In this part III (cf. [1] and [2]) we prove a functional limit theorem for random elements in the space $(D,\mathcal{D})$ constructed on the basis of the partial sums of a series $\sum_{j=1}^{\infty} a_{j}r_{j}(h) \psi_{j}(x)$ where $\{r_{j}(h)\}$ denotes the system of Rademacher functions, $a_{j} \in \mathbf{R}$, and $\{\psi_{j}(x)\}$ is a uniformly bounded orthonormal system in the interval [0,1]. We also consider the linear means of $\sum a_{j}\psi_{n_{j}}(x)$, where $\{\psi_{j}\}$ is the Walsh orthonormal system and $n_{j+1}/n_{j} \ge q > 1$.