  An iterated multistep forecasting scheme based on recurrent neural networks
(RNN) is proposed for the time series generated by causal chains with infinite
memory. This forecasting strategy contains, as a particular case, the iterative
prediction strategies for dynamical systems that are customary in reservoir
computing. Readily computable error bounds are obtained as a function of the
forecasting horizon, functional and dynamical features of the specific RNN
used, and the approximation error committed by it. The framework in the paper
circumvents difficult-to-verify embedding hypotheses that appear in previous
references in the literature and applies to new situations like the
finite-dimensional observations of functional differential equations or the
deterministic parts of stochastic processes to which standard embedding
techniques do not necessarily apply.
