## Properties of strictly $$\varphi$$-sub-Gaussian quasi-shot-noise processes.(English. Ukrainian original)Zbl 1455.60054

Theory Probab. Math. Stat. 101, 51-65 (2020); translation from Teor. Jmovirn. Mat. Stat. 101, 49-62 (2019).
Summary: Properties of $$\varphi$$-sub-Gaussian quasi-shot-noise processes $X(t)=\int_{-\infty }^{+\infty }g(t,u)\,d\xi (u), \quad t\in \mathbf{R},$ generated by a stochastic process $$\xi$$ and response function $$g$$ are studied in the paper. Sufficient conditions for quasi-shot-noise processes to belong to weighted spaces of continuous functions are obtained. Bounds for the distributions of supremums of strictly $$\varphi$$-sub-Gaussian quasi-shot-noise processes are established.

### MSC:

 60G07 General theory of stochastic processes 60K25 Queueing theory (aspects of probability theory) 60G15 Gaussian processes 60G99 Stochastic processes
Full Text:

### References:

 [1] bk V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TBiMC, Kiev, 1998; English translation American Mathematical Society, Providence, RI, 2000. · Zbl 0933.60031 [2] Campbell N. Campbell, The study of discontinuous phenomena, Proc. Cambr. Phil. Soc. 15, 117-136; Discontinuities in light emission, Proc. Cambr. Phil. Soc. 15, 310-328, 1909. [3] darkozper I. V. Dariychuk, Yu. V. Kozachenko, and M. M. Perestyuk, Stochastic Processes in Orlicz Spaces, “Zoloti lytavry”, Chernivtsi, 2011. (Ukrainian) [4] darkozSARD I. V. Dariychuk and Yu. V. Kozachenko, Some properties of pre-Gaussian shot noise processes, Stochastic Analysis and Random Dynamics, International Conference, Abstracts, Lviv, Ukraine, 2009, pp. 57-59. [5] darkozTPMS Yu. V. Kozachenko and I. V. Dariychuk, The distribution of the supremum of $$\Theta$$-pre-Gaussian shot noise processes, Teor. Imovirnost. Matem. Statyst. 80 (2009), 76-90; English translation in Theor. Probab. Math. Statist. 80 (2010), 85-100. [6] GikhSkor1977 I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, “Nauka”, Moscov, 1977; English translation of the 1st Russian edition W. B. Saunders, Philadelphia-London-Toronto, 1969. [7] giuliano R. Giuliano Antonini, Yu. V. Kozachenko, and T. Nikitina, Space of $$\varphi$$-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003), 92-124. [8] Koops D. T. Koops, O. J. Boxma, and M. R. H. Mandjes, Networks of $$\cdot /G/\infty$$ queues with shot-noise-driven arrival intensities, Queueing Systems 86 (2017), 301-325. · Zbl 1390.60327 [9] kozostr Yu. V. Kozachenko and E. I. Ostrovsky, Banach spaces of random variables of sub-Gaussian type, Teor. Veroyatnost. Matem. Statist. 32 (1985), 42-53; English translation in Theor. Probab. Math. Statist. 32 (1986), 42-53. [10] KozPash2016 Yu. Kozachenko and A. Pashko, Accuracy and Reliability of Simulation of Random Processes and Fields in Uniform Metrics, Kyiv, 2016. (Ukrainian) [11] kozpervasROSE Yu. V. Kozachenko, M. M. Perestyuk, and O. I. Vasylyk, On uniform convergence of wavelet expansions of $$\varphi$$-sub-Gaussian random process, Random Oper. Stoch. Equ. 14 (2006), no. 3, 209-232. · Zbl 1120.60036 [12] KVS2002 Yu. Kozachenko, O. Vasylyk, and T. Sottinen, Path space large deviations of a large buffer with Gaussian input traffic, Queueing Systems Theory Appl. 42 (2002), 113-129. · Zbl 1037.60082 [13] kozvasTSP Yu. V. Kozachenko and O. I. Vasilik, On the distribution of suprema of $$\textSub_\varphi (\Omega )$$ random processes, Theory Stoch. Process. 4(20) (1998), no. 1-2, 147-160. [14] kozvasTViMS2001 Yu. V. Kozachenko and O. I. Vasilik, Stochastic processes of the classes $$V(\varphi ,\psi )$$, Teor. Imovirnost. Matem. Statyst. 63 (2000), 100-111; English translation in Theor. Probab. Math. Statist. 63 (2001), 109-121. [15] kozvasVisnyk Yu. V. Kozachenko and O. I. Vasylyk, Sample paths continuity and estimates of distributions of the increments of separable stochastic processes from the class $$V(\varphi ,\psi )$$, defined on a compact set, Bull. Kiev Univ. Series: Physics and Mathematics (2004), no. 2, 45-50. (Ukrainian) · Zbl 1064.60057 [16] kozyamvasROSE Yu. Kozachenko, R. Yamnenko, and O. Vasylyk, Upper estimate of overrunning by $$\textSub_\varphi (\Omega )$$ random process the level specified by continuous function, Random Oper. Stoch. Equ. 13 (2005), no. 2, 111-128. · Zbl 1118.60025 [17] kyv Yu. V. Kozachenko, R. E. Yamnenko, and O. I. Vasylyk, $$\varphi$$-sub-Gaussian random process, Vydavnycho-Poligrafichnyi Tsentr “Kyivskyi Universytet”, Kyiv, 2008. (Ukrainian) [18] kr M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Moscow, 1958; English translation P. Noordhoff Ltd., G\"oningen, 1961. [19] Rice1944 S. O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. J. 23 (1944), 282-332. · Zbl 0063.06485 [20] Rice1945 S. O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. 24 (1945), 46-156. [21] Rice1977 J. Rice, On generalized shot noise, Adv. Appl. Probab. 9 (1977), 553-565. · Zbl 0379.60052 [22] Schmidt2014 T. Schmidt, Catastrophe insurance modeled by shot-noise processes, Risks, ISSN 2227-9091, MDPI, Basel 2 (2014), no. 1, 3-24. (http://dx.doi.org/10.3390/risks2010003) [23] Schmidt2016 T. Schmidt, Shot-noise processes in finance, in: From statistics to mathematical finance, Springer, Cham., 2017, pp. 367-385. · Zbl 1383.62253 [24] Schottky W. Schottky, \"Uber spontane Stromschwankungen in verschiedenen Elektrizit\"atsleitern, Ann. Physik 362(23) (1918), 541-567. [25] vasSOIC2017 O. I. Vasylyk, Strictly $$\varphi$$-sub-Gaussian quasi shot noise processes, Stat. Optim. and Inf. Comput. 5 (2017), 109-120.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.