  In this work, we apply the concept about operator connection to consider
bivariate random tensor means. We first extend classical Markov and Chebyshev
inequalities from a random variable to a random tensor by establishing Markov
inequality for tensors and Chebyshev inequality for tensors. These inequalities
are applied to establish tail bounds for bivariate random tensor means
represented by operator perspectives based on various types of connection
functions: tensor increasing functions, tensor decreasing functions, and tensor
concavity functions. We also consider tail bounds relations for the summation
and product of eigenvalues based on majorization ordering of eigenvalues of
bivariate random tensor means. This is Part I of our work about random tensor
inequalities and tail bounds for bivariate random tensor mean. In our Part II,
we will consider bivariate random tensor mean with respect to non-invertible
random tensors and their applications.
