First order autoregression is shown to satisfy a limit theory which is uniform over stationary values of the autoregressive coeﬀicient ρ = ρ n in [0,1) provided (1 - ρ n )n approaches inﬁnity. This extends existing Gaussian limit theory by allowing for values of stationary rho that include neighbourhoods of unity provided they are wider than ( n 1 ), even by a slowly varying factor. Rates of convergence depend on rho and are at least squareroot of / n but less than n . Only second moments are assumed, as in the case of stationary autoregression with ﬁxed ρ.
Giraitis, Liudas and Phillips, Peter C.B., "Uniform Limit Theory for Stationary Autoregression" (2004). Cowles Foundation Discussion Papers. 1754.