Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions
Pub. online: 13 February 2025
Type: Research Article
Open Access
Open Access
1
Part of this work was conducted while the author was a visiting scholar at the University of Cyprus.
Received
29 July 2024
29 July 2024
Revised
29 January 2025
29 January 2025
Accepted
29 January 2025
29 January 2025
Published
13 February 2025
13 February 2025
Abstract
In the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Lüroth digits and also covers some new cases that have never been studied before. Next, any assumptions concerning the structure of the Oppenheim expansions are dropped and a result concerning trimmed sums is proven when at least two summands are trimmed; combining this latter theorem with the asymptotic behavior of the r-th maximum term of the expansion, a convergence result is obtained for the case in which only the largest summand is deleted from the sum.
References
Aaronson, J., Nakada, H.: Trimmed sums for non-negative, mixing stationary processes. Stoch. Process. Appl. 104(2), 173–192 (2003). MR1961618. https://doi.org/10.1016/S0304-4149(02)00236-3
Berkes, I., Horvath, L., Schauer, J.: Asymptotic behavior of trimmed sums. Stoch. Dyn. 12(01), 1150002 (2012). MR2887914. https://doi.org/10.1142/S0219493712003602
Csörgo, S., Simons, G.: A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Stat. Probab. Lett. 26(1), 65–73 (1996). MR1385664. https://doi.org/10.1016/0167-7152(94)00253-3
Einmahl, J., Haeusler, E., Mason, D.: On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics. Probab. Theory Relat. Fields 79(1), 59–74 (1988). MR0952994. https://doi.org/10.1007/BF00319104
Galambos, J.: Futher ergodic results on the Oppenheim series. Q. J. Math. 25(1), 135–141 (1974). MR0347759. https://doi.org/10.1093/qmath/25.1.135
Giuliano, R.: Convergence results for Oppenheim expansions. Monatshefte Math. 187(3), 509–530 (2018). MR3858429. https://doi.org/10.1007/s00605-017-1126-y
Giuliano, R., Hadjikyriakou, M.: On exact laws of large numbers for Oppenheim expansions with infinite mean. J. Theor. Probab. 34, 1579–1606 (2021). MR4289895. https://doi.org/10.1007/s10959-020-01010-3
Giuliano, R., Hadjikyriakou, M.: Intermediately trimmed sums of Oppenheim expansions: a strong law. arXiv preprint arXiv:2310.00669 (2023)
Mori, T.: The strong law of large numbers when extreme terms are excluded from sums. Z. Wahrscheinlichkeitstheor. Verw. Geb. 36(3), 189–194 (1976). MR0423494. https://doi.org/10.1007/BF00532544
Mori, T.: Stability for sums of iid random variables when extreme terms are excluded. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40(2), 159–167 (1977). MR0458542. https://doi.org/10.1007/BF00532880
