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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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