Hi all, I am very new to this field. I am curious to know, what is a natural way to define point process on the space of positive integers(Z+) ? I saw Z+ valued point process, but I am not sure how one can define it On Z+. Is it well defined there ? Because in the definition of point process it seems there is no restriction on the domain itself. I might be wrong. Let me know. thanks, --Subhajit
1 Answer
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A point process on Z+ is just a random subset of Z+.
In general, a point process on a set $S$ is a random point pattern in $S$, which you can think of as a countable subset of $S$ without limit points. On Z+, all subsets are countable and don't have limit points (assuming you're working with the usual discrete topology, and take its Borel $\sigma$-algebra), so any subset will do.
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1$\begingroup$ In the litterature, "point process" allows for multiple points, meaning in some point, say 0, it is possible to have 0, 1, 2, 3, ... (thus a random point process is a random variable taking values in $\mathbb{N}^{\mathbb{Z}_+}$). It is a necessary assumption if for instance you define a Poisson point process on Z+. $\endgroup$ Sep 17, 2011 at 15:29
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$\begingroup$ Elena and Remark, thanks for your prompt reply. $\endgroup$– subhajitSep 19, 2011 at 12:25