  In this work, we try to build a theory for random double tensor integrals
(DTI). We begin with the definition of DTI and discuss how randomness structure
is built upon DTI. Then, the tail bound of the unitarily invariant norm for the
random DTI is established and this bound can help us to derive tail bounds of
the unitarily invariant norm for various types of two tensors means, e.g.,
arithmetic mean, geometric mean, harmonic mean, and general mean. By
associating DTI with perturbation formula, i.e., a formula to relate the
tensor-valued function difference with respect the difference of the function
input tensors, the tail bounds of the unitarily invariant norm for the
Lipschitz estimate of tensor-valued function with random tensors as arguments
are derived for vanilla case and quasi-commutator case, respectively. We also
establish the continuity property for random DTI in the sense of convergence in
the random tensor mean, and we apply this continuity property to obtain the
tail bound of the unitarily invariant norm for the derivative of the
tensor-valued function.
