  Testing covariance structure is of significant interest in many areas of
statistical analysis and construction of compressed sensing matrices is an
important problem in signal processing. Motivated by these applications, we
study in this paper the limiting laws of the coherence of an $n\times p$ random
matrix in the high-dimensional setting where $p$ can be much larger than $n$.
Both the law of large numbers and the limiting distribution are derived. We
then consider testing the bandedness of the covariance matrix of a high
dimensional Gaussian distribution which includes testing for independence as a
special case. The limiting laws of the coherence of the data matrix play a
critical role in the construction of the test. We also apply the asymptotic
results to the construction of compressed sensing matrices.
