Diagonal approximations by martingales

Let (Ω,A, μ) be a probability space with a bijective, bimeasurable and measure preserving transformation T . For a measurable function f on Ω, (f ◦ T )i is a (strictly) stationary process and reciprocally, any (strictly) stationary process can be represented in this way. We shall denote Uf = f ◦ T ; U is a unitary operator in L. The function f will be supposed to have a zero mean. (Fi)i∈Z, where T−1Fi = Fi+1, is an increasing filtration of sub-σ-fields of A. Hi denotes the space of Fi-measurable and square integrable random variables. Pi(f) is the projection of f onto the space Hi ⊖Hi−1, i.e. Pi(f) = E(f |T−i(F0)) − E(f |T−i+1(F0)). We suppose the function f to be regular, i.e. f = ∑+∞ i=−∞ Pi(f). Let σn = ||Sn(f)||, where || · || is the L2-norm and Sn(f) = ∑n i=1 f ◦ T . For simplicity of notation we write Q0(Sn(f)) for E(Sn(f)|F0) and Rn(Sn(f)) for Sn(f)− E(Sn(f)|Fn).