  In this paper, we study tail inequalities of the largest eigenvalue of a
matrix infinitely divisible (i.d.) series, which is a finite sum of fixed
matrices weighted by i.d. random variables. We obtain several types of tail
inequalities, including Bennett-type and Bernstein-type inequalities. This
allows us to further bound the expectation of the spectral norm of a matrix
i.d. series. Moreover, by developing a new lower-bound function for
$Q(s)=(s+1)\log(s+1)-s$ that appears in the Bennett-type inequality, we derive
a tighter tail inequality of the largest eigenvalue of the matrix i.d. series
than the Bernstein-type inequality when the matrix dimension is high. The
resulting lower-bound function is of independent interest and can improve any
Bennett-type concentration inequality that involves the function $Q(s)$. The
class of i.d. probability distributions is large and includes Gaussian and
Poisson distributions, among many others. Therefore, our results encompass the
existing work \cite{tropp2012user} on matrix Gaussian series as a special case.
Lastly, we show that the tail inequalities of a matrix i.d. series have
applications in several optimization problems including the chance constrained
optimization problem and the quadratic optimization problem with orthogonality
constraints. In addition, we also use the resulting tail bounds to show that
random matrices constructed from i.d. random variables satisfy the restricted
isometry property (RIP) when it acts as a measurement matrix in compressed
sensing.
