Probability Theory and Related Fields

, Volume 71, Issue 4, pp 517–552 | Cite as

Asymptotic behaviour of a class of stochastic approximation procedures

  • Erich Berger
Article

Summary

LetB be a real Banach space. Considering stochastic approximation procedures of Robbins-Monro and Kiefer-Wolfowitz type, one is led to recursion formulas of the form
$$U_{k + 1} = U_k - k^{ - 1} \Gamma _k U_k - k^{ - \rho } V_k - k^{ - \eta } T_k$$
(ϱ and η) positive real numbers,Uk, Vk, TkB-valued random variables, Γ k random linear operatorsBB, where — in addition to some further assumptions —Γ k Γ almost surely andTkT, say, almost surely. Letting (Y k,n )k∈{1,...,n},n∈N be an array ofB-valued random variables such that the partial sums\(\sum\limits_{k = 1}^m {V_k }\) and\(\sum\limits_{k = 1}^m {Y_{k,n} }\) do not differ “too much” (uniformly inm∈{1,...,n}), we shall investigate the distance of (U m )m∈{1,...,n} and (U (n) m )m∈{1,...,n}, where
$$U_{k + 1}^{\left( n \right)} = U_k^{\left( n \right)} - k^{ - 1} \Gamma U_k^{\left( n \right)} - k^{ - \rho } Y_{k,n} - k^{ - \eta } T$$
fork∈{1,...,n−1}. The proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems, weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence (Un).

Keywords

Real Number Stochastic Process Asymptotic Behaviour Probability Theory Limit Theorem 

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Erich Berger
    • 1
  1. 1.Institut für Mathematische StochastikUniversität GöttingenGöttingenGermany

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