• Bernoulli
  • Volume 15, Number 4 (2009), 1305-1334.

Small deviations of stable processes and entropy of the associated random operators

Frank Aurzada, Mikhail Lifshits, and Werner Linde

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We investigate the relation between the small deviation problem for a symmetric $α$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric $α$-stable random processes, including unbounded Riemann–Liouville processes, weighted Riemann–Liouville processes and the ($d$-dimensional)$α$-stable sheet.

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Bernoulli, Volume 15, Number 4 (2009), 1305-1334.

First available in Project Euclid: 8 January 2010

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Gaussian processes metric entropy random operators Riemann–Liouville processes small deviations stable processes


Aurzada, Frank; Lifshits, Mikhail; Linde, Werner. Small deviations of stable processes and entropy of the associated random operators. Bernoulli 15 (2009), no. 4, 1305--1334. doi:10.3150/09-BEJ212.

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  • Artstein, S., Milman, V. and Szarek, S.J. (2004). Duality of metric entropy. Ann. of Math. 159 1313–1328.
  • Aurzada, F. (2007a). On the lower tail probabilities of some random sequences in lp. J. Theoret. Probab. 20 843–858.
  • Aurzada, F. (2007b). Metric entropy and the small deviation problem for stable processes. Probab. Math. Statist. 27 261–274.
  • Aurzada, F. and Simon, T. (2007). Small ball probabilities for stable convolutions. ESAIM Probab. Stat. 11 327–343.
  • Aurzada, F. and Lifshits, M. (2008). Small deviation probability via chaining. Stochastic Process. Appl. 118 2344–2368.
  • Belinsky, E.S. (1998). Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative. J. Approx. Theory 93 114–127.
  • Borovkov, A.A. and Mogul’skiĭ, A.A. (1991). On probabilities of small deviations for stochastic processes. Siberian Adv. Math. 1 39–63.
  • Carl, B. and Stephani, I. (1990). Entropy, Compactness and the Approximation of Operators. Cambridge Tracts in Mathematics 98. Cambridge: Cambridge Univ. Press.
  • Dunker, T., Kühn, T., Lifshits, M.A. and Linde, W. (1999). Metric entropy of integration operators and small ball probabilities for the Brownian sheet. J. Approx. Theory 101 63–77.
  • Kühn, T. (2005). Entropy numbers of general diagonal operators. Rev. Mat. Complut. 18 479–491.
  • Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133–157.
  • Li, W.V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556–1578.
  • Li, W.V. and Linde, W. (2004). Small deviations of stable processes via metric entropy. J. Theoret. Probab. 17 261–284.
  • Lifshits, M.A. and Linde, W. (2002). Approximation and entropy of Volterra operators with application to Brownian motion. Mem. Amer. Math. Soc. 745 1–87.
  • Lifshits, M.A. and Simon, T. (2005). Small deviations for fractional stable processes. Ann. Inst. H. Poincaré Probab. Statist. 41 725–752.
  • Linde, W. (1986). Probability in Banach Spaces – Stable and Infinitely Divisible Distributions. Chichester: Wiley.
  • Marcus, M.B. and Pisier, G. (1984). Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152 245–301.
  • Mogul’skiĭ, A.A. (1974). Small deviations in a space of trajectories. Theory Probab. Appl. 19 726–736 (Russian), 755–765 (English).
  • Ryznar, M. (1986). Asymptotic behaviour of stable seminorms near the origin. Ann. Probab. 14 287–298.
  • Sztencel, R. (1984). On the lower tail of stable seminorms. Bull. Pol. Acad. Sci. Math. 32 715–719.
  • Samorodnitsky, G. and Taqqu, M.S. (1994). Stable non-Gaussian Random Processes. New York: Chapman & Hall.
  • Tortrat, A. (1976). Lois e(λ) dans les espaces vectoriels et lois stables. Z. Wahrsch. Verw. Gebiete 37 175–182.
  • Vakhania, N.N., Tarieladze, V.I. and Chobanjan, S.A. (1985). Probability Distributions in Banach Spaces. Moscow: Nauka.