Convergence is studied of a sequence (n_ξ) of irreducible positive-recurrent Markov chains to a Markov chain ξ in term of Fourier representation. If (ζ_∞,ω_c) is the representative class of ξ by directed circuits c with the weights having a probabilistic interpretation in term of Sample paths,then the sum of all ω_c for which the pair of states (i, j) is an arc of c is approximated by an almost surely convergent sequence of Fourier representations. When the weights have not a probabilistic interpretation then they can be explicitely given by an algorithm in term of the integral representation