Anisotropic scaling limits of long-range dependent linear random fields on Z3

Abstract

We provide a complete description of anisotropic scaling limits of stationary linear random field (RF) on Z 3 with long-range dependence and moving average coefficients decaying as O ( | t i | − q i ) in the i th direction, i = 1 , 2 , 3 . The scaling limits are taken over rectangles in Z 3 whose sides increase as O ( λ γ i ) , i = 1 , 2 , 3 when λ → ∞ , for any fixed γ i > 0 , i = 1 , 2 , 3 . We prove that all these limits are Gaussian RFs whose covariance structure is determined by the fulfillment or violation of the balance conditions γ i q i = γ j q j , 1 ≤ i < j ≤ 3 . The paper extends recent results in Puplinskaitė and Surgailis (2015, 2016) [27] , [28] , Pilipauskaitė and Surgailis (2016, 2017) [31] , [29] on anisotropic scaling of long-range dependent RFs from dimension 2 to dimension 3.