## 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 $$\{Z_n\}$$.

### MSC:

 60F99 Limit theorems in probability theory 62G20 Asymptotic properties of nonparametric inference
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### References:

 [1] Billingsley, P., Convergence of probability measures, (1968), 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
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