  Given a random sample from a multivariate normal distribution whose
covariance matrix is a Toeplitz matrix, we study the largest off-diagonal entry
of the sample correlation matrix. Assuming the multivariate normal distribution
has the covariance structure of an auto-regressive sequence, we establish a
phase transition in the limiting distribution of the largest off-diagonal
entry. We show that the limiting distributions are of Gumbel-type (with
different parameters) depending on how large or small the parameter of the
autoregressive sequence is. At the critical case, we obtain that the limiting
distribution is the maximum of two independent random variables of Gumbel
distributions. This phase transition establishes the exact threshold at which
the auto-regressive covariance structure behaves differently than its
counterpart with the covariance matrix equal to the identity. Assuming the
covariance matrix is a general Toeplitz matrix, we obtain the limiting
distribution of the largest entry under the ultra-high dimensional settings: it
is a weighted sum of two independent random variables, one normal and the other
following a Gumbel-type law. The counterpart of the non-Gaussian case is also
discussed. As an application, we study a high-dimensional covariance testing
problem.
