A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables - Fa-mei Zheng (Journal of Probability and Statistics)
Let {𝑋,𝑋𝑖;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 𝑆𝑛=∑𝑛𝑖=1𝑋𝑖,𝑆𝑛∑(𝑎)=𝑛𝑖=1𝑋𝑖𝐼(𝑀𝑛−𝑎<𝑋𝑖≤𝑀𝑛) and 𝑉2𝑛=∑𝑛𝑖=1(𝑋𝑖−𝑋)2, where 𝑀𝑛=max1≤𝑖≤𝑛𝑋𝑖, ∑𝑋=(1/𝑛)𝑛𝑖=1𝑋𝑖, and 𝑎>0 is a fixed constant. Under some suitable conditions, we show that (∏[𝑛𝑡]𝑘=1(𝑇𝑘(𝑎)/𝜇𝑘))𝜇/𝑉𝑛𝑑∫→exp{𝑡0(𝑊(𝑥)/𝑥)𝑑𝑥}𝑖𝑛𝐷[0,1], as 𝑛→∞, where 𝑇𝑘(𝑎)=𝑆𝑘−𝑆𝑘(𝑎) is the trimmed sum and {𝑊(𝑡);𝑡≥0} is a standard Wiener process.