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題名 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
作者 程嵩硯
Chen, Song-yen
貢獻者 陳隆奇
Lung-Chi Chen
程嵩硯
Chen, Song-yen
關鍵詞 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 theorems
Random sums
Slowly variations
Regularly variations
Self-normalization
Poisson processes
日期 2019
上傳時間 3-六月-2019 13:05:21 (UTC+8)
摘要 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.
參考文獻 [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. Statistics and Probability Letters. 52, 1206–1214.
[3] Bening, V. E. and Korolev, V. Y. (2012). Generalized Poisson Models and their Applications in Insurance and Finance. Walter de Gruyter.
[4] Bentkus, V., Bloznelis, M. and G¨otze, F. (1996). A Berry-Ess´een bound for Student’s statistic in the non-i.i.d. case. J. Theoret. Probab., 9, 765–796.
[5] Bentkus, V. and G¨otze, F. (1996). The Berry-Ess´een bound for Student’s statistic. Ann. Probab., 24, 491–503.
[6] Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987). Regular Variation, Cambridge
University Press, Cambridge.
[7] Babu, G.J., Singh, K. and Yang, Y. (2003). Edgeworth Expansions for Compound Poisson Processes and the Bootstrap. Ann. Inst. Statist. Math. 55, 83–94.
[8] Blum, J. R., Hanson, D. L. and Rosenblatt, J. I. (1963). On the Central Limit Theorem for the Sum of a Random Number of Independent Random Variables.
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 1, 389–393.
[9] Chen, X., Shao, Q.M., Wu, W.B. and Xu L. (2016). Self-normalized Cram´er-type Moderate Deviations under Dependence
Ann. Statist., 44, 1593–1617.
[10] Chistyakov, G. P. and G¨otze, F. (2004). Limit distributions of Studentized means.
Ann. Probab., 32, 28–77.
[11] Chung, K.L. (2002). A Course in Probability Theory. Academic Press.
[12] Cs¨org¨o and Rychlik. (1981). Asymptotic Properties of Randomly Indexed Sequences of Random Variables. The Canadian Journal of Statistics 9, 101–107.
[13] Cs¨org¨o, M., Szyszkowicz, B. and Wang, Q. (2003). Donsker’s theorem for self-normalized partial sums processes. Ann. Probab., 31, 1228– 1240.
[14] de la Pen´a, V.H., Klass, M.J. and Lai, T.L. (2007). Pseudo-maximization and self-normalized processes. Probability Surveys, 4, 172–192.
[15] de la Pena, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-normalized Processes: Limit Theorey and Statistical Applications. Springer, New York.
[16] Dembo, A. and Shao, Q.M. (1998a). Self-normalized Moderate Deviations and Lils. Stoch. Proc. and Appl, 75, 51–65.
[17] Dembo, A. and Shao, Q.M. (1998b). Self-normalized large deviations in vector spaces. Progress in Probability, 43, 27–32.
[18] Efron, B. (1969). Student’s t-test under symmetry conditions. J. Am. Statistist. Assoc. 89, 452–462.
[19] Egorov, V.A. (1996). On the Asymptotic Behavior of Self-normalized of Random Variables. Theory of Probability and its Applications, 41, 542–548.
[20] Embrechts, P ., Kl¨uppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance, Springer, Berlin.
[21] Feller, W. (1975). An Introduction to Probability Theory and its Applications. II New York: Wiley.
[22] Gin´e, E., G¨otze, F. and Mason, D. (1997). When is the Student t statistic asymptotically standard normal ? Ann. Probab., 25, 1514–1531.
[23] Gnedenko, B.V. and Kolmogorov, A.N. (1954). Limit distributions for sums of independent random variables. Addison Wesley, Cambridge, Massachusetts.
[24] Gut, A. (2006) Gnedenko-Raikov’s Theorem, Central Limit Theory, and the Weak Law of Large Numbers. Statistics and Probability Letters, 76, 1935–1939.
[25] Griffin, P. S. and Kuelbs, J. D. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab., 17, 1571–1601.
