A quenched weak invariance principle
  In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and non irreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, a theory which has interest in itself. 
J\'er\^ome Dedecker, Florence Merlev\`ede, Magda Peligrad
math.PR