a-finite measure for Marker processes which satisfy a certain recurrence condition. In [18] Om~Y obtained various limit theorems under the same recurrence condition. Additional limit theorems have been obtained by JAIN [13] and by OldEr and JA~IIso~ [14]. In [13] a limit theorem from the profound 1940 paper of DOEBLIN [7] was used to verify a conjecture of OREY ([18], page 816). One is led to suspect the existence of deeper connections between DOE]~LIN'S general theory of Marker processes (see [7] and [3]) and the theory of Marker processes satisfying tIAI~I~IS' recurrence condition. Our main purpose in this paper is to exhibit the full extent of this connection. The first section is devoted to notation, definitions and results of a preliminary nature, the most important of which, lemma 2, is a rederivation under weaker conditions of a lemma from DOEBLIN'S 1937 paper [6]. (This last paper is almost unobtainable. That portion of it dealing with processes satisfying what is called "Doeblin's condition" is generalized and presented in chapter V of DooB [8]. The lemma in question corresponds to lemma 5.3 in DOOB'S treatment. OREu was the first to realize that the conclusion of the lemma holds under conditions weaker than DOEBLIN'S: see Theorem 2.1 of [18].) The main results are in the second section. Sharpened forms of the limit theorems of DOWBLIN for the absolutely essential and indecomposable case (section III of [7]) are shown to follow from known results for processes satisfying ttAI~I~IS' recurrence condition. It is shown that what DOEBLIN calls the "anormal" case can arise only when the a-field of measurable sets of the state space is not countably generated. It is also shown that there exists a a-finite invaliant measure which is unique up to a constant multiple. In the third section we deal with the case where there is a afinite measure assigning positive measure to all stochastically closed sets of the state space. Under certain additional conditions invariant measures are shown to exist. In some cases, it is possible to describe completely the class of invariant measures. In the fourth section, it is shown that in the case where the state space is improperly essential an infinite but a-finite subinvariant measure exists. In the fifth section, we study the case where there is a a-finite measure which assigns positive measure to each perpetuable set of the state space. Our results here sharpen the results obtained by DOEBLIN under the same hypothesis. We close by discussing the connection of the results of this last section with some of the current literature.