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Fractional Poisson process. II. (English) Zbl 1086.60022
Let $$H\in(1/2,1)$$. For $$t>0$$ the process $W_{H}(t) =\frac{1}{(H-1/2)} \int_{0}^{t} u^{1/2-H} \biggl(\int_{u}^{t}\tau ^{H- 1/2}(\tau-u)^{H-3/2}\,d\tau \biggr)\, dq(u)$ is called a fractional Poisson process, where $$q(u) =N(u)/\sqrt{\lambda }-\sqrt{\lambda }u$$, and $$N(u)$$ is a homogeneous Poisson process with the intensity $$\lambda >0,$$ $$N(0) =0$$ a.s. It is self-similar in a wide sense, has the wide-sense stationary increments, exhibits long-range dependence and has continuous paths. Its correlation function is $E(W_{H}(t) W_{H}(s) ) =\frac{1}{2}\frac{(2-2H) \cos \pi H}{(2H-1) \pi H}(| s| ^{2H}+| t| ^{2H}-| t-s| ^{2H})$ and $$\text{Dim}(\operatorname{graph} W_{H})=2-H$$ with probability one. The Vigner-Vile spectrum of the process $$W_{H}(t)$$ is presented here. Denote $$\sigma _{H}(t) =\sqrt{\operatorname{Var}W_{H}(t) }.$$ It is proved that $$W_H(t)/\sigma_H(t)$$ converges to a normal $$N(0,1)$$ limit as $$t\to +\infty$$ or $$\lambda \to +\infty$$. The skewness and excess kurtosis of $$W_{H}(t)$$ are discussed as well. The process $$W_{H}(t)$$ is different from that of the fractional Poisson process defined by Jumarie through the Liouville-Rieman fractional derivative. The list of references contains 24 positions. In the introduction the authors recall the similar properties of the fractional Brownian process and explain their interest in the process $$W_{H}(t)$$ as caused by quantum physics and finance.

##### MSC:
 60G18 Self-similar stochastic processes 60G17 Sample path properties 60H05 Stochastic integrals
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##### References:
 [1] Adler, R.J., The geometry of random fields, (1981), John Wiley New York, p. 204 · Zbl 0478.60059 [2] Nottale, L., Chaos, solitons & fractals, 7, 6, 877, (1996) [3] Ash, R.B.; Gardner, M.F., Topics in stochastic process, (1975), Academic Press New York, p. 164 [4] Barton, R.J.; Poor, H.V., IEEE trans inform theory, 34, 5, 943, (1988) [5] Bouchaud, J.Ph., Physica A, 285, 18, (2000) [6] El-Wakil, S.A.; Elhanbaly, A.; Zahran, M.A., Chaos, solitons & fractals, 12, 1035, (2001) [7] Flandrin, P., IEEE trans inform theory, 35, 1, 197, (1989) [8] Havlin, S.; Ben-Avraham, D., Adv. phys., 36, 695, (1987) [9] Hurst, H.E.; Black, R.P.; Sinaika, Y.M., Proc IEEE, 3557, (1991) [10] Jumarie, G., Chaos, solitons & fractals, 12, 2577, (2001) [11] Lane, J.A., J. appl. prob., 21, 287, (1984) [12] Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D.V., ACM/SIGCOMM comput comm rev, 23, 183, (1993) [13] Lim, S.C.; Muniandy, S.V., Phys lett A, 266, 140, (2000) [14] Lim, S.C.; Sithi, V.M., Phys lett A, 206, 311, (1995) [15] Mandelbrot, B.B., Sci amer, Febr., 71, (1999) [16] Mandelbrot, B.B.; van Ness, J.W., SIAM review, 10, 4, 422, (1968) [17] Muniandy, S.V.; Lim, S.C., Phys rev E, 63, 046104, (2001) [18] Papoulis, A., Probability, random variables, and stochastic process, (1965), McGraw-Hill Book Company NY, p. 567 · Zbl 0191.46704 [19] Peng, C.K.; Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Sciortino, F.; Simons, M., Nature, 356, 168, (1992) [20] Stanley, H.E., Physics A, 285, 1, (2000) [21] Véhel, J.L., Fractals, 3, 4, 755, (1995) [22] El Naschie, M.S., Chaos, solitons & fractals, 14, 649, (2002) [23] El Naschie, M.S., Chaos, solitons & fractals, 17, 591, (2003) [24] Ord, G.N., Chaos, solitons & fractals, 17, 609, (2003)
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