Stochastic stability of adaptive quantizers for Markov sources

A stochastic stability result for a class of adaptive quantizers which were introduced by Goodman and Gersho is presented. We consider a case where the input process is a linear Markov source which is not necessarily stable. We present a stochastic stability result for the estimation error and the quantizer, thus generalizing the stability result of Goodman and Gersho to a Markovian, and furthermore to an unstable, setting. Furthermore, it is shown that, there exists a unique invariant distribution for the state and the quantizer parameters under mild irreducibility conditions. The second moment under the invariant distribution is finite, if the system noise is Gaussian.