Stability of Critical Cluster Fields

Abstract

The theory of critical spatially homogeneous cluster processes was developed in a series of papers by LIEMANT, DEBES, KERSTAN, fi'ZATTEtES, PREHN and others [la, 4, 131, and was summarized in Chapter 6 of [12]. A crucial role in this theory is played by the notion of stability, which induces a dualism similar to that of recurrence and transience in the theory of random walks. Given that a cluster process is based on a stable or unstable field, its asymptotic behavior has long been essentially understood. For some years, the principal problem of the theory has then been to decide whether a given cluster field is stable or not. Though some progress has been made by LIEMANT, MATTHES and FLEISCHMANN [ 14, 16,9, 15, 171, this problem has essentially remained open.The present approach t o stability is based on the idea, being developed in Q 2, of tracing the history of a randomly chosen pmticle of the n-th generation and of studying the asymptotic behavior of the arising backward tree as n becomes large. This approach leads in Q 3 t o a complete solution of the stability problem in the particular case when the branching and transition mechanisms are mutually independent. As indicated by results in 5 5 , no simple solution seems t o exist in general. However, it is shown in Q 4 that the stability criteria of 3 3 remain valid in the general case under mild additional assumptions. Throughout $5 3 -5 , our stability conditions are stated in terms of the generating function of the offspring distribution and the concentration functions of the successive convolution powers of the transition distribution. I n Q 6, we derive some general estimates for LAPLACE transforms, characteristic funct ions and concentration functions, which are then applied to yield restatements, under mild regularity assumptions, of our stability conditions. Among the tools, we further mention some contributions in §Q4 and 6 to the theory of critical GALTON-\VATSON processes.In the opening Q 1, it is shown that the dualism of stable and unstable cluster fields mentioned above, lwving been treated in the literature only for non-lattice transition distributions, extends in essence to the general case. As a by-product of this result, we obtain a general description of the cluster invariant distributions, and further a convergence theorem for reducible multi-type cluster processes. 6 relative to H . Note that, apart a non-singular linear transformation, H has the form RaZb with a f b s d , and that in this case G / H is equivalent to (R/Z)h RC where c =d -a -b , (cf. Q 1 in [3]). Theorem 1.1. Let t be a stationary point process on G whose distribution is invariant w.r.t. Rome stable cluster field [D]. Then there exists some a.s. unique stationary random measure p on G/H such that, giaen p , t is conditionally infinitelyKallenberg, Stability of Critical Cluster Fields 9 divisible with canoiLica1 measure E ik Since l B is stationary on H and cluster invariant w.r.t. [D], the sample intensity pA of ln is 5 . 9 . finite, and for given ps, l B is conditionally ...

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