Autoregressive sequences with covariance uncertainty under sublinear expectations

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{\text{-Brownian}} $ motion, and we derive the distribution of the related AR sequences.