On a functional contraction method

Details On a functional contraction method On a functional contraction method

2012-02-07

arxiv.org/abs/1202.1370v1

Mathematics

Probability

Scientific paper

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space $C[0,1]$ of continuous functions endowed with uniform topology and the space $D[0,1]$ of c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method originated form the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach's fixed-point theorem. We develop the use of the Zolotarev metrics on the spaces $C[0,1]$ and $D[0,1]$ in this context. As an application a short proof of Donsker's functional limit theorem is given.

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