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Almost sure continuity of stable moving average processes with index less than one. (English) Zbl 0634.60037

Let \((T_ k,X_ k)\), \(k=1,2,..\). be an enumeration of a Poisson point process on \(R\times R_+\) with mean measure dt \(\alpha\) \(x^{-1- \alpha}dx\), \(\alpha >0\). Let \(f: R\to R_+\) be a measurable function such that \(\int f^{\alpha}(s)ds<\infty\). Define \[ Z_ t\quad f=\sup_{k}X_ kf(T_ k+t),\quad S\quad f_ t=\sum_{k}X_ kf(T_ k+t). \] The authors prove following two assertions:
(i) Let \(\alpha\in (0,1)\). Then the process S f has a.s. continuous sample paths if and only if f is continuous and \[ (*)\quad \int^{\infty}_{-\infty}\sup_{0<t<1}| f(t+x)|^{\alpha}dx<\infty. \] (ii) Let \(\alpha >0\). Then Z f has a.s. continuous sample paths if and only if f is continuous and (*) holds.
Reviewer: J.Anděl

MSC:

60G17 Sample path properties
60G10 Stationary stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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