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In the case where the dimension of the data grows at the same rate as the
sample size we prove a central limit theorem for the difference of a linear
spectral statistic of the sample covariance and a linear spectral statistic of
the matrix that is obtained from the sample covariance matrix by deleting a
column and the corresponding row. Unlike previous works, we do neither require
that the population covariance matrix is diagonal nor that moments of all order
exist. Our proof methodology incorporates subtle enhancements to existing
strategies, which meet the challenges introduced by determining the mean and
covariance structure for the difference of two such eigenvalue statistics.
Moreover, we also establish the asymptotic independence of the difference-type
spectral statistic and the usual linear spectral statistic of sample covariance
matrices.