Let \(\xi_ 1,...,\xi_ n\) be i.i.d. r.v.'s with E \(\xi\) \({}_ i=0\), E \(\xi\) \({}^ 2_ i=1\), \(i=1,...,n\), \[ X_ n(t)=n^{- }\sum^{[nt]}_{i=1}\xi_ i+n^{-}(nt-[nt])\xi_{[nt]+1},\quad t\in [0,1],\quad X_ n(0)=0, \] and \(X_ n^{(a,\epsilon)}\) be conditional processes defined by the following way: \[ P\{X_ n^{(a,\epsilon)}\in A\}=P\{X_ n\in A| \quad X_ n\in B(a,\epsilon)\},\quad where \] \[ B(a,\epsilon)=\{x(\cdot)\in C[0,1]: | X(1)-a| \leq \epsilon \},\quad A\in {\mathcal B}_{C[0,1]},\quad a\in R,\quad \epsilon \geq 0. \] Convergence in variation of the distributions of the functionals from the processes \(X_ n^{(a,\epsilon_ n)}\) is proved as \(\epsilon_ n\to 0\), \(n\to \infty\).