The limiting distribution of the residual processes in nonstationary autoregressive processes

The weak limit of the partial sums of the normalized residuals in an AR(1) process y_t = py_t-1 + e_t is shown to be a standard Brownian motion W(x) when |ρ| ≠ 1. However, when |ρ| = 1, the weak limit is W(x) plus an extra term due to estimation of ρ. Asymptotic behaviour of the partial sums is investigated with ρ = exp(c/n) in the vicinity of unity, yielding a c-dependent weak limit as n → ∞, whose limit is again W(x) as |c| → ∞. An extension is made to nonstationary AR(p) processes with multiple characteristic roots on the unit circle. The weak limit of the partial sums has close resemblance to that for the polynomial regression.