Lindeberg’s Method for Moderate Deviations and Random Summation
- 105 Downloads
We apply Lindeberg’s method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation principles for martingales as well as for random sums, in the latter situation in both the cases when the limit distribution is Gaussian or non-Gaussian. Moreover, in the Gaussian case we show moderate deviations for random sums using bounds on cumulants, alternatively. Finally, we also prove a large deviation principle for certain random sums.
KeywordsRandom sums Moderate and large deviations Lindeberg’s method
Mathematics Subject Classification60F05 60F10 60G50
We are very grateful to an anonymous referee for a very careful reading of a first version of this manuscript. His comments helped to improve the correctness of the paper.
- 4.Deltuviene, D., Saulis, L.: Normal approximation for sum of random number of summands. Lith. Math. J. 47, 531–537 (2007)Google Scholar
- 7.Döbler, C.: On rates of convergence and Berry-Esseen bounds for random sums of centered random variables with finite third moments, preprint, arXiv:1212.5401 (2013)
- 16.Kalashnikov, V.: Geometric sums: bounds for rare events with applications. Mathematics and its Applications, vol. 413, Kluwer Academic Publishers Group, Dordrecht, Risk analysis, reliability, queueing (1997)Google Scholar
- 17.Kasparavičiūtė, A.: Theorems of large deviations for the sums of a random number of independent random variables, doctoral dissertation. vilnius: Technika (2013)Google Scholar
- 24.Saulis, L., Statulevičius, V.A.: Limit theorems for large deviations, Mathematics and its Applications (Soviet Series), vol. 73, Kluwer Academic Publishers Group, Dordrecht (1991). Translated and revised from the 1989 Russian originalGoogle Scholar
- 26.Toda, A.A.: Weak limit of the geometric sum of independent but not identically distributed random variables, arXiv:1111.1786v2 (2012)
- 27.Varadhan, S.R.S.: Large deviations and applications. École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, Lecture Notes in Math., vol. 1362. Springer, Berlin, pp. 1–49 (1988)Google Scholar