## Electronic Communications in Probability

### On the supremum of products of symmetric stable processes

Christophe Profeta

#### Abstract

We study the asymptotics, for small and large values, of the supremum of a product of symmetric stable processes. We show in particular that the lower tail exponent remains the same as for only one process, possibly up to some logarithmic terms. The proof relies on a path construction of stable bridges using last sign changes.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 97, 13 pp.

Dates
Received: 11 May 2018
Accepted: 13 November 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1545102493

Digital Object Identifier
doi:10.1214/18-ECP193

Mathematical Reviews number (MathSciNet)
MR3896835

Zentralblatt MATH identifier
07023483

Subjects
Primary: 60G52: Stable processes 60J65: Brownian motion [See also 58J65]

#### Citation

Profeta, Christophe. On the supremum of products of symmetric stable processes. Electron. Commun. Probab. 23 (2018), paper no. 97, 13 pp. doi:10.1214/18-ECP193. https://projecteuclid.org/euclid.ecp/1545102493

#### References

• [1] F. Aurzada and T. Simon: Persistence probabilities and exponents. In : Lévy Matters V, 183-224, Springer, 2015.
• [2] R. Bañuelos and K. Bogdan: Symmetric stable processes in cones. Potential Anal. 21 (3), 263-288, 2004.
• [3] R. Bañuelos and K. Bogdan: Symmetric stable processes in parabola-shaped regions. Proc. Amer. Math. Soc. 133 (12), 3581-3587, 2005.
• [4] J. Bertoin: Lévy processes. Cambridge University Press, Cambridge, 1996.
• [5] J. Bertoin and J. Pitman: Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (2), 147-166, 1994.
• [6] A. J. Bray, S. N. Majumdar and G. Schehr: Persistence and first-passage properties in non-equilibrium systems. Adv. Physics 62 (3), 225-361, 2013.
• [7] R.A. Doney and M. S. Savov: The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 (1), 316-326, 2010.
• [8] S. Janson: Moments of Gamma type and the Brownian supremum process area. Prob. Surveys 7, 1-52, 2010.
• [9] V.L. Wenbo and Q.-S. Shao: Lower tail probabilities for Gaussian processes. Ann. Probab. 32, 216-242, 2004.
• [10] P.J. Méndez-Hernández: Exit times of symmetric $\alpha$-stable processes from unbounded convex domains. Elect. J. Probab. 12, 100-121, 2007.
• [11] P.J. Méndez-Hernández: Exit times from cones in $\mathbb{R} ^n$ of symmetric stable processes. Illinois J. Math. 46 (1), 155-163, 2002.
• [12] C. Profeta and T. Simon: Windings of the stable Kolmogorov process. ALEA XII, 115-127, 2015.