Quantumq-white noise and aq-central limit theorem

We establish a connection between the Azéma martingales and certain quantum stochastic processes with increments satisfyingq-commutation relations. This leads to a theory ofq-white noise onq-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bialgebras. In particular, quantum Azéma noise,q-interpolations between Fermion and Boson quantum Brownian motion and unitary evolutions withq-independent multiplicative increments are studied. It follows from our results that the Azéma martingales and theq-interpolations are central limits of sums ofq-independent, identically distributed quantum random variables.