A strong law of large numbers for weighted sums of i.i.d. random variables under capacities. (English) Zbl 1442.60038

Zhang, Defei; He, Ping

J. Appl. Math. 2014, Article ID 412758, 5 p. (2014).

Summary: With the notion of independent identically distributed (i.i.d.) random variables under sublinear expectations initiated by Peng, a strong law of large numbers for weighted sums of i.i.d. random variables under capacities induced by sublinear expectations is obtained.

MSC:

60F15

Strong limit theorems

References:

[1]	Chow, Y. S.; Lai, T. L., Limiting behavior of weighted sums of independent random variables, Annals of Probability, 1, 5, 810-824 (1973) · Zbl 0303.60025
[2]	Stout, W. F., Almost Sure Convergence (1974), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0321.60022
[3]	Choi, B. D.; Sung, S. H., Almost sure convergence theorems of weighted sums of random variables, Stochastic Analysis and Applications, 5, 4, 365-377 (1987) · Zbl 0633.60049
[4]	Cuzick, J., A strong law for weighted sums of i.i.d. random variables, Journal of Theoretical Probability, 8, 3, 625-641 (1995) · Zbl 0833.60031
[5]	Rosalsky, A.; Sreehari, M., On the limiting behavior of randomly weighted partial sums, Statistics & Probability Letters, 40, 4, 403-410 (1998) · Zbl 0937.60014
[6]	Wu, W. B., On the strong convergence of a weighted sum, Statistics & Probability Letters, 44, 1, 19-22 (1999) · Zbl 0951.60027
[7]	Bai, Z. D.; Cheng, P. E., Marcinkiewicz strong laws for linear statistics, Statistics & Probability Letters, 46, 2, 105-112 (2000) · Zbl 0960.60026
[8]	Bai, Z. D.; Cheng, P. E.; Zhang, C. H., An extension of the Hardy-Littlewood strong law, Statistica Sinica, 7, 4, 923-928 (1997) · Zbl 1067.60501
[9]	Chen, Z.; Epstein, L., Ambiguity, risk, and asset returns in continuous time, Econometrica, 70, 4, 1403-1443 (2002) · Zbl 1121.91359
[10]	Huber, P.; Strassen, V., Minimax tests and the Neyman-Pearson Lemma for capacities, The Annals of Statistics, 1, 2, 251-263 (1973) · Zbl 0259.62008
[11]	Wakker, P., Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle, Econometrica, 69, 4, 1039-1059 (2001) · Zbl 1021.91016
[12]	Marinacci, M., Limit laws for non-additive probabilities and their frequentist interpretation, Journal of Economic Theory, 84, 2, 145-195 (1999) · Zbl 0921.90005
[13]	Maccheroni, F.; Marinacci, M., A strong law of large numbers for capacities, The Annals of Probability, 33, 3, 1171-1178 (2005) · Zbl 1074.60041
[14]	Peng, S., \(G\)-Expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications: The Abel Symposium 2005. Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, 2, 541-567 (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1131.60057
[15]	Peng, S., Law of large numbers and central limit theorem under nonlinear expectations · Zbl 1434.60075
[16]	Peng, S., A new central limit theorem under sublinear expectations
[17]	Denis, L.; Hu, M.; Peng, S., Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion paths, Potential Analysis, 34, 2, 139-161 (2011) · Zbl 1225.60057
[18]	Chen, Z.; Wu, P.; Li, B., A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54, 3, 365-377 (2013) · Zbl 1266.60051
[19]	Peng, S., Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and \(G\)-Brownian motion
[20]	Artzner, P.; Delbaen, F.; Eber, J.; Heath, D., Thinking coherently, Risk, 10, 1, 68-71 (1997)