Testing Independence for a Large Number of High–Dimensional Random Vectors

Capturing dependence among a large number of high dimensional random vectors is a very important and challenging problem. By arranging n random vectors of length p in the form of a matrix, we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not. Specifically, the proposed statistic can also be applied to n random vectors, each of whose elements can be written as either a linear stationary process or a linear combination of a random vector with independent elements. The asymptotic distribution of the proposed test statistic is established in the case where both p and n go to infinity at the same order. In order to avoid estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series, is constructed. The facts that the limiting distribution is a normal distribution and there is no need to know the inner structure of each investigated random vector result in simple implementation of the constructed test statistic. Simulation results demonstrate that the proposed test is powerful against many common dependent cases. An empirical application to detecting dependence of the closed prices from several stocks in S&P 500 also illustrates the applicability and effectiveness of our provided test.