  Deep Markov models (DMM) are generative models that are scalable and
expressive generalization of Markov models for representation, learning, and
inference problems. However, the fundamental stochastic stability guarantees of
such models have not been thoroughly investigated. In this paper, we provide
sufficient conditions of DMM's stochastic stability as defined in the context
of dynamical systems and propose a stability analysis method based on the
contraction of probabilistic maps modeled by deep neural networks. We make
connections between the spectral properties of neural network's weights and
different types of used activation functions on the stability and overall
dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we
propose a few practical methods for designing constrained DMMs with guaranteed
stability. We empirically substantiate our theoretical results via intuitive
numerical experiments using the proposed stability constraints.
