Asymptotic Behaviour of Truncated Stochastic Approximation Procedures

Abstract

We study asymptotic behaviour of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quite mild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable form of the central limit theorem can be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.

Portions of this text were previously a part of arXiv:1508.01902v1 which has been divided into two papers for publication at the request of the journal. The first part is now arXiv:1508.01902v2