Survey of Multivariate Aspects of the Contraction Method

We survey multivariate limit laws for sequences of random vectors which satisfy distributional recursions as they appear under various models of randomness for parameters of trees, characteristics of divideand-conquer algorithms, or, more generally, for quantities related to recursive structures or branching processes. While the area of probabilistic analysis of algorithms, since its introduction in the 60s of the last century by Knuth [25, 26, 27] has been dominated by analytic techniques based on generating functions, over the last decade, among other probabilistic techniques, the so called contraction method has been developed. This method was first introduced for the analysis of Quicksort in Rösler [45] and further on developed independently in Rösler [46] and Rachev and Rüschendorf [43], and later on in Rösler [48] and Neininger and Rüschendorf [40, 41], see also the survey article of Rösler and Rüschendorf [49]. In this survey we discuss multivariate aspects of the approach of the contraction method. In particular we study various conditions, under which multivariate limit laws can be established, mention applications to the probabilistic analysis of algorithms and connections to other areas as branching processes, and indicate as to which extend a multivariate point of view may also add flexibility to univariate studies.