Gaussian Approximation via Martingale Methods

Gordin’s seminal paper (1969) initiated a line of research in which limit theorems for stationary sequences are proved via appropriate approximations by stationary martingale difference sequences followed by an application of the corresponding limit theorem for such sequences. In this chapter, we first review different ways to get suitable martingale approximations and then derive the central limit theorem and its functional form for strictly stationary sequences under various types of projective criteria. More general normalizations than the traditional ones will be also investigated, as well as the functional moderate deviation principle. We shall also address the question of the functional form of the central limit theorem for not necessarily stationary sequences. The last part of this chapter is dedicated to the moderate deviations principle and its functional form for stationary sequences of bounded random variables satisfying projective-type conditions.