Geodesic
Renewal shot noise processes in the case of slowly varying tails
Teoriâ slučajnyh processov, Tome 21 (2016) no. 2, pp. 14-21.

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We investigate weak convergence of renewal shot noise processes in the case of slowly varying tails of the inter-shot times. We show that these processes, after an appropriate non-linear scaling, converge in the sense of finite-dimensional distributions to an inverse extremal process.
Keywords: Extremal process, random process with immigration, renewal theory, shot noise process. 
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Zakhar Kabluchko; Alexander Marynych. Renewal shot noise processes in the case of slowly varying tails. Teoriâ slučajnyh processov, Tome 21 (2016) no. 2, pp. 14-21. http://geodesic.mathdoc.fr/item/THSP_2016_21_2_a2/

[1] P. Billingsley, Convergence of probability measures, 2nd ed., Wiley, Chichester, 1999

[2] N. H. Bingham, “Limit theorems for occupation times of Markov processes”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 17 (1971), 1–22

[3] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987

[4] D. A. Darling, “The influence of the maximum term in the addition of independent random variables”, Trans. Amer. Math. Soc., 73 (1952), 95–107

[5] A. Dvoretzky, P. Erdős, “Some problems on random walk in space”, Proc. Berkeley Sympos. math. Statist. Probability, California, 1951, 353–367

[6] M. Dwass, “Extremal processes”, Ann. Math. Statist, 35 (1964), 1718–1725

[7] A. Iksanov, “Functional limit theorems for renewal shot noise processes with increasing response functions”, Stoch. Process. Appl., 123:6 (2013), 1987–2010

[8] A. Iksanov, Renewal theory for perturbed random walks and similar processes, Birkhäuser, 2016

[9] A. Iksanov, Z. Kabluchko, A. Marynych, “Weak convergence of renewal shot noise processes in the case of slowly varying normalization”, Stat. Probab. Lett., 114 (2016), 67–77

[10] A. Iksanov, A. Marynych, M. Meiners, “Limit theorems for renewal shot noise processes with eventually decreasing response functions”, Stoch. Process. Appl., 124:6 (2014), 2132–2170

[11] A. Iksanov, A. Marynych, M. Meiners, “Asymptotics of random processes with immigration I: scaling limits”, Bernoulli, 23:2 (2017), 1233–1278

[12] A. Iksanov, A. Marynych, M. Meiners, “Asymptotics of random processes with immigration {II}: convergence to stationarity”, Bernoulli, 23:2 (2017), 1279–1298

[13] A. Iksanov, M. Möhle, “On the number of jumps of random walks with a barrier”, Adv. Appl. Probab., 40:1 (2008), 206–228

[14] Y. Kasahara, “Extremal process as a substitution for "one-sided stable process with index $0$"”, Stochastic processes and their applications, Proc. Int. Conf., Lect. Notes Math., 1203, Nagoya, Japan, 1985, 90–100

[15] Y. Kasahara, “A limit theorem for sums of i.i.d. random variables with slowly varying tail probability”, J. Math. Kyoto Univ., 26:3 (1986), 437–443

[16] C. Klüppelberg, T. Mikosch, “Explosive Poisson shot noise processes with applications to risk reserves”, Bernoulli, 1:1-2 (1995), 125–147

[17] C. Klüppelberg, T. Mikosch, A. Schärf, “Regular variation in the mean and stable limits for Poisson shot noise”, Bernoulli, 9:3 (2003), 467–496

[18] J. Lane, “The central limit theorem for the Poisson shot-noise process”, J. Appl. Probab., 21:2 (1984), 287–301

[19] M. M. Meerschaert, H.-P. Scheffler, “Limit theorems for continuous time random walks with slowly varying waiting times”, Stat. Probab. Lett., 71:1 (2005), 15–22

[20] P. Nándori, Z. Shen, Logarithmic scaling of planar random walk's local times, 2016, arXiv: 1603.07587

[21] S. I. Resnick, “Inverses of extremal processes”, Adv. Appl. Probab., 6 (1974), 392–406

[22] S. I. Resnick, Extreme values, regular variation and point processes, Series in Operations Research and Financial Engineering, Springer, New York, 2008, Reprint of the 1987 original., Springer

[23] E. Seneta, Regularly varying functions, Lecture Notes in Mathematics, 508, Springer-Verlag, Berlin-New York, 1976

[24] H. Teicher, “Rapidly growing random walks and an associated stopping time”, Ann. Probab., 7:6 (1979), 1078–1081

[25] W. Whitt, “Weak convergence of first passage time processes”, J. Appl. Probab., 8 (1971), 417–422

[26] W. Whitt, Stochastic-process limits: An introduction to stochastic-process limits and their application to queues, Springer Series in Operations Research, Springer-Verlag, New York, 2002

[27] M. Yamazato, “On a $J_1$-convergence theorem for stochastic processes on $D[0,\infty)$ having monotone sample paths and its applications (The 8th workshop on stochastic numerics)”, RIMS Kôkyûroku, 1620 (2009), 109–118