Non-standard Limit Theorems for Stochastic Approximation Procedures and Their Applications for Urn Schemes

Kaniovski, Y.M. & Pflug, G.C. ORCID: https://orcid.org/0000-0001-8215-3550 (1992). Non-standard Limit Theorems for Stochastic Approximation Procedures and Their Applications for Urn Schemes. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-92-025

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Abstract

A limit theorem for the Robbins-Monro stochastic approximation procedure is proved in the case of a non-smooth regression function. Using this result a conditional limit theorem is given for the case when the regression function has several stable roots. The first result shows that the rate of convergence for the stochastic approximation-type procedures (including Monte-Carlo optimization algorithms and adaptive processes of growth being modelled by the generalized urn scheme) decreases as the smoothness increases. The second result demonstrates that in the case of several stable roots, there is no convergence rate for the procedure as whole, but for each of stable roots there exists its specific rate of convergence. The latter allows to derive several conceptual results for applied problems in biology, physical chemistry and economics which can be described by the generalized urn scheme.

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