Nonstationary INAR(1) Process with qth-Order Autocorrelation Innovation

and Applied Analysis 3 the autoregressive coefficient estimator. For the nonstationary of continuous-valued time series, we often need to examine whether the characteristic polynomial of AR(1) process has a unit root. Thus, we want to see if we can find the limiting distribution of the autoregressive coefficient estimator. Let us present a result that is needed later on. Lemma 3. Suppose that Z t follows a random walk without drift, Z t = Z t−1 + w t , (7) whereZ 0 = 0 and {w t } is an i.i.d. sequence withmean zero and variance σ2 w > 0. Let “⇒” denote converges in distribution, and letW(r) denote the standard Brownian motion. Then, one has the following properties: (i) T−1/2 ∑T t=1 w t ⇒ σ w W(1), (ii) T−1 ∑T t=1 Z t−1 w t ⇒ (1/2)σ2 w [W2(1) − 1], (iii) T−3/2 ∑T t=1 tw t ⇒ σ w W(1) − σ w ∫ 1 0 W(r)dr, (iv) T−3/2 ∑T t=1 Z t−1 ⇒ σ w ∫ 1