Annals of Probability

On a functional contraction method

Ralph Neininger and Henning Sulzbach

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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 càdlàg 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.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1777-1822.

Received: April 2013
Revised: February 2014
First available in Project Euclid: 3 June 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 60G18: Self-similar processes 60C05: Combinatorial probability

Functional limit theorem contraction method recursive distributional equation Zolotarev metric Donsker’s invariance principle


Neininger, Ralph; Sulzbach, Henning. On a functional contraction method. Ann. Probab. 43 (2015), no. 4, 1777--1822. doi:10.1214/14-AOP919.

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