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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
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\centerline{ Sequoia Hall, Room 200}
\centerline{4:15 p.m., Thursday, August 8, 2002}


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\centerline{\sl Hermann Thorisson}
\centerline{\sl University of Iceland}
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\centerline{\bf  Point-Stationarity}

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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
$P(T ^Â 0)$ 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" can 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.



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