Conditions and speed of convergence of distributions of the increasing sums of random elements of the final field to uniform distribution

Conditions of exponential convergence of distributions of the sums $S_{l}=a_{1}y_{1}+...+a_{l}y_{l}$ are brought to uniform distribution, where $a_{j}$ - random independent elements, and $y_{j}$ - the set nonzero elements of final field $K=\mathrm{GF}(p^{s}) $. It is supposed that distributions $\mathcal{P}_{j}$ elements $a_{j}$ can be various. It is shown that the exponential convergence in the parameter $l$ is carried out under quite wide conditions of distributions of $\mathcal{P}_{j}$ , $j=1,...,l$. In particular, if $\mathcal{P}_{1}=...=\mathcal{P}_{l}=\mathcal{P}$ and $K=\mathrm{GF}(p)$ - the simple field, then $\mathcal{P}$ can be any nondegenerate distribution.