  Stochastic approximation is a framework unifying many random iterative
algorithms occurring in a diverse range of applications. The stability of the
process is often difficult to verify in practical applications and the process
may even be unstable without additional stabilisation techniques. We study a
stochastic approximation procedure with expanding projections similar to
Andrad\'{o}ttir [Oper. Res. 43 (1995) 1037-1048]. We focus on Markovian noise
and show the stability and convergence under general conditions. Our framework
also incorporates the possibility to use a random step size sequence, which
allows us to consider settings with a non-smooth family of Markov kernels. We
apply the theory to stochastic approximation expectation maximisation with
particle independent Metropolis-Hastings sampling.
