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Survival Exponents for Some Gaussian Processes

DOI: 10.1155/2012/137271

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Abstract:

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , . 1. The Problem Let be a real-valued stochastic process with the following asymptotics: where is the so-called survival exponent of . Below we focus on estimating for some self-similar Gaussian processes in extended intervals and , . Usually the estimation of the survival exponents is based on Slepian’s lemma. The estimation requires reference processes with explicit or almost explicit values of . Unfortunately, the list of such processes is very short. This includes the fractional Brownian motion (FBM), , of order both with one- and multidimensional time. According to Molchan ([1]) Another important example is the integrated Brownian motion with the exponent (Sinai [2]). The nature of this result is best understood in terms of a series of generalizations where the integrand is a random walk with discrete or continuous time (see, e.g., Isozaki and Watanabe [3]; Isozaki and Kotani [4]; Simon [5]; Vysotsky [6, 7]; Aurzada and Dereich [8]; Dembo et al. [9]; Denisov and Wachtel [10]. The extension of (1.3) to include the case of the integrated fractional Brownian motion, , remains an important; but as yet unsolved problem. Below we consider the survival exponents for the following Gaussian processes: ; FBM in , ; the Laplace transform of white noise with ; the fractional Slepian’s stationary process whose correlation function is , . Our approach to the estimation of is more or less traditional. Namely, any self-similar process in generates a dual stationary process , , where is the self-similarity index of . For a large class of Gaussian processes, relation (1.1) induces the dual asymptotics with the same exponent , [1, 11]. More generally, the dual exponent is defined by the asymptotics To formulate the simplest condition for the exponents to be equal, we define one more exponent by means of the asymptotics where is the position of the maximum of in , that is, . Lemma 1.1 (see [1, 11]). Let be a self-similar continuous Gaussian process in and be the reproducing kernel Hilbert space associated with . Suppose that there exists such an element of that and . Then , and can exist simultaneously only; moreover, the exponents are equal to each other. The equality reduces the original problem to the estimation of . Nonnegativity of the correlation function of guarantees the

References

[1]  G. M. Molchan, “Maximum of a fractional Brownian motion: probabilities of small values,” Communications in Mathematical Physics, vol. 205, no. 1, pp. 97–111, 1999.
[2]  Ya. G. Sina?, “Distribution of some functionals of the integral of a random walk,” Rossi?skaya Akademiya Nauk. Teoreticheskaya i Matematicheskaya Fizika, vol. 90, no. 3, pp. 323–353, 1992.
[3]  Y. Isozaki and S. Watanabe, “An asymptotic formula for the Kolmogorov diffusion and a refinement of Sinai's estimates for the integral of Brownian motion,” Proceedings of the Japan Academy, Series A, vol. 70, no. 9, pp. 271–276, 1994.
[4]  Y. Isozaki and S. Kotani, “Asymptotic estimates for the first hitting time of fluctuating additive functionals of Brownian motion,” in Séminaire de Probabilités, 34, vol. 1729 of Lecture Notes in Mathematics, pp. 374–387, Springer, Berlin, Germany, 2000.
[5]  T. Simon, “The lower tail problem for homogeneous functionals of stable processes with no negative jumps,” ALEA. Latin American Journal of Probability and Mathematical Statistics, vol. 3, pp. 165–179, 2007.
[6]  V. Vysotsky, “On the probability that integrated random walks stay positive,” Stochastic Processes and Their Applications, vol. 120, no. 7, pp. 1178–1193, 2010.
[7]  V. Vysotsky, “Positivity of integrated random walks,” http://arxiv.org/abs/1107.
[8]  F. Aurzada and S. Dereich, “Universality of the asymptotics of the one-sided exit problem for integrated processes,” Annales de l'Institut Henri Poincaré (B), http://arxiv.org/abs/1008.0485.
[9]  A. Dembo, J. Ding, and F. Gao, “Persistence of iterated partial sums,” http://arxiv.org/abs/1205.5596.
[10]  D. Denisov and V. Wachtel, “Exit times for integrated random walks,” http://arxiv.org/pdf/1207.2270v1.
[11]  G. Molchan, “Unilateral small deviations of processes related to the fractional Brownian motion,” Stochastic Processes and Their Applications, vol. 118, no. 11, pp. 2085–2097, 2008.
[12]  W. V. Li and Q.-M. Shao, “Lower tail probabilities for Gaussian processes,” The Annals of Probability, vol. 32, no. 1, pp. 216–242, 2004.
[13]  G. Molchan and A. Khokhlov, “Unilateral small deviations for the integral of fractional Brownian motion,” http://arxiv.org/abs/math/0310413.
[14]  G. Molchan and A. Khokhlov, “Small values of the maximum for the integral of fractional Brownian motion,” Journal of Statistical Physics, vol. 114, no. 3-4, pp. 923–946, 2004.
[15]  W. V. Li and Q.-M. Shao, “A normal comparison inequality and its applications,” Probability Theory and Related Fields, vol. 122, no. 4, pp. 494–508, 2002.
[16]  A. Dembo, B. Poonen, Q.-M. Shao, and O. Zeitouni, “Random polynomials having few or no real zeros,” Journal of the American Mathematical Society, vol. 15, no. 4, pp. 857–892, 2002.
[17]  T. Newman and W. Loinaz, “Critical dimensions of the diffusion equation,” Physical Review Letters, vol. 86, no. 13, pp. 2712–2715, 2001.
[18]  L. A. Shepp, “First passage time for a particular Gaussian process,” Annals of Mathematical Statistics, vol. 42, pp. 946–951, 1971.
[19]  J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, and C. Sire, “Persistence exponents for fluctuating interfaces,” Physical Review E, vol. 56, no. 3, pp. 2702–2712, 1997.
[20]  M. A. Lifshits, Gaussian Random Functions, vol. 322 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.

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