Abstract
The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.
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A. Benassi, S. Cohen, and J. Istas, “Identification and properties of real harmonizable fractional Lévy motions,” Bernoulli vol. 8 pp. 97–115, 2002.
A. Benassi, S. Cohen, and J. Istas, “On roughness indexes for fractional fields,” Bernoulli vol. 10(2) pp. 357–373, 2004
R. Cioczek-Georges and B. B. Mandelbrot, “Alternative micropulses and fractional Brownian motion,” Stochastic Processes and their Applications vol. 64 pp. 143–152, 1996.
W. Feller, An Introduction to Probability Theory and its Applications vol. 1. Wiley: New York, 3rd edition, 1957.
R. Gaigalas, An integral representation for the asymptotic arrival process under intermediate connection intensity. Preprint, 2003.
R. Gaigalas and I. Kaj, “Convergence of scaled renewal processes and a packet arrival model,” Bernoulli vol. 9(4) pp. 671–703, 2003.
W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of Ethernet traffic (Extended version),” IEEE/ACM Transactions on Networking vol. 2 pp. 1–15, 1994.
J. B. Levy and M. S. Taqqu, “Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards,” Bernoulli vol. 6(1) pp. 23–44, 2000.
V. Pipiras and M. S. Taqqu, “The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion,” Bernoulli vol. 6(4) pp. 607–614, 2000.
V. Pipiras and M. S. Taqqu, “Decomposition of self-similar stable mixed moving averages,” Probability Theory and Related Fields vol. 123(3) pp. 412–452, 2002a.
V. Pipiras and M. S. Taqqu, “Deconvolution of fractional Brownian motion,” Journal of Time Series Analysis vol. 23(4) pp. 487–501, 2002b.
V. Pipiras and M. S. Taqqu, “The structure of self-similar stable mixed moving averages,” The Annals of Probability vol. 30(2) pp. 898–932, 2002c.
V. Pipiras and M. S. Taqqu, “Dilated fractional stable motions,” Journal of Theoretical Probability vol. 17(1) pp. 51–84, 2004.
V. Pipiras, M. S. Taqqu, and J. B. Levy, “Slow, fast and arbitrary growth conditions for renewal reward processes when both the renewals and the rewards are heavy-tailed,” Bernoulli vol. 10(1) pp. 121–163, 2004.
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Chapman and Hall: New York, London, 1994.
M. S. Taqqu, W. Willinger, and R. Sherman, “Proof of a fundamental result in self-similar traffic modeling,” Computer Communications Review vol. 27(2) pp. 5–23, 1997.
W. Willinger, M. S. Taqqu, R. Sherman, and D. V. Wilson, “Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level,” IEEE/ACM Transactions in Networking, vol. 5(1) pp. 71–86, 1997. Extended Version of the paper with the same title that appeared in Computer Communications Review vol. 25 pp. 100–113, 1995.
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Cohen, S., Taqqu, M.S. Small and Large Scale Behavior of the Poissonized Telecom Process. Methodology and Computing in Applied Probability 6, 363–379 (2004). https://doi.org/10.1023/B:MCAP.0000045085.17224.82
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DOI: https://doi.org/10.1023/B:MCAP.0000045085.17224.82