[26] Griffin, P. S. and Kuelbs, J. D. (1989). Some Extensions of the Laws of the Iterated Logarithm via Self-normalizations. Ann. Probab., 19, 380–395.
[27] Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Math. Proc. Cambridge Philos. Soc., 109, 597–610.
[28] Griffin, P.S. (2002). Tightness of the Student t-statistic. Elect. Comm. Probab. 7, 181–190.
[29] Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press.
[30] Helmers, R. and Tarigan, B. (2003). Compound sums and their applications in finance. Working Paper.
[31] Ibragimov, I.A. and Y.V. Linnik (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff: Groningen.
[32] Jing, B.Y., Shao, Q.M. and Wang, Q.Y. (2003). Self-normalized Cram´er type large deviations for independent random variables. Ann. Probab., 31, 2167–2215.
[33] Jing, B.Y., Wang, Q.Y., Wang, X.P. and Zhou, W. (2009). Saddlepoint Approximation for Studentized Compound Poisson Sums. Working Paper.
[34] Jing, B.Y., Wang, Q.Y. and Zhou, W. (2015). Cram´er-Type Moderate Deviation for Studentized Compound Poisson Sum. J. Theor. Probab. 28, 1556–1570.
[35] Kallenberg, O. (2002). Foundations of Modern Probability. 2nd ed., Springer, New York.
[36] Kl¨uppelberg, C. and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications to insurance and finance. J. Appl. Probab. 34, 293–308.
[37] LaPage, R.,Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 713–752.
[38] Logan, B. F., Mallows, C. L., Rice, S. O. and Sheep, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab., 1, 788–809.
[39] Mason, D.M. (2005). The Asymptotic Distribution of Self-normalized triangular arrays. Journal of Theoret. Probab., 18, 853–870.
[40] Mikosch, T. and Nagaev, A.V. (1998). Large deviations of heavy-tailed sums with applications to insurance. Extremes. 1, 81–110.
[41] Mikosch, T. (1999). Regular Variation, Subexponentiality and their applications
in probability theory. Lecture notes for the workshop ”Heavy Tails and Queques,”
EURANDOM, Eindhoven, Netherlands.
[42] O’Brien, G.L. (1980). A Limit Theorem for Sample Maximum and Heavy Branches in Galton-Watson Trees. Journal of Appl. Probab., 17, 539–545.
[43] Petrov, V.V. (1975). Sums of independent random variables Ergebnisse der Math-ematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York, Heidelberg, Berlin.
[44] R´enyi, A. (1957). On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 8, 193– 199.
[45] Resnick, S.I. (1987). Extreme Vallues, Regular Variation, and Point Processes. New York: Springer-Verlang.
[46] Resnick, S.I. (2007). Probabilistic and Statistical Modeling of Heavy Tailed Phenomena. New York: Springer-Verlag.
[47] Robinson, J. and Wang, Q. (2005). On the Self-normalized Cram´er-type Large Deviation. Journal of Theoret. Probab., 18, 891–909.
[48] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman-Hall.
[49] Sato, K. (1999). L´evy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68, Cambridge University Press.
[50] Senta, E. (1976). Regularly Varying Functions, Lecture Notes in Mathematics 508.
Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[51] Seri R, Choirat C. (2015) Comparison of Approximations for Compound Poisson Processes. ASTIN Bulletin. 45, 601–637.
[52] Shao, Q.M. (1997). Self-normalized Large Deviations. Ann. Probab., 25, 285–328.
[53] Shao, Q.M. (1998). Recent Developments on Self-normalized Limit Theorems. Asymptotic Methods in Probability and Statistics, A Volume in Honour of Mikl´os
Cs¨org¨o , Elsevier Science, 467–480.
[54] Shao, Q.M. (1999). A Cram´er Type Large Deviation Result for Student’s t-Statistic. J. Theoret. Probab., 12, 385–398.
[55] Shao, Q.M. (2004). Recent Progress on Self-normalized Limit Theorems. Probability, Finance and Insurance, World Sci. Publ., River Edge, NJ, 50–68.
