Contributions to the theoretical analysis of the algorithms with adversarial and dependent data In this work I present the concentration inequalities of Bernsteins type for the norms of Banach valued random sums under a general functional weak dependency assumption (the so called $\\cC $mixing). The latter is then used to prove in the asymptotic framework excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here in particular a question of robustness of the kernel based forecaster is investigated. Afterwards in the framework of sequential learning the multi armed bandit problem under $\\cC $mixing assumption on the arms outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma Hoeffdings and of Burkholders type) to the partial sums of real valued weakly dependent random fields (under the type of projective dependence condition).