On a functional contraction method

TL;DR

This paper extends the contraction method to function spaces like continuous and ce0dle0g functions, providing a unified approach to prove functional limit laws using fixed-point equations and Zolotarev metrics.

Contribution

It generalizes the contraction method to the spaces c5[0,1] and d8[0,1], enabling new proofs of functional limit theorems and analysis of recursive stochastic processes.

Findings

Provides a short proof of Donsker's functional limit theorem.

Develops the use of Zolotarev metrics in function spaces.

Applies the method to recurrences in algorithm analysis.

Abstract

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 $\mathcal{C}[0,1]$ of continuous functions endowed with uniform topology and the space $\mathcal {D}[0,1]$ of c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method originated from 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 $\mathcal{C}[0,1]$ and $\mathcal{D}[0,1]$ in this context. Applications are given, in particular, a short proof of Donsker's functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.