Abstract
This paper provides sharp sufficient conditions for almost sure summability of the maximal normed partial sums of m-dependent random elements in stable type p Banach spaces, complementing recent results of Li, Qi, and Rosalsky (Trans Amer Math Soc 368(1):539–561, 2016) and Thành (Math Nachr 296(1):402–423, 2023). The main theorems are new even when the underlying random elements are independent. Our proof is direct and can be extended to other dependence structures. Two illustrative examples are also presented.
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References
Hechner, F., Heinkel, B.: The Marcinkiewicz–Zygmund LLN in Banach spaces: a generalized martingale approach. J. Theor. Probab. 23(2), 509–522 (2010)
Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11(1), 286–295 (1983)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer, Berlin (1991)
Li, D., Qi, Y., Rosalsky, A.: An extension of theorems of Hechner and Heinkel. In: Asymptotic Laws and Methods in Stochastics, pp. 129–147. Fields Inst. Commun., 76. Fields Inst. Res. Math. Sci., Toronto, ON (2015)
Li, D., Qi, Y., Rosalsky, A.: A characterization of a new type of strong law of large numbers. Trans. Amer. Math. Soc. 368(1), 539–561 (2016)
Rosalsky, A., Thành, L.V.: A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers. Stat. Probab. Lett. 178, Paper No. 109181, 10 pp. (2021)
Rosalsky, A., Thành, L.V.: Optimal moment conditions for complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements in Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117(3), Paper No. 108, 15 pp. (2023)
Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343–356 (2000)
Thành, L.V.: On the Brunk–Chung type strong law of large numbers for sequences of blockwise \(m\)-dependent random variables. ESAIM Probab. Stat. 10, 258–268 (2006)
Thành, L.V.: On the \((p, q)\)-type strong law of large numbers for sequences of independent random variables. Math. Nachr. 296(1), 402–423 (2023)
Wu, Y., Wang, X.: Strong laws for weighted sums of \(m\)-extended negatively dependent random variables and its applications. J. Math. Anal. Appl. 494(2), Paper No. 124566, 23 pp. (2021)
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The author is grateful to the reviewer for carefully reading the manuscript and for offering useful and detailed comments and suggestions which enabled him to improve the paper.
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Thành, L.V. Almost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spaces. Arch. Math. 122, 203–212 (2024). https://doi.org/10.1007/s00013-023-01929-z
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DOI: https://doi.org/10.1007/s00013-023-01929-z
Keywords
- Maximal normed partial sum
- Almost sure summability
- Rademacher type p Banach space
- Stable type p Banach space
- (p
- q)-type strong law of large numbers