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

Random Tensor Inequalities and Tail bounds for Bivariate Random Tensor Means, Part I