[56] Shao, Q.M. (2005). An explicit Berry-Ess´een bound for Student’s t-statistic via Stein’s method. Stein’s Method and Applications, Lecture Notes Series, Institute
for Mathematical Sciences, National University of Singapore, 143–155.
[57] Shao, Q.M. (2018). On necessary and sufficient conditions for the self-normalized central limit theorem. SCIENCE CHINA Mathematics. 61, 1741.
[58] Shao, Q.M. and Wang, Q.Y. (2013). Self-normalized Limit Theorems: A Survey. Probability Surveys. 10, 69–93.
[59] Steutel, F.W. (1974). On the tails of infinitely divisible distributions. Zeitschrift f¨ur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, 28, 273–276.
[60] Tang, Q., Su, C., Jiang, T., and Zhang J. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statistics and Probability Letters. 52
91–100
[61] Wang, Q. and Jing, B.-Y. (1999). An Exponential Nonuniform Berry- Ess´een Bound for Self-normalized Sums. Ann. Probab., 27, 2068–2088.
[62] Zolotarev, V.M, (1986). One-dimensional Stable Distributions. American Mathematical Society, Providence, RI.
描述 碩士
國立政治大學
應用數學系
104751011
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104751011
資料類型 thesis
dc.contributor.advisor 陳隆奇zh_TW
dc.contributor.advisor Lung-Chi Chenen_US
dc.contributor.author (作者) 程嵩硯zh_TW
dc.contributor.author (作者) Chen, Song-yenen_US
dc.creator (作者) 程嵩硯zh_TW
dc.creator (作者) Chen, Song-yenen_US
dc.date (日期) 2019en_US
dc.date.accessioned 3-六月-2019 13:05:21 (UTC+8)-
dc.date.available 3-六月-2019 13:05:21 (UTC+8)-
dc.date.issued (上傳時間) 3-六月-2019 13:05:21 (UTC+8)-
dc.identifier (其他 識別碼) G0104751011en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/123683-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 104751011zh_TW
dc.description.abstract (摘要) 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。zh_TW
dc.description.abstract (摘要) 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.en_US
dc.description.tableofcontents 1 Introduction 2
2 Reviews on Self-normalized Asymptotics with Heavy tails 10
3 Preliminary and Main Results 20
3.1 Preliminaries 20
3.1.1 Stable Laws 20
3.1.2 Continuous Mapping Theorems 24
3.1.3 Ragularly Varying Functions 25
3.2 Statement of the Results 27
4 Proofs of the Main Theorems 29
4.1 Proof of Theorem 3.1 29
4.2 Proof of Theorem 3.2 38
4.3 Proofs of Lemmas 40
A Appendices 58
References 61		zh_TW
dc.format.extent 805143 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104751011en_US
dc.subject (關鍵詞) 中央極限定理zh_TW
dc.subject (關鍵詞) 隨機和zh_TW
dc.subject (關鍵詞) 緩變函數zh_TW
dc.subject (關鍵詞) 常變函數zh_TW
dc.subject (關鍵詞) 自我正規化zh_TW
dc.subject (關鍵詞) 卜瓦松過程zh_TW
dc.subject (關鍵詞) central limit theoremsen_US
dc.subject (關鍵詞) Random sumsen_US
dc.subject (關鍵詞) Slowly variationsen_US
dc.subject (關鍵詞) Regularly variationsen_US
dc.subject (關鍵詞) Self-normalizationen_US
dc.subject (關鍵詞) Poisson processesen_US
dc.title (題名) 不存在變異數的複合 Poisson 過程之自正規化中央極限定理zh_TW
dc.title (題名) On the self-normalized central limit theorems for compound Poisson processes under infinite variancesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [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. Statistics and Probability Letters. 52, 1206–1214.
[3] Bening, V. E. and Korolev, V. Y. (2012). Generalized Poisson Models and their Applications in Insurance and Finance. Walter de Gruyter.
[4] Bentkus, V., Bloznelis, M. and G¨otze, F. (1996). A Berry-Ess´een bound for Student’s statistic in the non-i.i.d. case. J. Theoret. Probab., 9, 765–796.
