  In this work, we extend double tensor integrals (DTI) from our previous work
to parametrization double tensors integrals (PDTI) by applying integral kernel
transform bounds to upper bound PDTI norm and establishing a new perturbation
formula. Besides, the convergence property of random PDTI is investigated and
this property is utilized to characterize the relation between the original
derivative tensor and the action result of PDTI to the original derivative
tensor. These tools help us to derive new tail bounds for random tensors
according to more general operator inequalities, e.g., Heinz inequality and
Birman-Koplienko-Solomyak inequality. Moreover, new tail bounds about random
tensors are also obtained according to our new derived perturbation formula and
integral kernel transform bounds.
