In this paper, we investigate the law of the logarithm for delayed sums under sublinear expectation. The necessary and sufficient conditions for the law of the logarithm for delayed sums under sublinear expectation are established which differ from those for partial sums. The law of the logarithm for delayed sums can be treated as a tool to prove the convergence rates of logarithm form for partial sums under sublinear expectation. These results extend the results of Lai [13] from the classical probability setting to the sublinear expectation framework.
A theory of autoregressive (AR) sequences under sublinear expectations is developed in this paper. AR sequences play a significant role in linear time series analysis. The traditional AR time series model requires the data to be stationary; however, data in real world often obey different distributions at different time points, which makes the traditional AR model invalid. To characterize the AR property of this kind of time series without stationarity, we introduce AR sequences under sublinear expectations, taking the distribution uncertainty into account by virtue of the sublinear expectation theory. To give the well-definedness of AR sequences under sublinear expectations, we prove the existence and uniqueness of the solution to AR equations under sublinear expectations. Furthermore, we introduce the Yule-Walker equation of our AR sequences, which gives the relationship between the coefficients and the autocovariance functions. At last, we give an example of white noise, which is generated by a G-Brownian motion, and we derive the distribution of the related AR sequences.
In this paper, we examine two types of nonlinear expectations: Choquet expectation and sublinear expectation. Similar to the role played by the normal distribution and Wiener space in classical probability theory, the G-normal distribution and G-expectation to investigate the relationship between moments of variables following a distribution under Choquet expectation and sublinear expectation. We provide bounds for the p-order absolute moment (where p > 0) of a G-normal distribution, specifically under Choquet expectation. Notably, we derive a formula for the Choquet expectation of phi(X), where phi is a strictly monotonic function and X is a G-normally distributed random variable. As a special case, we obtain the explicit expressions for the odd moments of a G-normal distribution corresponding to Choquet expectation. Consequently, we establish that if G-expectation coincides with a Choquet expectation, then the G-expectation space are crucial in the theory of sublinear expectation. Our objective is G-normal is linear.
Scalar invariance is a fundamental property in the theory of risk measures. In this paper, we investigate scalar invariant maps on throughtwomaindirections.First, L-infinity we introduce the notions of scalar closedness and scalar solidity for pseudo-acceptance sets and show that these properties provide a necessary and sufficient characterization of acceptance sets without requiring convexity or . weak*-closedness Second, we develop a generalized duality framework between and the class of L-infinity scalar invariant maps, based on the evaluation pairing (h, phi) -> phi(h) . This leads to a bipolar-type representation for acceptance sets under minimal structural assumptions. Finally, we introduce a hull operator associated with abstract properties on scalar invariant maps, interpret it as the largest minorant satisfying a given property, and study consistency and representation results. Several classical properties (convexity, subadditivity, monotonicity, etc.) are revisited within this framework, and explicit formulas for the associated hulls are provided.
