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In this work, we study the problem of aggregating a finite number ofpredictors for nonstationary sub-linear processes. We provide oracleinequalities relying essentially on three ingredients: (1) a uniform bound ofthe $\ell^1$ norm of the time varying sub-linear coefficients, (2) a Lipschitzassumption on the predictors and (3) moment conditions on the noise appearingin the linear representation. Two kinds of aggregations are considered givingrise to different moment conditions on the noise and more or less sharp oracleinequalities. We apply this approach for deriving an adaptive predictor forlocally stationary time varying autoregressive (TVAR) processes. It is obtainedby aggregating a finite number of well chosen predictors, each of them enjoyingan optimal minimax convergence rate under specific smoothness conditions on theTVAR coefficients. We show that the obtained aggregated predictor achieves aminimax rate while adapting to the unknown smoothness. To prove this result, alower bound is established for the minimax rate of the prediction risk for theTVAR process. Numerical experiments complete this study. An important featureof this approach is that the aggregated predictor can be computed recursivelyand is thus applicable in an online prediction context.

Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes