  We consider the problem of adaptive estimation of the regression function in
a framework where we replace ergodicity assumptions (such as independence or
mixing) by another structural assumption on the model. Namely, we propose
adaptive upper bounds for kernel estimators with data-driven bandwidth
(Lepski's selection rule) in a regression model where the noise is an increment
of martingale. It includes, as very particular cases, the usual i.i.d.
regression and auto-regressive models. The cornerstone tool for this study is a
new result for self-normalized martingales, called ``stability'', which is of
independent interest. In a first part, we only use the martingale increment
structure of the noise. We give an adaptive upper bound using a random rate,
that involves the occupation time near the estimation point. Thanks to this
approach, the theoretical study of the statistical procedure is disconnected
from usual ergodicity properties like mixing. Then, in a second part, we make a
link with the usual minimax theory of deterministic rates. Under a beta-mixing
assumption on the covariates process, we prove that the random rate considered
in the first part is equivalent, with large probability, to a deterministic
rate which is the usual minimax adaptive one.
