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

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