  Since Kilmer et al. introduced the new multiplication method between two
third-order tensors around 2008 and third-order tensors with such
multiplication structure are also called as T-product tensors, T-product
tensors have been applied to many fields in science and engineering, such as
low-rank tensor approximation, signal processing, image feature extraction,
machine learning, computer vision, and the multi-view clustering problem, etc.
However, there are very few works dedicated to exploring the behavior of random
T-product tensors. This work considers the problem about the tail behavior of
the unitarily invariant norm for the summation of random symmetric T-product
tensors. Majorization and antisymmetric Kronecker product tools are main
techniques utilized to establish inequalities for unitarily norms of
multivariate T-product tensors. The Laplace transform method is integrated with
these inequalities for unitarily norms of multivariate T-product tensors to
provide us with Bernstein Bounds estimation of Ky Fan $k$-norm for functions of
the symmetric random T-product tensors summation.
