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A Bound for Norms in Lp(T) of Deviations of φ-sub-Gaussian Stochastic Processes

Abstract

We study deviations of φ-sub-Gaussian stochastic process from a measurable function and generalize the results of [6]. We obtain a bound for the distributions of norms in the space Lp(\( \mathbb{T} \)). As an example, the obtained result is applied for an aggregate of independent processes of generalized ϕ-sub-Gaussian fractional Brownian motions.

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References

  1. V.V. Buldygin and Yu.V. Kozachenko, Metric Characterization of Random Variables and Random Processes, Amer. Math. Soc., Providence, RI, 2000.

  2. R. Giuliano-Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of ϕ-subgaussian random variables, Rend. Accad. Naz. Sci. XL, Mem. Mat. Appl. (5), 27(1):95–124, 2003.

    MathSciNet  Google Scholar 

  3. Yu. Kozachenko, T. Sottinen, and O. Vasilik, Weakly self-similar stationary increment processes from the space SSub φ (Ω), Theory Probab. Math. Stat., 65:77–88, 2002.

    MathSciNet  Google Scholar 

  4. Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, Upper estimate of overrunning by Sub φ (Ω)random process the level specified by continuous function, Random Oper. Stoch. Equ., 13(2):111–128, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  5. Yu. Kozachenko and R. Yamnenko, Application of φ-sub-Gaussian random processes in queuing theory, in V. Korolyuk, N. Limnios, Yu. Mishura, L. Sakhno, and G. Shevchenko (Eds.), Modern Stochastics and Applications, Springer Optimization and Its Applications, Vol. 90, Springer, 2014, pp. 21–38.

  6. Yu.V. Kozachenko and O.E. Kamenshchikova, Approximation of SSub φ (Ω)stochastic processes in the space Lp(T), Theory Probab. Math. Stat., 79:83–88, 2009.

    Article  MathSciNet  Google Scholar 

  7. Yu.V. Kozachenko and Yu.A. Kovalchuk, Boundary value problems with random initial conditions, and functional series from Sub φ (ω). I, Ukr. Math. J., 50(4):504–515, 1998.

    MATH  MathSciNet  Google Scholar 

  8. Yu.V. Kozachenko and E.I. Ostrovskij, Banach spaces of random variables of sub-Gaussian type, Theory Probab. Math. Stat., 32:45–56, 1986.

    MATH  Google Scholar 

  9. M.A. Krasnoselskii and Ya.B. Rutitskii, Convex Functions in the Orlicz spaces, Noordhoff, Gröningen, 1961.

  10. R. Yamnenko, Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion, Theory Stoch. Process., 12(28)(3–4):261–275, 2006.

  11. R. Yamnenko, Yu. Kozachenko, and D. Bushmitch, Generalized sub-Gaussian fractional Brownian motion queueing model, Queueing Syst., 77(1):75–96, 2014.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Yamnenko and O. Vasylyk, Random process from the class v(φ,ψ): exceeding a curve, Theory Stoch. Process., 13(29)(4):219–232, 2007.

  13. R.E. Yamnenko, Bounds for the distribution of some functionals of processes with φ-sub-Gaussian increments, Theory Probab. Math. Stat., 85:181–197, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  14. R.E. Yamnenko and O.S. Shramko, On the distribution of storage processes from the class υ(φ, ψ), Theory Probab. Math. Stat., 83:191–206, 2011.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Rostyslav E. Yamnenko.

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Yamnenko, R.E. A Bound for Norms in Lp(T) of Deviations of φ-sub-Gaussian Stochastic Processes. Lith Math J 55, 291–300 (2015). https://doi.org/10.1007/s10986-015-9281-0

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  • DOI: https://doi.org/10.1007/s10986-015-9281-0

MSC

  • 60G07
  • 60G18

Keywords

  • φ-sub-Gaussian stochastic process
  • norm of process
  • generalized fractional Brownian motion