Power Spectra of Point Processes

A simple point process on the positive half-line is, roughly speaking, a strictly increasing sequence \(\left\{ T_n\right\} _{n\in {\mathbb {N}}_+}\) of random variables taking their values in \([0, +\infty ]\) and called the event times.

1. 1.

   For instance: [Kallenberg], [Neveu], [Daley and Vere-Jones].

2. 2.

   In particular, for any \(C\in \mathcal{B}(E)\), the mapping \(p_C:\,\mu \rightarrow \mu (C)\) is measurable.

3. 3.

   The \(\infty \) represents the number of servers. This model is sometimes called a “queuing” system, although in reality there is no queuing, since customers are served immediately upon arrival and without interruption. It is in fact a “pure delay” system.

4. 4.

   See for instance, Baccelli and Brémaud, Elements of Queuing Theory, Springer, New York, 1994 (2nd edition 2003).

5. 5.

   This proof is borrowed from [Neveu]. See also [Daley and Vere-Jones]. The latter has also the equivalent of the Cramér–Khinchin decomposition.

6. 6.

   For applications of the formulas of the present subsection to communications systems, see [Ridolfi].

7. 7.

   Cioczek-Georges, R., Mandelbrot, B.B.: “A class of micropulses and antipersistent fractal Brownian motion”, Stochastic Processes and their Applications, 60, pp. 1–18, (1995).

Authors and Affiliations

1. Inria, Paris, France

   Pierre Brémaud

2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

   Pierre Brémaud

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Correspondence to Pierre Brémaud .

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