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This paper compiles several aspects of the dynamics of stochasticapproximation algorithms with Markov iterate-dependent noise when the iteratesare not known to be stable beforehand. We achieve the same by extending thelock-in probability (i.e. the probability of convergence of the iterates to aspecific attractor of the limiting o.d.e. given that the iterates are in itsdomain of attraction after a sufficiently large number of iterations (say) n 0) framework to such recursions. Specifically, with the more restrictiveassumption of Markov iterate-dependent noise supported on a bounded subset ofthe Euclidean space we give a lower bound for the lock-in probability. We usethese results to prove almost sure convergence of the iterates to the specifiedattractor when the iterates satisfy an asymptotic tightness condition. Thenovelty of our approach is that if the state space of the Markov process iscompact we prove almost sure convergence under much weaker assumptions comparedto the work by Andrieu et al. which solves the general state space case undermuch restrictive assumptions. We also extend our single timescale results tothe case where there are two separate recursions over two different timescales.This, in turn, is shown to be useful in analyzing the tracking ability ofgeneral adaptive algorithms. Additionally, we show that our results can be usedto derive a sample complexity estimate of such recursions, which then can beused for step-size selection

Dynamics of stochastic approximation with iterate-dependent Markov noise  under verifiable conditions in compact state space with the stability of  iterates not ensured