A complex mathematical topic, the asymptotic behavior of sequences of random variables, or the behavior of indefinitely long sequences of random variables, has significant ramifications for the statistical analysis of data from large samples.
The asymptotic behavior of the sample estimators of the eigenvalues and eigenvectors of covariance matrices is examined in this claim. This work focuses on limited sample size scenarios where the number of accessible observations is comparable in magnitude to the observation dimension rather than usual high sample-size asymptotic .
Under the presumption that both the sample size and the observation dimension go to infinity while their quotient converges to a positive value, the asymptotic behavior of the conventional sample estimates is examined using methods from random matrix theory.
Closed form asymptotic expressions of these estimators are obtained, demonstrating the inconsistency of the conventional sample estimators in these asymptotic conditions, assuming that an asymptotic eigenvalue splitting condition is satisfied.
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