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Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

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Abstract

We study the persistence probability for some two-sided, discrete-time Gaussian sequences that are discrete-time analogues of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous time in Molchan (Commun Math Phys 205(1):97–111, 1999) and Molchan (J Stat Phys 167(6):1546–1554, 2017) to a wide class of discrete-time processes.

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References

  1. Ash, R.B., Gardner, M.E.: Topics in Stochastic Processes. Probability and Mathematical Statistics, vol. 27. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Aurzada, F., Baumgarten, C.: Persistence of fractional Brownian motion with moving boundaries and applications. J. Phys. A: Math. Theor. 46, 12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aurzada, F., Dereich, S.: Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49(1), 236–251 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Aurzada, F., Guillotin-Plantard, N., Pène, F.: Persistence probabilities for stationary increment processes. Stoch. Process. Their Appl. https://doi.org/10.1016/j.spa.2017.07.016

  5. Aurzada, F., Simon, T.: Persistence probabilities and exponents. In: Lévy Matters V, vol. 2149 of Lecture Notes in Mathematics, pp. 183–224. Springer, Cham (2015)

  6. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  7. Bray, A.J., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in nonequilibrium systems. Adv. Phys. 62(3), 225–361 (2013)

    Article  ADS  Google Scholar 

  8. Castell, F., Guillotin-Plantard, N., Pène, F., Schapira, B.: On the one-sided exit problem for stable processes in random scenery. Electron. Commun. Probab. https://doi.org/10.1214/ECP.v18-2444 (2013)

  9. Majumdar, S.N.: Persistence in nonequilibrium systems. Curr. Sci. 77(3), 370–375 (1999)

    Google Scholar 

  10. Majumdar, S.N.: Persistence of a particle in the Matheron–de Marsily velocity field. Phys. Rev. E 68(5), 050101 (2003)

    Article  ADS  Google Scholar 

  11. Molchan, G.: Maximum of a fractional Brownian motion: probabilities of small values. Commun. Math. Phys. 205(1), 97–111 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Molchan, G.: The inviscid burgers equation with fractional brownian initial data: the dimension of regular lagrangian points. J. Stat. Phys. 167(6), 1546–1554 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Molchan, G., Khokhlov, A.: Small values of the maximum for the integral of fractional brownian motion. J. Stat. Phys. 114(3–4), 923–946 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Oshanin, G., Rosso, A., Schehr, G.: Anomalous fluctuations of currents in sinai-type random chains with strongly correlated disorder. Phys. Rev. Lett. 110(10), 100602 (2013)

    Article  ADS  Google Scholar 

  15. Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–316 (1971)

    MathSciNet  MATH  Google Scholar 

  16. Redner, S.: Survival probability in a random velocity field. Phys. Rev. E 56, 4967–4972 (1997)

    Article  ADS  Google Scholar 

  17. Samorodnitsky, G.: Long range dependence. Found. Trends® Stoch. Syst. 1 3, 163–257 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. She, Z.-S., Aurell, E., Frisch, U.: The inviscid burgers equation with initial data of brownian type. Commun. Math. Phys. 148(3), 623–641 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Sinai, Y.G.: Statistics of shocks in solutions of inviscid burgers equation. Commun. Math. Phys. 148(3), 601–621 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Whitt, W.: Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research and Financial Engineering. Springer, New York (2002)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Technische Universität Darmstadt, Darmstadt, Germany

    Frank Aurzada & Micha Buck

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  1. Frank Aurzada
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  2. Micha Buck
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Correspondence to Micha Buck.

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Aurzada, F., Buck, M. Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences. J Stat Phys 170, 784–799 (2018). https://doi.org/10.1007/s10955-018-1954-8

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  • DOI: https://doi.org/10.1007/s10955-018-1954-8

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