| 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. 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. |
