Reverse submartingale property arising from exchangeable random variables |
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| Authors: | Wen-Jang Huang Jyh-Cherng Su |
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| Institution: | Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, 80424, R.O.C. (e-mail: huangwj@math.nsysu.edu.tw), TW
Department of Mathematics, Chinese Military Academy, FengShan, Kaohsiung, Taiwan, 830, R.O.C. (e-mail: sujc@cc.cma.edu.tw), TW
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| Abstract: | In this work, for an exchangeable sequence of random variables {Xi, i̿}, and two nondecreasing sequences of positive integers {hn, ǹ} and {kn, ǹ}, where hn+knhn, Qǹ, we prove that {Rn,hn,kn/n, ǹ} forms a reverse submartingale sequence, where R_{n,hn,kn}={\displaystyle {1\over kn}} ~^{kn-1}_{j=0} X_{n-j,n}-{\displaystyle {1\over hn}} ~^{hn}_{j=1} X_{j,n}$R_{n,hn,kn}={\displaystyle {1\over kn}} ~^{kn-1}_{j=0} X_{n-j,n}-{\displaystyle {1\over hn}} ~^{hn}_{j=1} X_{j,n}, and X1,nhX2,nh?hXn,n are the order statistics based on {X1,?,Xn}. |
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