**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series B, Pt. 3, pp. 389--395

**ASYMPTOTIC DISTRIBUTION OF PERIODOGRAM ORDINATE FOR A NEARLY NONSTATIONARY AR(1) PROCESS**

By

B. B. BHATTACHARYYA, *North Carolina State University, Raleigh*

and

G. D. RICHARDSON, *University of Central Florida, Orlando*

SUMMARY. A first-order autoregressive model $Y_{t}-\mu = \alpha_n(Y_{t-1}-\mu)+\epsilon_t$, $1 \leq t \leq n$, is called nearly nonstationary when the sequence {$\alpha_n$} converges to one. Reparameterizing $\alpha_n=e^c/n$, the limiting distribution of the normalized periodogram ordinate of a time series obeying this model is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c. This determines the limiting distribution of a unit root test in the frequency domain proposed by Akdi (1995) under Pitman-type alternatives of the form e^{c/n}

*AMS (1991) subject classification.* 62M10, 62F12.

*Key words and phrases.* Nearly nonstationary, first-order autoregressive process, periodogram ordinate.