One-shot bounds for various information theoretic problems using smooth min and max Rényi divergences

One-shot analogues for various information theory results known in the asymptotic case are proven using smooth min and max Rényi divergences. In particular, we prove that smooth min Rényi divergence can be used to prove one-shot analogue of the Stein´s lemma. Using smooth min Rényi divergence we prove a special case of packing lemma in the one-shot setting. Furthermore, we prove a one-shot analogue of covering lemma using smooth max Rényi divergence. We also propose one-shot achievable rate for source coding under maximum distortion criterion. This achievable rate is quantified in terms of smooth max Rényi divergence.