Multidimensional compound Poisson approximations for symmetric distributions
Pub. online: 26 May 2026
Type: Research Article
Open Access
Open Access
Received
29 January 2026
29 January 2026
Revised
13 May 2026
13 May 2026
Accepted
14 May 2026
14 May 2026
Published
26 May 2026
26 May 2026
Abstract
Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström-type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O({n^{-1}})$.
References
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon Press, Oxford (1992). MR1163825
Barbour, A.D., Chryssaphinou, O., Vaggelatou, E.: Applications of compound Poisson approximation. In: Charalambides, C.A., Koutras, M.V., Balakrishnan, N. (eds.) Probability and Statistical Models with Applications, pp. 41–62. Chapman & Hall (2001). MR1865030. https://doi.org/10.1214/aoap/1015345355
Barman, K., Upadhye, N.S., Vellaisamy, P.: Approximations related to tempered stable distributions. Mod. Stoch. Theory Appl. 12(3), 325–346 (2025). MR4917904. https://doi.org/10.15559/25-vmsta275
Barbour, A.D., Luczak, M.J., Xia, A.: Multivariate approximation in total variation II: discrete normal approximation. Ann. Probab. 46, 1405–1440 (2018). MR3785591. https://doi.org/10.1214/17-AOP1205
Bergström, H.: On asymptotic expansion of probability functions. Skand. Aktuarietidskr. 1, 1–34 (1951). MR0043403. https://doi.org/10.1080/03461238.1951.10432122
Bobkov, S.G., Götze, F.: Berry–Esseen bounds in the local limit theorems. Lith. Math. J. 65, 50–66 (2025). MR4885697. https://doi.org/10.1007/s10986-025-09659-1
Čekanavičius, V., Roos, B.: Compound binomial approximations. Ann. Inst. Stat. Math. 58, 187–210 (2006). MR2281210. https://doi.org/10.1007/s10463-005-0018-4
Genest, C., Marceau, E., Mesfioui, M.: Compound Poisson approximations for individual models with dependent risks. Insur. Math. Econ. 32(1), 73–91 (2003). MR3600678. https://doi.org/10.1016/j.insmatheco.2016.10.012
Götze, F., Yu, Z.A.: On alternative approximating distributions in the multivariate version of Kolmogorov’s second uniform limit theorem. Teor. Veroâtn. Primen. 61(1), 3–22 (2022). Transl.: Theory Probab. Appl., 67, 1, 1–16. MR4466409. https://doi.org/10.1137/s0040585x97t99071x
Hipp, C.: Improved approximations for the aggregate claims distribution in the individual model. ASTIN Bull. 16(2), 89–100 (1986). https://doi.org/10.2143/AST.16.2.2015001
Jokubauskienė, S., Čekanavičius, V.: Multivariate Hipp-type compound Poisson approximations for lattice distributions I. Lith. Math. J. 65, 337–356 (2025). MR4953667. https://doi.org/10.1007/s10986-025-09686-y
Kruopis, J.: Precision of approximations of the generalized Binomial distribution by convolutions of Poisson measures. Litovsk. Mat. Sb. 26(1), 53–69 (1986). (Russian). Transl.: Lith. Math. J., 26(1), 37–49. MR0847204
Kruopis, J., Čekanavičius, V.: Compound Poisson approximations for symmetric vectors. J. Multivar. Anal. 123, 30–42 (2014). MR3130419. https://doi.org/10.1016/j.jmva.2013.08.017
Lishamol, T., Veena, G.: A retrospective study of Skellam and related distributions. Austrian J. Stat. 51, 102–111 (2022). https://doi.org/10.17713/ajs.v51i1.1224
Lu, D., Chen, Y., Wang, L.: Asymptotics for the joint tail probability of bidimensional randomly weighted sums of dependent and heavy-tailed random variables. Lith. Math. J. 66, 79–92 (2026). MR5022834. https://doi.org/10.1007/s10986-026-09709-2
Novak, S.Y.: Extreme value methods with applications to finance. Chapman & Hall/CRC Press, London (2011). MR2933280
Roos, B.: Binomial approximation to the Poisson binomial distribution: the Krawtchouk expansion. Teor. Veroâtn. Primen. 45(2), 328–344 (2000). Reprinted in: Theory Probab. Appl., 45(2), 258–272. MR1967760. https://doi.org/10.4213/tvp466
Roos, B.: Multinomial and Krawtchouk approximations to the generalized multinomial distribution. Teor. Veroâtn. Primen. 46(1), 117–133 (2001). Reprinted in: Theory Probab. Appl. 46(1), 103–117. MR1968708. https://doi.org/10.4213/tvp3954
Roos, B.: Kerstan’s method in the multivariate Poisson approximation: an expansion in the exponent. Teor. Veroâtn. Primen. 43(2), 397–402 (2002). Reprinted in: Theory Probab. Appl., 47(2), 358–363. MR2003208. https://doi.org/10.4213/tvp3672
Roos, B.: On variational bounds in the compound Poisson approximation of the individual risk model. Insur. Math. Econ. 40(3), 403–414 (2007). MR2310979. https://doi.org/10.1016/j.insmatheco.2006.06.003
Roos, B.: Refined total variation bounds in the multivariate and compound Poisson approximation. ALEA Lat. Am. J. Probab. Math. Stat. 14, 337–360 (2017). MR3647296. https://doi.org/10.30757/alea.v14-19
Sengar, A.S., Upadhye, N.S.: Subordinated compound Poisson processes of order k. Mod. Stoch. Theory Appl. 7(4), 395–413 (2020). MR4195643. https://doi.org/10.15559/20-vmsta165
Sundt, B., Vernic, R.: Recursions for Convolutions and Compound Distributions with Insurance Applications. EAA Lecture Notes. Springer, Dordrecht, Heidelberg, London, New York (2009). MR2502743. https://doi.org/10.1007/978-3-540-92900-0
Vellaisamy, P., Chaudhuri, B.: Poisson and compound Poisson approximations for random sums or random variables. J. Appl. Probab. 33, 127–137 (1996). MR1371960. https://doi.org/10.2307/3215270
Yu, Z.A.: Estimates for the closeness of successive convolutions of multidimensional symmetric distributions. Probab. Theory Relat. Fields 79, 175–200 (1988). MR0958287. https://doi.org/10.1007/BF00320918
Yu, Z.A.: Approximation of convolutions of multi-dimensional symmetric distributions by accompanying laws. J. Sov. Math. 177(1), 55–72 (1989). (Russian). Transl, 61, 1859–1872, 1992. MR1053124. https://doi.org/10.1007/BF01362793
Yu, Z.A.: On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance. Teor. Veroâtn. Primen. 69(2), 272–284 (2024). (Russian). Transl.: Theory Prob. Appl. 69(2), 217–226. MR4912064. https://doi.org/10.4213/tvp5690
