Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-10-31T22:48:53.191Z Has data issue: false hasContentIssue false

Infinitely divisible approximations for discrete nonlattice variables

Published online by Cambridge University Press:  01 July 2016

V. Čekanavičius*
Affiliation:
Vilnius University
*
Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Email address: vydas.cekanavicius@maf.vu.lt

Abstract

Sums of independent random variables concentrated on discrete, not necessarily lattice, set of points are approximated by infinitely divisible distributions and signed compound Poisson measures. A version of Kolmogorov's first uniform theorem is proved. Second-order asymptotic expansions are constructed for distributions with pseudo-lattice supports.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arak, T. V. (1981a). On the convergence rate in Kolmogorov's uniform limit theorem I. Theory Prob. Appl. 26, 219239.CrossRefGoogle Scholar
Arak, T. V. (1981b). On the convergence rate in Kolmogorov's uniform limit theorem II. Theory Prob. Appl. 26, 437451.Google Scholar
Arak, T. V. and Zaitsev, A. Yu. (1988). Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174, 1222.Google Scholar
Barbour, A. D. and Čekanavičius, V. (2002). Total variation asymptotics for sums of independent integer random variables. Ann. Prob. 30, 509545.CrossRefGoogle Scholar
Barbour, A. D. and Chryssaphinou, O. (2001). Compound Poisson approximation: a user's guide. Ann. Appl. Prob. 11, 9641002.Google Scholar
Barbour, A. D. and Xia, A. (1999). Poisson perturbations. ESAIM Prob. Statist. 3, 131150.Google Scholar
Bergström, H., (1951). On asymptotic expansion of probability functions. Skand. Aktuar. 1, 134.Google Scholar
Bikelis, A. (1996). Asymptotic expansions for distributions of statistics. In Proc. 36th Conf. Lithuanian Math. Soc. (Vilnius, 1995), eds, Kudžma, R. and Mackevičius, V., Vilnius University Press, pp. 528 (in Russian).Google Scholar
Booth, J. G., Hall, P. and Wood, A. T. A. (1994). On the validity of Edgeworth and saddlepoint approximations. J. Multivariate Analysis 51, 121138.CrossRefGoogle Scholar
Borisov, I. S. and Ruzankin, P. S. (2002). Poisson approximation for expectations of unbounded functions of independent random variables. Ann. Prob. 30, 16571680.Google Scholar
Borovkov, K. and Pfeifer, D. (1996). Pseudo-Poisson approximation for Markov chains. Stoch. Process. Appl. 61, 163180.Google Scholar
Brown, T. C., Weinberg, G. V. and Xia, A. (2000). Removing logarithms from Poisson process error bounds. Stoch. Process. Appl. 87, 149165.CrossRefGoogle Scholar
Čekanavicius, V., (1998). Estimates in total variation for convolutions of compound distributions. J. Lond. Math. Soc. 58, 748760.Google Scholar
Čekanavicius, V., (1999). On compound Poisson approximations under moment restrictions. Theory Prob. Appl. 44, 7486.Google Scholar
Čekanavičius, V. and Wang, Y. H. (2003). Compound Poisson approximations for sums of discrete non-lattice variables. Adv. Appl. Prob. 35, 228250.Google Scholar
Chen, L. H. Y. (1998). Stein's method: some perspectives with applications. In Probability Towards 2000 (Lecture Notes Statist. 128), eds Accardi, L. and Heyde, C. C., Springer, New York, pp. 97122.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit distributions for Sums of Independent Random Variables, revised edn. Addison-Wesley, Reading, MA.Google Scholar
Hipp, C. (1986). Improved approximations for the aggregate claims distribution in the individual model. ASTIN Bull. 16, 89100.Google Scholar
Ibragimov, I. A. and Presman, E. L. (1973). On the rate of approach of the distributions of sums of independent random variables to accompanying distributions. Theory Prob. Appl. 18, 713727.Google Scholar
Kolmogorov, A. N. (1956). Two uniform limit theorems for sums of independent random variables. Theory Prob. Appl. 1, 384394.Google Scholar
Kolmogorov, A. N. (1963). On the approximation of distributions of sums of independent summands by infinitely divisible distributions. Sankhy=a A 25, 159174.Google Scholar
Kornya, P. (1983). Distribution of aggregate claims in the individual risk theory model. Soc. Actuar. Trans. 35, 823858.Google Scholar
Kruopis, J. (1986). Approximations for distributions of sums of lattice random variables I. Lithuanian Math. J. 26, 234244.CrossRefGoogle Scholar
Le Cam, L. (1965). On the distribution of sums of independent random variables. In Bernoulli 1713, Bayes 1763, Laplace 1813. Anniversary Volume. Springer, Berlin, pp. 179202.Google Scholar
Michel, R. (1988). An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. ASTIN Bull. 17, 165169.CrossRefGoogle Scholar
Presman, E. L. (1983). Approximation of binomial distributions by infinitely divisible ones. Theory Prob. Appl. 28, 393403.Google Scholar
Prohorov, Yu. V. (1952). Some refinements of Lyapunov's theorem. Izvest. Akad. Nauk SSSR Ser. Mat. 16, 281292 (in Russian).Google Scholar
Prohorov, Yu. V. (1953). Asymptotic behaviour of the binomial distribution. Uspek-hi Mate. Nauk. 8, 135142 (in Russian). English translation: Select. Transl. Math. Statist. Prob. 1 (1961), 87–95.Google Scholar
Prohorov, Yu. V. (1960). On a uniform limit theorem of A. N. Kolmogorov. Theory Prob. Appl. 5, 98106.Google Scholar
Roos, B. (1999). Asymptotics and sharp bounds in the Poisson approximation to the Poisson–binomial distribution. Bernoulli 5, 10211034.CrossRefGoogle Scholar
Roos, B. (2002). Kerstan's method in the multivariate Poisson approximation: an expansion in the exponent. Teorya Veroyatn. Primen. 47, 397401.CrossRefGoogle Scholar
Roos, B. (2003). Kerstan's method for compound Poisson approximation. Ann. Prob. 31.Google Scholar
Studnev, Yu. P. (1960). An approximation to the distribution of sums by infinitely divisible laws. Theory Prob. Appl. 5, 421425.Google Scholar
Zaitsev, A. Yu. (1991). An example of a distribution whose set of n-fold convolutions is uniformly separated from the set of infinitely divisible laws in the sense of the variation distance. Theory Prob. Appl. 36, 419425.CrossRefGoogle Scholar
Zaitsev, A. Yu. (1992). On the approximation of convolutions of multidimensional symmetric distributions by accompanying laws. J. Soviet. Math. 61, 18591872.CrossRefGoogle Scholar
Zaitsev, A. Yu. (1996). Approximation of convolutions by accompanying laws under the existence of moments of low order. Zapiski Nauchn. Semin. POMI 228, 135141, 359 (in Russian).Google Scholar