30, 361-367 (2007) · Zbl 1141.60309
[6] Jurlewicz, A.; Weron, K., Relaxation of dynamically correlated clusters, J. Non-Cryst. Solids, 305, 112-121 (2002)
[7] Kaluszka, M.; Okolewski, A., On Fatou-type lemma for monotone moments of weakly convergent random variables, Statist. Probab. Lett., 66, 45-50 (2004) · Zbl 1116.60308
[8] Marciniak, E.; Wesolowski, J., Asymptotic Eulerian expansions for binomial and negative binomial reciprocals, Proc. Amer. Math. Soc., 127, 3329-3338 (1999) · Zbl 0930.60004
[9] Mendenhall, W.; Lehman, E. H., An approximation to the negative moments of the positive binomial useful in life-testing, Technometrics, 2, 227-242 (1960) · Zbl 0105.12305
[10] Pittenger, A. O., Sharp mean-variance bounds for Jensen-type inequalities, Statist. Probab. Lett., 10, 91-94 (1990) · Zbl 0705.60017
[11] Ramsay, C. M., A note on random survivorship group benefits, ASTIN Bull., 23, 149-156 (1993)
[12] Wooff, D. A., Bounds on reciprocal moments with applications and developments in Stein estimation and post-stratification, J. R. Stat. Soc. Ser. B, 47, 362-371 (1985) · Zbl 0603.62016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.