Abstract
In this chapter, we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. It implies a general approach to get the non-asymptotic bounds for accuracy of approximation of nonlinear forms in random elements in terms of Lyapunov type ratios. Applications to classical and free probability theory are discussed. In particular, we apply the general results to the central limit theorem for weighted sums, including the case of dependent summands and the case when the distributions of weighted sums are approximated by the normal distribution with accuracy of order \(\text{ O }(n^{-1})\). We consider also applications for distributions of U-statistics of the second order and higher.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bentkus, V., & Götze, F. (1999). Optimal bounds in non-Gaussian limit theorems for U-statistics. The Annals of Probability, 27(1), 454–521.
Bergström, H. (1944). On the central limit theorem. Scandinavian Actuarial Journal, 1944(3–4), 139–153.
Bhattacharya, R. N., & Rao, R. R. (2010). Normal approximation and asymptotic expansions. SIAM-Society for Industrial and Applied Mathematics.
Bloznelis, M., & Götze, F. (2002). An Edgeworth expansion for symmetric finite population statistics. The Annals of Probability, 30(3), 1238–1265.
Chaterjee, S. (2006). A generalization of the Lindeberg principle. The Annals of Probability, 34(6), 2061–2076.
Chistyakov, G. P., & Götze, F. (2013). Asymptotic expansions in the CLT in free probability. Probability Theory and Related Fields, 157(1–2), 107–156.
Gnedenko, B. V., Koroluk, V. S., & Skorohod, A. V. (1960). Asymptotic expansions in probability theory. In Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability (Vol. II, pp. 153–170). Berkeley,: University of California Press.
Götze, F. (1981). On Edgeworth expansions in Banach spaces. The Annals of Probability, 9(5), 852–859.
Götze, F. (1985). Asymptotic expansions in functional limit theorems. Journal of Multivariate analysis, 16, 1–20.
Götze, F. (1989). Edgeworth expansions in functional limit theorems. The Annals of Probability, 17(4), 1602–1634.
Götze, F. (2004). Lattice point problems and values of quadratic forms. Inventiones Mathematicae, 157, 195–226.
Götze, F., & Hipp, C. (1978). Asymptotic expansions in the central limit theorem under moment conditions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 42(1), 67–87.
Götze, F., & Reshetenko, A. (2014). Asymptotic expansions in free limit theorems. arXiv:1408.1360.
Götze, F., & Zaitsev, A Yu. (2014). Explicit rates of approximation in the CLT for quadratic forms. The Annals of Probability, 42(1), 354–397.
Götze, F., Naumov, A. A., & Ulyanov, V. V. (2017). Asymptotic analysis of symmetric functions. Journal of Theoretical Probability, 30(3), 876–897.
Jing, B.-Y., & Wang, Q. (2010). A unified approach to Edgeworth expansions for a general class of statistics. Statistica Sinica, 20, 613–636.
Klartag, B., & Sodin, S. (2012). Variations on the Berry–Esseen theorem. Theory of Probability & Its Applications, 56(3), 403–419.
Lauwerier, H. A. (1963). The asymptotic expansion of the statistical distribution of N. V. Smirnov. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 61–68.
Lindeberg, J. W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 15, 211–225.
Petrov, V. V. (1975). Sums of independent random variables. New York: Springer.
Prokhorov, Y. V., & Ulyanov, V. V. (2013). Some approximation problems in statistics and probability. Limit theorems in probability, statistics and number theory (Vol. 42, pp. 235–249)., Springer proceedings in mathematics & statistics Heidelberg: Springer.
Ulyanov, V. V. (1986). Normal approximation for sums of nonidentically distributed random variables in Hilbert spaces. Acta Scientiarum Mathematicarum, 50(3–4), 411–419.
Ulyanov, V. V. (1987). Asymptotic expansions for distributions of sums of independent random variables in H. Theory of Probability and its Applications, 31(1), 25–39.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Fujikoshi, Y., Ulyanov, V.V. (2020). General Approach to Constructing Non-Asymptotic Bounds. In: Non-Asymptotic Analysis of Approximations for Multivariate Statistics. SpringerBriefs in Statistics(). Springer, Singapore. https://doi.org/10.1007/978-981-13-2616-5_11
Download citation
DOI: https://doi.org/10.1007/978-981-13-2616-5_11
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2615-8
Online ISBN: 978-981-13-2616-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)