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Order statistics for jumps of normalised subordinators. (English) Zbl 0777.60070

From the author’s introduction: Let \(V_ 1\geq V_ 2\geq V_ 3\geq\dots\) be the points, ranked by size, of a Poisson counting process \(N\) on \((0,\infty)\) with a Lévy mean measure \(\Lambda\). The assumptions on \(\Lambda\) are \(\Lambda(0,\infty)=\infty\), \(\Lambda(1,\infty)<\infty, \int^ 1_ 0 x\Lambda(dx)<\infty\). The author considers \(T_ 1=\sum_ i V_ i\) and the random vector \((D_ 1,D_ 2,\dots)=(V_ 1/T_ 1,V_ 2/T_ 1,\dots)\). This random vector takes values in the infinite simplex of sequences with positive terms that add up to 1. The aim of this paper is to describe the finite-dimensional distributions of this random vector, and we apply the results in some particular cases.

MSC:

60J99 Markov processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J55 Local time and additive functionals
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