Abstract: In this paper we prove a conjecture raised by Tanaka on the first moment of the limiting distribution of the least squares estimator (LSE) of the unit root I(d) process. The limiting random variable is a ratio of quadratic functionals of the d-fold integrated Brownian motion. Its expectation can be found by using Karhunen-Loéve expansion and a property of the eigenfunctions of its covariance kernel.

Key words and phrases: Discrete Fourier matrix, integrated Brownian motion, Karhunen-Loéve expansion, nonstationary time series, unit root.