Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors

Consider the likelihood ratio test (LRT) statistics for the independence of sub-vectors from a $p$-variate normal random vector. We are devoted to deriving the limiting distributions of the LRT statistics based on a random sample of size $n$. It is well known that the limit is chi-square distribution when the dimension of the data or the number of the parameters are fixed. In a recent work by Qi, Wang and Zhang (Ann Inst Stat Math (2019) 71: 911--946), it was shown that the LRT statistics are asymptotically normal under condition that the lengths of the normal random sub-vectors are relatively balanced if the dimension $p$ goes to infinity with the sample size $n$. In this paper, we investigate the limiting distributions of the LRT statistic under general conditions. We find out all types of limiting distributions and obtain the necessary and sufficient conditions for the LRT statistic to converge to a normal distribution when $p$ goes to infinity. We also investigate the limiting distribution of the adjusted LRT test statistic proposed in Qi, Wang and Zhang (2019). Moreover, we present simulation results to compare the performance of classical chi-square approximation, normal and non-normal approximation to the LRT statistics, chi-square approximation to the adjusted test statistic, and some other test statistics.