This paper develops a unified framework, based on iterated random operator\ntheory, to analyze the convergence of constant stepsize recursive stochastic\nalgorithms (RSAs). RSAs use randomization to efficiently compute expectations,\nand so their iterates form a stochastic process. The key idea of our analysis\nis to lift the RSA into an appropriate higher-dimensional space and then\nexpress it as an equivalent Markov chain. Instead of determining the\nconvergence of this Markov chain (which may not converge under constant\nstepsize), we study the convergence of the distribution of this Markov chain.\nTo study this, we define a new notion of Wasserstein divergence. We show that\nif the distribution of the iterates in the Markov chain satisfy a contraction\nproperty with respect to the Wasserstein divergence, then the Markov chain\nadmits an invariant distribution. We show that convergence of a large family of\nconstant stepsize RSAs can be understood using this framework, and we provide\nseveral detailed examples.\n