Non-standard limit theorems for urn models and stochastic approximation procedures

Kaniovski, Y.M. & Pflug, G.C. ORCID: (1995). Non-standard limit theorems for urn models and stochastic approximation procedures. Stochastic Models 11 (1) 79-102. 10.1080/15326349508807332.

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The adaptive processes of growth modeled by a generalized urn scheme have proved to be an efficient tool for the analysis of complex phenomena in economics, biology and physical chemistry. They demonstrate non-ergodic limit behavior with multiple limit states. There are two major sources of complex feedbacks governing these processes: nonlinearity (even local, which is caused by nondifferentiability of the functions driving them) and multiplicity of limit states stipulated by the nonlinearity.We suggest an analytical approach for studying some of the patterns of complex limit behavior. The approach is based on conditional limit theorems. The corresponding limits are, in general, not infinitely divisible. We show that convergence rates could be different for different limit states. The rates depend upon the smoothness (in neighborhoods of the limit states) of the functions governing the processes.
Since the mathematical machinery allows us to treat a quite general class of recursive stochastic discrete-time processes, we also derive corresponding limit theorems for stochastic approximation procedures. The theorems yield new insight into the limit behavior of stochastic approximation procedures in the case of nondifferentiable regression functions with multiple roots

Item Type: Article
Uncontrolled Keywords: generalized urn scheme, conditional limit theorems, stochastic approximation
Research Programs: Optimization under Uncertainty (OPT)
Technological and Economic Dynamics (TED)
Bibliographic Reference: Stochastic Models; 11(1):79-102 [1995]
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 02:05
Last Modified: 27 Aug 2021 17:15

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