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In the first part of this paper are considered real polynomials$P$(ζ), ζ∈$R$^{n}, complete and nondegenerate in the sense that there is a set of (even) multi-indices α^{$j$},$j$=1,...,$N$, such that, for |ζ|>$K$, ζ real, $$cP(\xi ) \leqslant \sum {\xi ^{\alpha j} } \leqslant CP(\xi ).$$ (See V. P. Mihailov,$Soviet Math. Dokl. 164$(1965), MR 32: 6047).

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We shall consider point systems in$R$_{1}which are stationary renewal distributed. We let the points undergo random translations which are assumed to be independent identically distributed random variables with a non-degenerate distribution function. The translations are also independent of the starting positions. It is shown in theorem 3.1 that the only distribution of the points which is conserved after the random translations is the Poisson one. Finally in section 4 we give a characterization of renewal processes on the positive semiaxis in terms of conditional mean values.