Limits of compound and thinned point processes

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Abstract
Letη=Σjδτjbe a point process on some spaceSand letβ,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent ofη.We can then form the compound point processξ= Σjβjδτjwhich is a random measure onS.The purpose of this paper is to study the limiting behaviour ofξas. In the particular case whenβtakes the values 1 and 0 with probabilitiespand 1 –prespectively,ξbecomes ap-thinning ofηand our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

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