On a functional contraction method

Ralph Neininger, Henning Sulzbach

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)
124 Downloads (Pure)


Methods for proving functional limit laws are developed for se-
quences 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 continu-
ous functions endowed with uniform topology and the space D[0, 1]
of c`adl`ag functions with the Skorokhod topology. The contraction
method originated from the probabilistic analysis of algorithms and
random trees where characteristics satisfy natural distributional re-
currences. It is based on stochastic fixed-point equations, where prob-
ability 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 con-
text. 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.
Original languageEnglish
Pages (from-to)1777-1822
Number of pages48
JournalAnnals of Probability
Issue number4
Publication statusPublished - 3 Jun 2015


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


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