Abstract: Under some nonstationary conditions, it is proved that all the finite dimensional distributions of the stochastic process {ξ_n(t); 0≤t≤1} weakly converge to the finite dimensional distribution of the Wiener measure under the conditional probability measure PB(·). At last, it is proved that the process {ξ_νn(u);0≤u≤1} weakly converges to the Wiener measure, where {ν_n;n∈N} is a sequence of positive integer-valued random variables statisfying some conditions.

Key words: functional central limit theorem, linear process, martingale difference