Limiting Distribution of the Score Statistic under Moderate Deviation from a Unit Root in MA(1)

This paper derives the asymptotic distribution of Tanaka's score statistic under moderate deviation from a unit root in a moving average model of order one or MA(1). We classify the limiting distribution into three types depending on the order of deviation. In the fastest case, the convergence order of the asymptotic distribution continuously changes from the invertible process to the unit root. In the slowest case, the limiting distribution coincides with the invertible process in a distributional sense. This implies that these cases share an asymptotic property. The limiting distribution in the intermediate case provides the boundary property between the fastest and slowest cases.

Spectral norm of random tensors

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor nuclear norm (the tightest convex relaxation of the set of rank one tensors), the bound implies that the convex relaxation yields sample complexity that is linear in (the sum of) the number of dimensions, which is much smaller than other recently proposed convex relaxations of tensor rank that use unfolding.Comment: 5 page