Complete convergence type theorems for sums of elements of linear autoregression sequences
DOI:
https://doi.org/10.20535/mmtu-2019.2-011Keywords:
Linear autoregression models, SLLN, Sums of independent random variables, Weighted sums, Complete convergence, Hsu–Robbins–Erdős series, Spitzer series, Baum–Katz seriesAbstract
In this paper, we study limit behaviour of sums $(S_n)$ whose terms are elements of linear autoregression sequences, in the sense of complete convergence and convergence of so-called Spitzer–Baum–Katz series $\sum_{n=1}^\infty n^{\frac{r}{p}-2}\mathbb{P} \Big\{\frac{|S_n|}{n^{1\!/\!p}}>\varepsilon \Big\}$, for any $\varepsilon>0$, where $0References
Baum, L., & Katz, M. (1965). Convergence rates in the law of large numbers. Transactions of the American Mathematical Society, 120, 108–123. https://doi.org/10.1090/S0002-9947-1965-0198524-1
Buraczewski, D., Damek, E., & Mikosch, T. (2016). Stochastic models with power-law tails. The equation $X=AX+B.$ Springer International Publishing. https://doi.org/10.1007/978-3-319-29679-1
Choi, B., & Sung, S. (1987). Almost sure convergence theorems of weighted sums of random variables. Stochastic Anal. Appl., 5(4), 365–377. https://doi.org/10.1080/07362998708809124
Cuzick, J. (1995). A strong law for weighted sums of i.i.d. random variables. Journal of Theoretical Probability, 8(3), 625–641. https://doi.org/10.1007/BF02218047
Erdős, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist., 20, 286–291. https://doi.org/10.1214/aoms/1177730037
Gut, A. (1978). Marcinkiewicz laws and convergence rates in the law of large numbersfor random variables with multidimensional indices. Ann. Probab., 6(3), 469–482. https://doi.org/10.1214/aop/1176995531
Gut, A. (1992). Complete convergence for arrays. Periodica Mathematica Hungarica, 25(1), 51–75. https://doi.org/10.1007/BF02454383
Hsu, P., & Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A., 33(2), 25–31. https://doi.org/10.1073/pnas.33.2.25
Hu, T.-C., Szynal, D., & Volodin, A. (1998). A note on complete convergence for arrays. Statistics & Probability Letters, 38(1), 27–31. https://doi.org/10.1016/S0167-7152(98)00150-3
Lin, Z., & Bai, Z. (2010). Probability inequalities. Science Press Beijing and Springer Verlag.
Marcinkiewicz, J., & Zygmund, A. (1937). Sur les fonctions indépendantes. Fund. Math., 29(1), 60–90. http://eudml.org/doc/212925
Pruitt, W. (1966). Summability of independent random variables. Journal of Mathematics and Mechanics, 15(5), 769–776. https://www.jstor.org/stable/24901430
Rosalsky, A., & Sreehari, M. (1998). On the limiting behavior of randomly weighted partial sums. Statistics & Probability Letters, 40(4), 403–410. https://doi.org/10.1016/S0167-7152(98)00153-9
Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82(2), 323–339. https://doi.org/10.2307/1993051
Sung, S. (2001). Strong laws for weighted sums of i.i.d. random variables. Statistics & Probability Letters, 52(4), 413–419. https://doi.org/10.1016/S0167-7152(01)00020-7
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