This paper is concerned with the derivation of a functional scaling limit theorem for a certain class of discrete time Markov chains, each consisting of a one dimensional reference process and an $L^2_{loc}$-valued volume process, for which the conditional probability distributions of the increments are assumed to depend only on the current value of the reference process and on the volumes standing to the left of it. Such dynamics appear for example in the modeling of state dependent limit order books. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we then construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense.
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