Applications of a Kushner and Clark lemma to general classes of stochastic algorithms

Two general classes of stochastic algorithms are considered, including algorithms considered by Ljung as well as algorithms of the form \theta_{n+1} = \theta_{n} - \gamma_{n+1} V_{n+1}(\theta_{n}, Z) , where Z is a stationary ergodic process. It is shown how one can apply a lemma of Kushner and Clar...

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Bibliographic Details
Published in:IEEE transactions on information theory 1984-03, Vol.30 (2), p.140-151
Main Authors: Metivier, M., Priouret, P.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematics
Probability and statistics
Sciences and techniques of general use
Sequential methods
Statistics
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Summary:Two general classes of stochastic algorithms are considered, including algorithms considered by Ljung as well as algorithms of the form \theta_{n+1} = \theta_{n} - \gamma_{n+1} V_{n+1}(\theta_{n}, Z) , where Z is a stationary ergodic process. It is shown how one can apply a lemma of Kushner and Clark to obtain properties of these algorithms. This is done by using in particular Martingale arguments in the generalized Ljung case. In these various situations the convergence is obtained by the method of the associated ordinary differential equation, under the classical boundedness assumptions. In the case of linear algorithms, the boundedness assumptions are dropped.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.1984.1056894