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Abstract.Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches oo . The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct.

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Summary. Regenerative subsets of ~ constitute an analog of classical renewal processes. Limits and intersections of independent regenerative sets are discussed. These ideas are related to the usual quantities associated with subordinators. O. Introduction: Discussion of the Basic DefinitionA regenerative set is the analog for random subsets of N of the classical renewal process. Roughly speaking, if we split a regenerative set at a stopping time T in the set, then the right-hand portion (as viewed from T) and the lefthand portion are independent, the distribution of the right-hand portion being independent of the particular stopping time T.The precise notion of regenerative set used in this paper is due to Maisonneuve [14] and is somewhat weaker than Hoffmann-Jorgensen's [6] notion of a strong Markov set (see also [9-11, 16, 17]). This weaker notion is of interest in itself and is more suitable when considering several regenerative sets simultaneously.For ease of reference we now recall Hoffmann-Jorgensen's definition. Let (~2,o ~,4,P) be a filtered probability space endowed with a family (0t)ta 0 of shift operators which is compatible with (4) in that 01-1~ ~4+~ for t, sEN+. Let M be a progressive subset of N+ x ~2 with right-closed sections M(~o). We assume that P(OeM)= 1 and that M o 0 t = (M-t)~ IR+ for each t > 0. Here we follow the usual convention of using the letter M to denote both a subset of N+ x~? and the associated map (o~--~M(co) from O into 2 m. The collection (0, o ~,4, Or, M, P)