A NOTE ON STRONG CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATION
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초록
The sub-linear expectation space is a nonlinear expecta- tion space having advantages of modeling the uncertainty of probability and statistics. Strong convergence for non-additive probabilities or non- linear expectation are challenging issues which have raised progressive interest recently. In the sub-linear expectation space, we use capac- ity and sub-linear expectation to replace probability and expectation of classical probability theory. Recently Zhang has proved very important theorems in the sub-linear expectation space which can be looked upon as being extensions of Kolmogorov's three series theorem in classical probability theory. Zhang and Lin obtained Marcinkiewicz's strong law of large numbers for sums of independent random variables under sub- linear expectation space. In addition, they have given a theorem about strong convergence of a random series for independent random variables under sub-linear expectation. In this paper we investigate the strong convergence for sums of inde- pendent random variables under sub-linear expectation and gives also almost surely convergence of sums of independent random variables in capacity. They are extensions of strong convergence of a random series and almost surely convergence of sums for independent random variables in the framework of sub-linear expectation. By using Zhang's result, the proof rests on the methods of proof due to Petrov's result concerning the almost sure convergence of series of independent random variable. As an application Marcinkiewicz's strong law of large number for nonlinear expectation are obtained.
Abstract
The sub-linear expectation space is a nonlinear expecta- tion space having advantages of modeling the uncertainty of probability and statistics. Strong convergence for non-additive probabilities or non- linear expectation are challenging issues which have raised progressive interest recently. In the sub-linear expectation space, we use capac- ity and sub-linear expectation to replace probability and expectation of classical probability theory. Recently Zhang has proved very important theorems in the sub-linear expectation space which can be looked upon as being extensions of Kolmogorov's three series theorem in classical probability theory. Zhang and Lin obtained Marcinkiewicz's strong law of large numbers for sums of independent random variables under sub- linear expectation space. In addition, they have given a theorem about strong convergence of a random series for independent random variables under sub-linear expectation. In this paper we investigate the strong convergence for sums of inde- pendent random variables under sub-linear expectation and gives also almost surely convergence of sums of independent random variables in capacity. They are extensions of strong convergence of a random series and almost surely convergence of sums for independent random variables in the framework of sub-linear expectation. By using Zhang's result, the proof rests on the methods of proof due to Petrov's result concerning the almost sure convergence of series of independent random variable. As an application Marcinkiewicz's strong law of large number for nonlinear expectation are obtained.
참고문헌 (20)
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