  In recent years, tensors have been applied to different applications in
science and engineering fields. In order to establish theory about tail bounds
of the tensors summation behavior, this work extends previous work by
considering the tensors summation tail behavior of the top $k$-largest singular
values of a function of the tensors summation, instead of the largest/smallest
singular value of the tensors summation directly (identity function) explored
in Shih Yu's work: Convenient tail bounds for sums of random tensors.
Majorization and antisymmetric tensor product tools are main techniques
utilized to establish inequalities for unitarily norms of multivariate tensors.
The Laplace transform method is integrated with these inequalities for
unitarily norms of multivariate tensors to give us tail bounds estimation for
Ky Fan $k$-norm for a function of the tensors summation. By restricting
different random tensor conditions, we obtain generalized tensor Chernoff and
Bernstein inequalities.
