Hermann Thorisson
University of Iceland

Title:

More on Point-Stationarity

Abstract:

Let N^o be the Palm version of a stationary Poisson process N in R^d,
that is, N^o has the same distribution as N + \delta_0.
Consider the following problem: when d>1, is there some non-randomized
way of shifting the origin of N^o from the point at the origin to another
point T so that the distribution of N^o does not change?

This is clearly possible when d=1, since then the intervals between
points are i.i.d. exponential and remain so when the origin is shifted to
the n-th point on the right (or on the left) of the point at the origin.
And when d>1$, it is shown in Thorisson (2000) that such a T - with
P(T \ne 0) arbitrarily close to 1, - exists if external randomization is
allowed. But is there a strictly non-zero non-randomized T?

We shall show that the answer is yes. There is actually a sequence
(T_n : n \in Z) of such points, and for d=2 and d=3
this sequence strings up the points of N^o.

If we go beyond the Poisson case, a more general problem concerns the
concept of "point-stationarity". Intuitively, point-stationarity means that
the behaviour of a point process N^o relative to a given point of the
process is independent of the point selected as origin. Formally, this
concept is defined in Thorisson (2000) to be distributional invariance
under bijective point-shifts "against any independent stationary
background" and shown to be the characterizing property of the Palm version
N^o of any stationary point process N in R^d. A natural question is
whether the definition of "point-stationarity" be reduced to distributional
invariance under non-randomized bijective point-shifts. An approach to this
problem will be outlined.

Reference:

Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, NY.