Motivated by our recent extension of the Two-Stage Sequential Algorithm (eTSSO), we propose an adaptation of the framework in Pasupathy et al. (2015) for the study of convergence of kriging-based procedures. Specifically, we extend the proof scheme in Pasupathy et al. (2015) to the class of kriging-based simulation-optimization algorithms. In particular, the asymptotic convergence and the convergence rate of eTSSO are investigated by interpreting the kriging-based search as a stochastic recursion. We show the parallelism between the two paradigms and exploit the deterministic counterpart of eTSSO, the more famous Efficient Global Optimization (EGO) procedure, in order to derive eTSSO structural properties. This work represents a first step towards a general proof framework for the asymptotic convergence and convergence rate analysis of meta-model based simulation-optimization.