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Self-similar processes with stationary increments generated by point processes. (English) Zbl 0567.60052
A point process $$\Pi$$ on $${\mathbb{R}}\times ({\mathbb{R}}\setminus \{0\})$$ is defined to be a Poincaré point process if $$\Pi$$ is locally finite and if $$\Pi$$ $$=^{d}\Pi f^{-1}_{a,b}$$ for each a and b, where $$f_{a,b}(t, x)=(at+b, ax)$$. The authors study processes $$X_ H$$, $$H>0$$, admitting representations $$X_ H(t)=\int | x|^ H(sgn x)\Pi ((0,t],dx)$$, where $$\Pi$$ is Poincaré. Such processes are H-self- similar with stationary increments.
Properties studied include the set $${\mathcal H}_ a$$ of H-values for which the integral defining $$X_ H(t)$$ converges absolutely for all t: it is shown that almost surely, $${\mathcal H}_ a\subset (1,\infty)$$, with equality a.s. if and only if the intensity measure E[$$\Pi$$ ] is finite. Related issues of conditional convergence are also examined.
Several examples are presented; these include the case that $$\Pi$$ is a Poisson process - and $$X_ H$$ is strictly stable - and a new class of ”g-adic lattice processes” that are, in a certain sense, minimally random among Poincaré processes.
Reviewer: A.Karr

MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G10 Stationary stochastic processes 60E07 Infinitely divisible distributions; stable distributions 60K99 Special processes
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