The Annals of Probability

On a functional contraction method

Ralph Neininger and Henning Sulzbach

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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 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

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

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

Permanent link to this document
http://projecteuclid.org/euclid.aop/1433341320

Digital Object Identifier
doi:10.1214/14-AOP919

Mathematical Reviews number (MathSciNet)
MR3353815

Zentralblatt MATH identifier
06457511

Subjects
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

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

Citation

Neininger, Ralph; Sulzbach, Henning. On a functional contraction method. Ann. Probab. 43 (2015), no. 4, 1777--1822. doi:10.1214/14-AOP919. http://projecteuclid.org/euclid.aop/1433341320.


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