On a generalised typicality with respect to general probability distributions

The method of typical sequences is a fundamental tool in asymptotic analyses of information theory. The conditional typicality lemma is one of the most commonly used lemmas in the method of typical sequences. Recent works have generalised the definition of typicality to general alphabets or general probability distributions. However, there is still a lack of the conditional typicality lemma based on the definition of typicality with respect to general distributions on the product space. In this paper, we propose a generalised joint typicality for general alphabets and with respect to general probability distributions, and obtain the counterpart of conventional conditional and joint typicality lemmas based on the generalised typicality. As applications of the typicality lemmas, we prove the packing and coverings for the proposed generalised typicality, and then recover the direct part of the capacity theorem on the general Gelfand-Pinsker coding. We also prove a mutual covering lemma for the generalised typicality, and then obtain the Marton-type inner bound to the capacity region of the general broadcast channel.