Journal of Probability and Statistics
Volume 2011 (2011), Article ID 181409, 13 pages
http://dx.doi.org/10.1155/2011/181409
Research Article

A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Fa-mei Zheng

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

Received 13 May 2011; Revised 25 July 2011; Accepted 11 August 2011

Academic Editor: Man Lai Tang

Copyright © 2011 Fa-mei Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let { 𝑋 , 𝑋 𝑖 ; 𝑖 ≥ 1 } be a sequence of independent and identically distributed positive random variables with a continuous distribution function 𝐹 , and 𝐹 has a medium tail. Denote 𝑆 𝑛 = ∑ 𝑛 𝑖 = 1 𝑋 𝑖 , 𝑆 𝑛 ∑ ( 𝑎 ) = 𝑛 𝑖 = 1 𝑋 𝑖 𝐼 ( 𝑀 𝑛 − 𝑎 < 𝑋 𝑖 ≤ 𝑀 𝑛 ) and 𝑉 2 𝑛 = ∑ 𝑛 𝑖 = 1 ( 𝑋 𝑖 − 𝑋 ) 2 , where 𝑀 𝑛 = m a x 1 ≤ 𝑖 ≤ 𝑛 𝑋 𝑖 , ∑ 𝑋 = ( 1 / 𝑛 ) 𝑛 𝑖 = 1 𝑋 𝑖 , and 𝑎 > 0 is a fixed constant. Under some suitable conditions, we show that ( ∏ [ 𝑛 𝑡 ] 𝑘 = 1 ( 𝑇 𝑘 ( 𝑎 ) / 𝜇 𝑘 ) ) 𝜇 / 𝑉 𝑛 𝑑 ∫ → e x p { 𝑡 0 ( 𝑊 ( 𝑥 ) / 𝑥 ) 𝑑 𝑥 } 𝑖 𝑛 𝐷 [ 0 , 1 ] , as 𝑛 → ∞ , where 𝑇 𝑘 ( 𝑎 ) = 𝑆 𝑘 − 𝑆 𝑘 ( 𝑎 ) is the trimmed sum and { 𝑊 ( 𝑡 ) ; 𝑡 ≥ 0 } is a standard Wiener process.