  This work prepares new probability bounds for sums of random, independent,
Hermitian tensors. These probability bounds characterize large-deviation
behavior of the extreme eigenvalue of the sums of random tensors. We extend
Lapalace transform method and Lieb's concavity theorem from matrices to
tensors, and apply these tools to generalize the classical bounds associated
with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor
setting. Tail bounds for the norm of a sum of random rectangular tensors are
also derived from corollaries of random Hermitian tensors cases. The proof
mechanism can also be applied to tensor-valued martingales and tensor-based
Azuma, Hoeffding and McDiarmid inequalities are established.
