Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations

The paper proves an LDP for W_n=(n^{-1}∑X_k^2, n^{-1}∑_{k=2}^n X_kX_{k-1}) in a stationary Gaussian AR(1) via Toeplitz/eigenvalue methods, deriving the limiting cumulant generating function L(λ1,λ2)=−1/2 log((1+θ^2−2λ1+√((1+θ^2−2λ1)^2−4(θ+λ2)^2))/2) on the domain 2|θ+λ2|<1+θ^2−2λ1, and then applying Gärtner–Ellis to obtain a good rate function given by the Legendre–Fenchel transform; they explicitly use J(x,y)=1/2[x(1+θ^2)−1−2θy+log(x/(x^2−y^2))] on {x>0,|y|<x} in subsequent contractions (e.g., their (3.3)), confirming the closed form that the model derives by Gaussian determinant methods and tridiagonal recursions. Thus, results and expressions coincide; the approaches differ (spectral Toeplitz vs. precision-matrix determinant recursion), but both are correct and consistent with Gärtner–Ellis (essential smoothness/steepness verified) .