REFERENCES
H. G. Tucker, "Convolutions of distributions attracted to stable laws," Ann. Math. Statist., 39, 1381-1390 (1968).
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Massachusetts (1954).
B. V. Gnedenko and V. Yu. Korolev, Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton, Florida (1996).
S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, Chichester-New York (1991).
V. M. Zolotarev, Modern Theory of Summation of Random Variables, VSP, Utrecht (1997).
T. R. Kashaev and V. Yu. Korolev, "Natural estimates of the accuracy of approximation of distributions of random sums by location mixtures of stable laws," J. Math. Sci. (2003) (to appear).
G. Englund, "A remainder term estimate in a random-sum central limit theorem," Theor. Probab. Appl., 28, No. 1, 143-149 (1983).
V. M. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums [in Russian], Moscow State Univ. Press, Moscow (1990).
G. Christoph and W. Wolf, Convergence Theorems with a Stable Limit Law, Akademie Verlag, Berlin (1992).
V. Paulauskas, "On a strengthening of the Lyapunov theorem," Lit. Math. J., 9, No. 2, 323-328 (1969).
A. A. Mitalauskas, "An estimate of convergence rate in the integral limit theorem in the case of convergence to a stable law," Lit. Math. J., 11, No. 3, 627-639 (1971).
I. I. Banis, "Convergence rate estimates in the integral limit theorem," Lit. Math. J., 12, No. 1, 41-46 (1972).
V. M. Zolotarev, "An estimate of the closeness of two convolutions of distributions," in: International Vilnius Conference on Probability Theory and Mathematical Statistics. Abstracts of Communications, Vol. 1, Vilnius (1973), pp. 257-259.
G. Christoph, "Über notwendige und hinreichende Bedingungen für Konvergenzaussagen im Falle einer stabilen Grenzverteilung," Z. Wahrsch. verw. Geb., 54, No. 1, 29-40 (1980).
A. Karoblis, "On approximation to the distributions of sums of independent random variables," Lit. Math. J., 23, No. 1, 101-107 (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kashaev, T.R., Korolev, V.Y. Natural Estimates of the Accuracy of Approximation of the Distributions of Random Sums by Location Mixtures of Stable Laws. Journal of Mathematical Sciences 123, 3741–3750 (2004). https://doi.org/10.1023/B:JOTH.0000036315.45324.70
Issue Date:
DOI: https://doi.org/10.1023/B:JOTH.0000036315.45324.70