Let \((\xi^{(n)})_{n\geq1}\) be a sequence of square integrable martingale-differences such that for all \(i\geq1\), \(\lim_{n\to\infty}n(\xi_{i}^{(n)})^2=1\) a.s. and for some \(C\geq1\), \(\max_{1\leq i\leq n}|\xi_{i}^{(n)}|\leq C/\sqrt n\) a.s. Let us define \(W_{t}^{n}:=\sum_{i=1}^{[nt]}\xi_{i}^{(n)}\), \(0\leq t\leq1\), and \(Z_{t}^{n}:=\int_{0}^{t}z^{(n)}(t,s)\,dW_{s}^{n}\), where \[ z^{(n)}(t,s):=n\int_{s-1/n}^{s}z\left([nt]/n,u\right)\,du \] for \(s\in[1/n,1]\) and \(t\in[0,1]\). The author proves that under the considered conditions, if \(H>1/2\), the processes \(Z^{n}\) converge in distribution, as \(n\to\infty\), to the fractional Brownian motion \(Z\) with Hurst parameter \(H>1/2\).