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 Title: 不存在變異數的複合 Poisson 過程之自正規化中央極限定理On the self-normalized central limit theorems for compound Poisson processes under infinite variances Authors: 程嵩硯Chen, Song-yen Contributors: 陳隆奇Lung-Chi Chen程嵩硯Chen, Song-yen Keywords: 中央極限定理隨機和緩變函數常變函數自我正規化卜瓦松過程central limit theoremsRandom sumsSlowly variationsRegularly variationsSelf-normalizationPoisson processes Date: 2019 Issue Date: 2019-06-03 13:05:21 (UTC+8) Abstract: 在本篇論文，我們探討厚尾隨機過程的自正規化漸近分佈行為。目前為止，文獻上在關於隨機過程的自正規化漸近行為的討論仍相當稀少。在本文中，考慮一個不具二階動差的複合 Poisson 過程，分別為緩變厚尾及常變厚尾兩種情況，建立其自正則中央極限定理。我們的結果部份推廣了對於隨機和的古典中央極限定理結果至自正規化的情況下。在緩變厚尾的截斷二階動差限制，得到與一般古典中央極限定理下一致的極限常態結果;在常變厚尾的截斷二階動差限制，其極限分佈有別於傳統廣義中央極限定理之穩定分佈結果。In this thesis, we investigate the self-normalized asymptotic distributions coupled with the heavy tails for random processes. To the best of our knowledge, the discussions on self-normalized asymptotic behaviors for random processes are fairly rare. Consider a compound Poisson process relaxing the second moment with slowly varying tails and regularly varying tails, respectively, the self-normalized central limit theorems of random sums are specifically established. Our results partly extend the classical central limit theorems for random sums, under Poisson randomly-indexed sums to the self-normalized versions. The first result is consistent with the classical central limit theorem for random sums, but provides more flexible applications to normal approximations in practices, without needing the consistent estimators of variances. The second result shows that the limiting distribution of the self-normalized random sums is a ratio of two strictly stable laws when the law is in the domain of a stable law. Reference: [1] Asmussen, S. (2000) Ruin probabilities. World Scientific Press.[2] Baltr˜unas, A., Leipus, R., ˘ Siaulys, J. (2008). Precise large deviation results for the total claim amount under subexponential claim sizes. 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