Multivariate large deviations for sums of i.i.d. random vectors with compactly supported distribution

• Autorzy: Zaigraev A.
• Języki publikacji: EN

Abstrakty

EN

The sums of i.i.d. random vectors with compactly supported and absolutely continuous distribution are considered. Under some conditions the strong form of the local limit theorem for large deviations is proved. In passing the asymptotic behaviour of the moment generating function as well as possible non-degenerate limit laws for the natural exponential family of distributions are established.

Słowa kluczowe

EN

Abel theorem; Cramér transformation; Fenchel-Legendre transformation; Laplace method; local limit theorem; natural exponential family of distributions

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2003
• Tom: Vol. 23, Fasc. 2
• Strony: 315--335
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Zaigraev A.

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] A. Balkema, C. Klüppelberg and S. Resnick, Limit laws for exponential families, Bernoulli 5 (6) (1999), pp. 951-968.
• [2] A. Balkema, C. Klüppelberg and S. Resnick, Stability for multivariate exponential families, J. Math. Sci. 106 (2) (2001), pp. 2777-2791.
• [3] R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, Wiley, New York 1976.
• [4] A. A. Borovkov and A. A. Mogulskii, Large deviations and testing statistical hypotheses. I. Large deviations of sums of random vectors, Siberian Adv. Math. 2 (3) (1992), pp. 52-120.
• [5] A. A. Borovkov and B. A. Rogozin, On the central limit theorem in multidimensional case (in Russian), Theory Probab. Appl. 10 (1965), pp. 61-69.
• [6] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston 1993.
• [7] J. D. Deuschel and D. W. Strook, Large Deviations, Academic Press, Boston 1989.
• [8] R. J. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer, New York 1985.
• [9] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York 1971.
• [10] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge Univ. Press, Cambridge 1996.
• [11] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen 1971.
• [12] A. V. Nagaev and A. Zaigraev, Abelian theorems for a class of probability distributions in Rd and their application, J. Math. Sci. 99 (4) (2000), pp. 1454-1462.
• [13] A. V. Nagaev and A. Zaigraev, Abelian theorems, limit properties of conjugate distributions and large deviations for sums of independent random vectors (in Russian), Theory Probab. Appl. 48 (2003).
• [14] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York 1987.