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Large deviations of infinite intersections of events in Gaussian processes

Author

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  • Mandjes, Michel
  • Mannersalo, Petteri
  • Norros, Ilkka
  • van Uitert, Miranda

Abstract

Consider events of the form {Zs>=[zeta](s),s[set membership, variant]S}, where Z is a continuous Gaussian process with stationary increments, [zeta] is a function that belongs to the reproducing kernel Hilbert space R of process Z, and is compact. The main problem considered in this paper is identifying the function [beta]*[set membership, variant]R satisfying [beta]*(s)>=[zeta](s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when [zeta](s)=s for s[set membership, variant][0,1] and Z is either a fractional Brownian motion or an integrated Ornstein-Uhlenbeck process.

Suggested Citation

  • Mandjes, Michel & Mannersalo, Petteri & Norros, Ilkka & van Uitert, Miranda, 2006. "Large deviations of infinite intersections of events in Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1269-1293, September.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:9:p:1269-1293
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    References listed on IDEAS

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    1. Dieker, A.B., 2005. "Conditional limit theorems for queues with Gaussian input, a weak convergence approach," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 849-873, May.
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    Cited by:

    1. Lucia Caramellino & Barbara Pacchiarotti & Simone Salvadei, 2015. "Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 383-401, June.
    2. Norros, Ilkka & Saksman, Eero, 2009. "Local independence of fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3155-3172, October.
    3. Braunsteins, Peter & Mandjes, Michel, 2023. "The Cramér-Lundberg model with a fluctuating number of clients," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 1-22.
    4. Martin Zubeldia & Michel Mandjes, 2021. "Large deviations for acyclic networks of queues with correlated Gaussian inputs," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 333-371, August.

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