A Proof of Theorem 1, y

Let M be the recurrent neural network predicting H-step forecasts using the direct strategy. For a target coverage level α (0, 1), the intervals obtained by the ICP-based conformal forecasting algorithm satisfy P ( h {1,..., H}. y Due to the direct forecasting strategy, every step in the horizon can be treated as a separate inductive conformal predictor that uses the same underlying model M (with the final predictions derived from the internal state being independent) and the same dataset D. The independent validity of each of the H ICPs follows from Proposition 1 in Vovk [51]. Setting the error rate of each of the H ICPs to (1 α)/H and applying Boole's inequality we obtain that the combined error rate of the H-step forecaster is 1 α, as required. B.1 Qualitative results for static and time-dependent noise settings See Figure 4. From top to bottom: 1) CF-RNN for the static noise variance setting; 2) CF-RNN for the time-dependent noise variance setting; 3) MQ-RNN for the time-dependent noise variance setting; 4) DP-RNN for the time-dependent noise variance setting. The periodic component is defined following the quasi-random walk model in Durbin and Koopman [55] (Equations (3.7) and (3.8)): we define γ Table 8: Empirical joint coverage of CF-RNN for the datasets with asynchronous, out-of-phase examples with dynamic series lengths, averaged across prediction horizons and reported as mean std over five random seeds.