Abstract
We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+⋯+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t −β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.
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References
Borovkov A. A. and Borovkov K. A., Asymptotic Analysis of Random Walks. Part I: Slowly Decreasing Jumps [in Russian], Nauka, Moscow (to appear).
Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge (1987).
Borovkov A. A. and Mogul’skii A. A., “Integro-local limit theorems including large deviations for sums of random vectors. I,” Teor. Veroyatnost. i Primenen., 43, No. 1, 3–17 (1998).
Rozovskii L. V., “Probabilities of large deviations of sums of independent random variables with a common distribution function that belongs to the domain of attraction of a normal law,” Theory Probab. Appl., 34, No. 4, 625–644 (1989).
Nagaev S. V., “Large deviations for sums of independent random variables,” Ann. Probab., 7, No. 5, 745–789 (1979).
Nagaev A. V., “Limit theorems accounting for large deviations with Cramér’s condition violated,” Izv. Akad. Nauk UzSSSR Ser. Fiz.-Mat. Nauk, 6, 17–22 (1969).
Mikosh T. and Nagaev A. V., “Large deviations of heavy-tailed sums with applications in insurance,” Extremes, 1, No. 1, 81–110 (1998).
Pinelis I. F., “A problem on large deviations in a space of trajectories,” Theory Probab. Appl., 26, No. 1, 69–84 (1981).
Stone C., “On local and ratio limit theorems,” in: Proc. of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. Vol. II, part II, Univ. of California Press, Berkeley; Los Angeles, 1966, pp. 217–224.
Stone C., “A local limit theorem for nonlattice multi-dimensional distribution functions,” Ann. Math. Statist., 36, 546–551 (1965).
Borovkov A. A. and Mogul’skii A. A., “Integro-local theorems for sums of independent random vectors in the series scheme,” Math. Notes, 79, No. 4, 468–482 (2006).
Borovkov A. A. and Mogul’skii A. A., “Limit theorems for the sums of random variables with semiexponential distributions on the whole axis,” Siberian Math. J., 47, No. 6, 990–1026 (2006).
Borovkov A. A., “Large deviation probabilities for random walks with semiexponential distributions,” Siberian Math. J., 41, No. 6, 1061–1093 (2000).
Borovkov A. A., “On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums,” Siberian Math. J., 43, No. 6, 995–1022 (2002).
Borovkov A. A. and Mogul’skii A. A., “Large deviations and superlarge deviations of the sums of independent random variables with the Cramér condition. I,” Teor. Veroyatnost. i Primenen., 51, No. 2, 260–294 (2006).
Borovkov A. A. and Mogul’skii A. A., “Large deviations and superlarge deviations of the sums of independent random variables with the Cramér condition. II,” Teor. Veroyatnost. i Primenen., 51, No. 4, 641–673 (2006).
Gnedenko B. D., “On a local limit theorem of probability,” Uspekhi Mat. Nauk, 3, No. 3, 187–194 (1948).
Shepp L. A., “A local limit theorem,” Ann. Math. Statist., 35, 419–423 (1964).
Borovkov A. A. and Mogul’skii A. A., Large Deviations and Testing of Statistical Hypotheses [in Russian], Nauka, Novosibirsk (1992).
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Original Russian Text Copyright © 2008 Mogul’skiĭ A. A.
The author was supported by the Russian Foundation for Basis Research (Grants 05-01-00810, 07-01-00595, and 08-01-00962) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh-8980.2006.1 and RNSh.2.1.1.1379).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 837–854, July–August, 2008.
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Mogul’skii, A.A. An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions. Sib Math J 49, 669–683 (2008). https://doi.org/10.1007/s11202-008-0064-2
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DOI: https://doi.org/10.1007/s11202-008-0064-2