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