A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables. (9th October 2011)

• Record Type: Journal Article
• Authors: Zheng, Fa-mei
• Other Names: Tang Man Lai Academic Editor.
• Is Part Of: Journal of probability and statistics. Volume 2011(2011)
• Publication Date: 2011-10-09
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; Mathematical statistics; Probabilities; Periodicals; 519
• DOI: 10.1155/2011/181409
• Languages: English
• ISSNs: 1687-952X

Abstract

Abstract : Let{ X, X i ; i ≥ 1 } be a sequence of independent and identically distributed positive random variables with a continuous distribution functionF, andF has a medium tail. DenoteS n = ∑ i = 1 n X i, S n ( a ) = ∑ i = 1 n X i I ( M n - a < X i ≤ M n ) andV n 2 = ∑ i = 1 n ( X i - X ̅ ) 2, whereM n = max 1 ≤ i ≤ n X i, X ̅ = (1/ n) ∑ i = 1 n X i, anda > 0 is a fixed constant. Under some suitable conditions, we show that( ∏ k = 1 [ n t ] (T k ( a ) / μ k) ) μ / V n → d exp { ∫ 0 t (W ( x ) / x) d x } i n D [ 0, 1 ], asn → ∞, whereT k ( a ) = S k - S k ( a ) is the trimmed sum and{ W ( t ) ; t ≥ 0 } is a standard Wiener process.