## Journal of Applied Probability

### Some inequalities of linear combinations of independent random variables. I.

#### Abstract

In this paper we provide some sufficient conditions to stochastically compare linear combinations of independent random variables. The main results extend those given in Proschan (1965), Ma (1998), Zhao et al. (2011), and Yu (2011). In particular, we propose a new sufficient condition to compare the peakedness of linear combinations of independent random variables which may have heavy-tailed properties.

#### Article information

Source
J. Appl. Probab., Volume 48, Number 4 (2011), 1179-1188.

Dates
First available in Project Euclid: 16 December 2011

https://projecteuclid.org/euclid.jap/1324046026

Digital Object Identifier
doi:10.1239/jap/1324046026

Mathematical Reviews number (MathSciNet)
MR2896675

Zentralblatt MATH identifier
1230.60019

Subjects
Primary: 60E15: Inequalities; stochastic orderings

#### Citation

Xu, Maochao; Hu, Taizhong. Some inequalities of linear combinations of independent random variables. I. J. Appl. Probab. 48 (2011), no. 4, 1179--1188. doi:10.1239/jap/1324046026. https://projecteuclid.org/euclid.jap/1324046026

#### References

• An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization. J. Econom. Theory 80, 350–369.
• Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econom. Theory 26, 445–469.
• Birnbaum, Z. W. (1948). On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.
• Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
• Hu, T., Nanda, A. K., Xie, H. and Zhu, Z. (2004). Properties of some stochastic orders: A unified study. Naval Res. Logistics 51, 193–216.
• Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Prob. Appl. 1, 255–280.
• Ibragimov, R. (2007). Efficiency of linear estimators under heavy-tailedness: convolutions of $\alpha$-symmetric distributions. Econometric Theory 23, 501–517.
• Jensen, D. R. (1997). Peakedness of linear forms in ensembles and mixtures. Statist. Prob. Lett. 35, 277–282.
• Ma, C. (1998). On peakedness of distributions of convex combinations. J. Statist. Planning Infer. 70, 51–56.
• Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York.
• Pan, X., Xu, M. and Hu, T. (2011). Some inequalities of linear combinations of independent random variables: II. Tech. Rep., University of Science and Technology of China.
• Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist. 36, 1703–1706.
• Purkayastha, S. (1998). Simple proofs of two results on convolutions of unimodal distributions. Statist. Prob. Lett. 39, 97–100.
• Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
• Tong, Y. L. (1994). Some recent developments on majorization inequalities in probability and statistics. Linear Algebra Appl. 199, 69–90.
• Wintner, A. (1938). Asymptotic Distributions and Infinite Convolutions. Edwards Brothers, Ann Arbor, MI.
• Yu, Y. (2011). Some stochastic inequalities for weighted sums. To appear in Bernoulli.
• Zhao, P., Chan, P. S. and Ng, H. K. T. (2011). Peakedness for weighted sums of symmetric random variables. J. Statist. Planning Infer. 141, 1737–1743.