In section 2 we introduce and study the independence property for a sequence of two-dimensional random variables and by means of this property we define independent motion in section 3. Section 4 is mainly a survey of known results about the convergence of the spatial distribution of the point system as the time t→∞. In theorem 5.1 we show that the only distributions which are time-invariant under given reversible motion of non-degenerated type are the weighted Poisson ones. Lastly in section 6 we study a more general type of random motion where the position of a point after translation is a function f of its original position and its motion ability. We consider functions f which are monotone in the starting position. Limiting ourselves to the case when the point system initially is weighted Poisson distributed with independent motion abilities, we prove in theorem 6.1 that this is the case also after the translations, if and only if the function f is linear in the starting position. In the paper also some implications of our results to the theory of road traffic with free overtaking are given.

We shall consider point systems in R₁ 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.

In the first part of this paper are considered real polynomials P(ζ), ζ∈Rⁿ, 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).

It is then proved by an explicit computation, for every given even multi-index γ, that there are a real number θ>0 and an integer r, 0≤r< n, depending only on γ and {α¹}, and such that $\int {\xi ^\gamma } \exp \{ - tP(\xi )\} d\xi - K\gamma (P)t^{ - \theta } \left| {\log t} \right|^\gamma (1 + o(1))$ as t→+0. A Tauberian argument then leads to an asymptotic estimate of the integral $e_0^{(\beta ,\beta )} (\lambda ,0) = \int {{}_{P(\xi \leqslant \lambda )}\xi ^{2\beta } d\xi ,} $ , where e ${}_{0}^{(β, β)}$ is a derivative of a certain spectral function. Less explicit results for a larger class of polynomials were given by N. Nilsson, Ark. f. Mat. 5 (1965). In the second part of the paper, the explicit computations are extended to the larger class considered by Nilsson but under the restriction n=2.