[5] Bentkus, V. and G¨otze, F. (1996). The Berry-Ess´een bound for Student’s statistic. Ann. Probab., 24, 491–503.
[6] Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987). Regular Variation, Cambridge
University Press, Cambridge.
[7] Babu, G.J., Singh, K. and Yang, Y. (2003). Edgeworth Expansions for Compound Poisson Processes and the Bootstrap. Ann. Inst. Statist. Math. 55, 83–94.
[8] Blum, J. R., Hanson, D. L. and Rosenblatt, J. I. (1963). On the Central Limit Theorem for the Sum of a Random Number of Independent Random Variables.
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 1, 389–393.
[9] Chen, X., Shao, Q.M., Wu, W.B. and Xu L. (2016). Self-normalized Cram´er-type Moderate Deviations under Dependence
Ann. Statist., 44, 1593–1617.
[10] Chistyakov, G. P. and G¨otze, F. (2004). Limit distributions of Studentized means.
Ann. Probab., 32, 28–77.
[11] Chung, K.L. (2002). A Course in Probability Theory. Academic Press.
[12] Cs¨org¨o and Rychlik. (1981). Asymptotic Properties of Randomly Indexed Sequences of Random Variables. The Canadian Journal of Statistics 9, 101–107.
[13] Cs¨org¨o, M., Szyszkowicz, B. and Wang, Q. (2003). Donsker’s theorem for self-normalized partial sums processes. Ann. Probab., 31, 1228– 1240.
[14] de la Pen´a, V.H., Klass, M.J. and Lai, T.L. (2007). Pseudo-maximization and self-normalized processes. Probability Surveys, 4, 172–192.
[15] de la Pena, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-normalized Processes: Limit Theorey and Statistical Applications. Springer, New York.
[16] Dembo, A. and Shao, Q.M. (1998a). Self-normalized Moderate Deviations and Lils. Stoch. Proc. and Appl, 75, 51–65.
[17] Dembo, A. and Shao, Q.M. (1998b). Self-normalized large deviations in vector spaces. Progress in Probability, 43, 27–32.
[18] Efron, B. (1969). Student’s t-test under symmetry conditions. J. Am. Statistist. Assoc. 89, 452–462.
[19] Egorov, V.A. (1996). On the Asymptotic Behavior of Self-normalized of Random Variables. Theory of Probability and its Applications, 41, 542–548.
[20] Embrechts, P ., Kl¨uppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance, Springer, Berlin.
[21] Feller, W. (1975). An Introduction to Probability Theory and its Applications. II New York: Wiley.
[22] Gin´e, E., G¨otze, F. and Mason, D. (1997). When is the Student t statistic asymptotically standard normal ? Ann. Probab., 25, 1514–1531.
[23] Gnedenko, B.V. and Kolmogorov, A.N. (1954). Limit distributions for sums of independent random variables. Addison Wesley, Cambridge, Massachusetts.
[24] Gut, A. (2006) Gnedenko-Raikov’s Theorem, Central Limit Theory, and the Weak Law of Large Numbers. Statistics and Probability Letters, 76, 1935–1939.
[25] Griffin, P. S. and Kuelbs, J. D. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab., 17, 1571–1601.
[26] Griffin, P. S. and Kuelbs, J. D. (1989). Some Extensions of the Laws of the Iterated Logarithm via Self-normalizations. Ann. Probab., 19, 380–395.
[27] Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Math. Proc. Cambridge Philos. Soc., 109, 597–610.
[28] Griffin, P.S. (2002). Tightness of the Student t-statistic. Elect. Comm. Probab. 7, 181–190.
[29] Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press.
[30] Helmers, R. and Tarigan, B. (2003). Compound sums and their applications in finance. Working Paper.
[31] Ibragimov, I.A. and Y.V. Linnik (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff: Groningen.
[32] Jing, B.Y., Shao, Q.M. and Wang, Q.Y. (2003). Self-normalized Cram´er type large deviations for independent random variables. Ann. Probab., 31, 2167–2215.
