## Electronic Journal of Probability

### Small deviations for time-changed Brownian motions and applications to second-order chaos

#### Abstract

We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.

#### Article information

Source
Electron. J. Probab. Volume 19 (2014), paper no. 85, 23 pp.

Dates
Accepted: 16 September 2014
First available in Project Euclid: 4 June 2016

http://projecteuclid.org/euclid.ejp/1465065727

Digital Object Identifier
doi:10.1214/EJP.v19-2993

Mathematical Reviews number (MathSciNet)
MR3263642

Zentralblatt MATH identifier
1329.60087

Rights

#### Citation

Dobbs, Daniel; Melcher, Tai. Small deviations for time-changed Brownian motions and applications to second-order chaos. Electron. J. Probab. 19 (2014), paper no. 85, 23 pp. doi:10.1214/EJP.v19-2993. http://projecteuclid.org/euclid.ejp/1465065727.

#### References

• Aurzada, Frank. A short note on small deviations of sequences of i.i.d. random variables with exponentially decreasing weights. Statist. Probab. Lett. 78 (2008), no. 15, 2300–2307.
• Aurzada, Frank; Lifshits, Mikhail. On the small deviation problem for some iterated processes. Electron. J. Probab. 14 (2009), no. 68, 1992–2010.
• Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2.
• Bogachev, Vladimir I. Gaussian measures. Mathematical Surveys and Monographs, 62. American Mathematical Society, Providence, RI, 1998. xii+433 pp. ISBN: 0-8218-1054-5.
• Borell, Christer. Tail probabilities in Gauss space. Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977), I, pp. 73–82, Lecture Notes in Math., 644, Springer, Berlin-New York, 1978.
• C. Borell. On the Taylor series of a Wiener polynomial. Seminar notes on multiple stochastic integration, polynomial chaos and their integration. Case Western Reserve University, Cleveland, 1984.
• Borovkov, A. A.; Ruzankin, P. S. On small deviations of series of weighted random variables. J. Theoret. Probab. 21 (2008), no. 3, 628–649.
• Chen, Xia; Kuelbs, James; Li, Wenbo. A functional LIL for symmetric stable processes. Ann. Probab. 28 (2000), no. 1, 258–276.
• Chung, Kai Lai. On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. 64, (1948). 205–233.
• Driver, Bruce K.; Gordina, Maria. Heat kernel analysis on infinite-dimensional Heisenberg groups. J. Funct. Anal. 255 (2008), no. 9, 2395–2461.
• Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3.
• Kuelbs, J.; Li, W. V. A functional LIL and some weighted occupation measure results for fractional Brownian motion. J. Theoret. Probab. 15 (2002), no. 4, 1007–1030.
• Kuelbs, James; Li, Wenbo. A functional LIL for stochastic integrals and the Lévy area process. J. Theoret. Probab. 18 (2005), no. 2, 261–290.
• Ledoux, Michel. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics (Saint-Flour, 1994), 165–294, Lecture Notes in Math., 1648, Springer, Berlin, 1996.
• Li, Wenbo V. Small ball probabilities for Gaussian Markov processes under the $L_ p$-norm. Stochastic Process. Appl. 92 (2001), no. 1, 87–102.
• Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533–597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001.
• Lifshits, M. A. On the lower tail probabilities of some random series. Ann. Probab. 25 (1997), no. 1, 424–442.
• M. A. Lifshits. Asymptotic behavior of small ball probabilities. In Theory of Probability and Mathematical Statistics. Proceedings of VII International Vilnius Conference, pages 453–468, 1999.
• Lifshits, Mikhail A.; Linde, Werner. Small deviations of weighted fractional processes and average non-linear approximation. Trans. Amer. Math. Soc. 357 (2005), no. 5, 2059–2079 (electronic).
• Mayer-Wolf, Eddy; Zeitouni, Ofer. The probability of small Gaussian ellipsoids and associated conditional moments. Ann. Probab. 21 (1993), no. 1, 14–24.
• Nelson, Edward. The free Markoff field. J. Functional Analysis 12 (1973), 211–227.
• Nelson, Edward. Quantum fields and Markoff fields. Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), pp. 413–420. Amer. Math. Soc., Providence, R.I., 1973.
• Rémillard, Bruno. On Chung's law of the iterated logarithm for some stochastic integrals. Ann. Probab. 22 (1994), no. 4, 1794–1802.
• Rozovskiĭ, L. V. On small deviation probabilities for sums of independent positive random variables. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 341 (2007), Veroyatn. i Stat. 11, 151–167, 232; translation in J. Math. Sci. (N. Y.) 147 (2007), no. 4, 6935–6945
• Rozovskiĭ, L. V. On small deviations of sums of weighted positive random variables. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 384 (2010), Veroyatnost i Statistika. 16, 212–224, 313; translation in J. Math. Sci. (N. Y.) 176 (2011), no. 2, 224–231
• Shi, Z. Liminf behaviours of the windings and Lévy's stochastic areas of planar Brownian motion. Séminaire de Probabilités, XXVIII, 122–137, Lecture Notes in Math., 1583, Springer, Berlin, 1994.
• Zhang, Rongmao; Lin, Zhengyan. A functional LIL for $m$-fold integrated Brownian motion. Chinese Ann. Math. Ser. B 27 (2006), no. 4, 459–472.
• Zhang, Rong Mao; Lin, Zheng Yan. A functional LIL for integrated $\alpha$ stable process. Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 2, 393–404.