This paper focuses on the analyticity of the limiting behavior of a class of dynamical systems defined by iteration of non expansive random operators. The analyticity is understood in function of the parameters which govern the law of the operators. The proofs are based on contraction with respect to certain projective semi norms. We first concentrate on the analyticity of Lyapunov exponents. Such exponents can be seen as functionals of a Bernoulli process whenever each operator is sampled independently between two possible values. Our main concern is then the domain of analyticity of these exponents seen as functions of the parameter of the Bernoulli law. Several examples are considered, including Lyapunov exponents associated with products of randoms matrices both in the conventional algebra, and in the $(\max, +)$ semi-field, and Lyapunov exponents associated with non--linear dynamical systems arising in stochastic control. For the class of reducible operators (defined in the paper), we also address the issue of analyticity of the expectation of functionals of the limiting behavior in function of the parameters of the law, and connect this with contraction properties with respect to the supremum norm. We give several applications to the analyticity of stationary response times in certain queueing networks in function of the intensity of the arrival process and the paramaters of the law of the service times.