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

Fa-mei Zheng

Journal of Probability and Statistics, 2011, vol. 2011, 1-13

Abstract:

Let { ð ‘‹ , ð ‘‹ ð ‘– ; ð ‘– ≥ 1 } be a sequence of independent and identically distributed positive random variables with a continuous distribution function ð ¹ , and ð ¹ has a medium tail. Denote 𠑆 ð ‘› = ∑ ð ‘› ð ‘– = 1 ð ‘‹ ð ‘– , 𠑆 ð ‘› ∑ ( ð ‘Ž ) = ð ‘› ð ‘– = 1 ð ‘‹ ð ‘– ð ¼ ( ð ‘€ ð ‘› − ð ‘Ž < ð ‘‹ ð ‘– ≤ ð ‘€ ð ‘› ) and 𠑉 2 ð ‘› = ∑ ð ‘› ð ‘– = 1 ( ð ‘‹ ð ‘– − ð ‘‹ ) 2 , where ð ‘€ ð ‘› = m a x 1 ≤ ð ‘– ≤ ð ‘› ð ‘‹ ð ‘– , ∑ ð ‘‹ = ( 1 / ð ‘› ) ð ‘› ð ‘– = 1 ð ‘‹ ð ‘– , and ð ‘Ž > 0 is a fixed constant. Under some suitable conditions, we show that ( ∠[ ð ‘› ð ‘¡ ] 𠑘 = 1 ( 𠑇 𠑘 ( ð ‘Ž ) / 𠜇 𠑘 ) ) 𠜇 / 𠑉 ð ‘› ð ‘‘ ∫ → e x p { ð ‘¡ 0 ( ð ‘Š ( ð ‘¥ ) / ð ‘¥ ) ð ‘‘ ð ‘¥ } ð ‘– ð ‘› ð · [ 0 , 1 ] , as ð ‘› → ∞ , where 𠑇 𠑘 ( ð ‘Ž ) = 𠑆 𠑘 − 𠑆 𠑘 ( ð ‘Ž ) is the trimmed sum and { ð ‘Š ( ð ‘¡ ) ; ð ‘¡ ≥ 0 } is a standard Wiener process.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljps:181409

DOI: 10.1155/2011/181409

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