Strong Consistency of Log-Likelihood-Based Information Criterion in High-Dimensional Canonical Correlation Analysis

We consider the strong consistency of a log-likelihood-based information criterion in a normality-assumed canonical correlation analysis between q- and p-dimensional random vectors for a high-dimensional case such that the sample size n and number of dimensions p are large but p/n is less than 1. In general, strong consistency is a stricter property than weak consistency; thus, sufficient conditions for the former do not always coincide with those for the latter. We derive the sufficient conditions for the strong consistency of this log-likelihood-based information criterion for the high-dimensional case. It is shown that the sufficient conditions for strong consistency of several criteria are the same as those for weak consistency obtained by Yanagihara et al. (J. Multivariate Anal. 157, 70-86: 2017).