[33] Jing, B.Y., Wang, Q.Y., Wang, X.P. and Zhou, W. (2009). Saddlepoint Approximation for Studentized Compound Poisson Sums. Working Paper.
[34] Jing, B.Y., Wang, Q.Y. and Zhou, W. (2015). Cram´er-Type Moderate Deviation for Studentized Compound Poisson Sum. J. Theor. Probab. 28, 1556–1570.
[35] Kallenberg, O. (2002). Foundations of Modern Probability. 2nd ed., Springer, New York.
[36] Kl¨uppelberg, C. and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications to insurance and finance. J. Appl. Probab. 34, 293–308.
[37] LaPage, R.,Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 713–752.
[38] Logan, B. F., Mallows, C. L., Rice, S. O. and Sheep, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab., 1, 788–809.
[39] Mason, D.M. (2005). The Asymptotic Distribution of Self-normalized triangular arrays. Journal of Theoret. Probab., 18, 853–870.
[40] Mikosch, T. and Nagaev, A.V. (1998). Large deviations of heavy-tailed sums with applications to insurance. Extremes. 1, 81–110.
[41] Mikosch, T. (1999). Regular Variation, Subexponentiality and their applications
in probability theory. Lecture notes for the workshop ”Heavy Tails and Queques,”
EURANDOM, Eindhoven, Netherlands.
[42] O’Brien, G.L. (1980). A Limit Theorem for Sample Maximum and Heavy Branches in Galton-Watson Trees. Journal of Appl. Probab., 17, 539–545.
[43] Petrov, V.V. (1975). Sums of independent random variables Ergebnisse der Math-ematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York, Heidelberg, Berlin.
[44] R´enyi, A. (1957). On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 8, 193– 199.
[45] Resnick, S.I. (1987). Extreme Vallues, Regular Variation, and Point Processes. New York: Springer-Verlang.
[46] Resnick, S.I. (2007). Probabilistic and Statistical Modeling of Heavy Tailed Phenomena. New York: Springer-Verlag.
[47] Robinson, J. and Wang, Q. (2005). On the Self-normalized Cram´er-type Large Deviation. Journal of Theoret. Probab., 18, 891–909.
[48] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman-Hall.
[49] Sato, K. (1999). L´evy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68, Cambridge University Press.
[50] Senta, E. (1976). Regularly Varying Functions, Lecture Notes in Mathematics 508.
Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[51] Seri R, Choirat C. (2015) Comparison of Approximations for Compound Poisson Processes. ASTIN Bulletin. 45, 601–637.
[52] Shao, Q.M. (1997). Self-normalized Large Deviations. Ann. Probab., 25, 285–328.
[53] Shao, Q.M. (1998). Recent Developments on Self-normalized Limit Theorems. Asymptotic Methods in Probability and Statistics, A Volume in Honour of Mikl´os
Cs¨org¨o , Elsevier Science, 467–480.
[54] Shao, Q.M. (1999). A Cram´er Type Large Deviation Result for Student’s t-Statistic. J. Theoret. Probab., 12, 385–398.
[55] Shao, Q.M. (2004). Recent Progress on Self-normalized Limit Theorems. Probability, Finance and Insurance, World Sci. Publ., River Edge, NJ, 50–68.
[56] Shao, Q.M. (2005). An explicit Berry-Ess´een bound for Student’s t-statistic via Stein’s method. Stein’s Method and Applications, Lecture Notes Series, Institute
for Mathematical Sciences, National University of Singapore, 143–155.
[57] Shao, Q.M. (2018). On necessary and sufficient conditions for the self-normalized central limit theorem. SCIENCE CHINA Mathematics. 61, 1741.
[58] Shao, Q.M. and Wang, Q.Y. (2013). Self-normalized Limit Theorems: A Survey. Probability Surveys. 10, 69–93.
[59] Steutel, F.W. (1974). On the tails of infinitely divisible distributions. Zeitschrift f¨ur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, 28, 273–276.
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dc.identifier.doi (DOI) 10.6814/THE.NCCU.MATH.001.2019.B01en_US