December 2016

Bibliography of Small Deviation

Probabilities

compiled by

M. A. Lifshits

Attention!! This version of the bibliography is not maintained any-more. This file is obsolete. For a newer, searchable version, followthe link

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1 Selection criteria

This bibliography contains ”all” known published and not yet published ar-ticles concerning estimates and asymptotic behavior of small deviation prob-abilities.

We tried also to mention the articles that contain small deviation resultsimplicitly, e.g. those on entropy numbers of operators related to stochasticprocesses.

Some general theoretical works on Gaussian measures are also includedwhenever they treat inequalities widely used in small deviation theory.

Also included are the relevant surveys of small deviation theory.The articles on applications of small deviation theory (e.g. to the laws

of iterated logarithm or to quantization problems) are NOT included unlessthey contain original small deviation results. This could well be a subject ofseparate bibliographic search.

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2 Keywords

The BibTeX source code of every item of the bibliography contains a list ofkeywords chosen from the following list.

Bessel process, Bogoliubov process, branching processes, Brownian bridge,Brownian excursion, Brownian meander, Brownian motion, Brownian sheet,diffusion processes, fractional Brownian motion, fractional processes, gen-eral processes, Gaussian processes, Gaussian Markov processes, integratedprocesses, iterated processes, Levy processes, local time, Mandelbrot cas-cades, multi-parametric processes, Ornstein–Uhlenbeck process, Poisson pro-cess, processes with stationary increments, series of independent variables,stationary processes, stable processes, stochastic integrals, sums of i.i.d. vari-ables; general norms, Hilbert norm, Holder norm, L2-norm, Lp-norm, Sobolevnorm, sup-norm, weighted norms; comparison results, dilatation, fractal sets,one-sided lower tails, Onsager-Machlup functional, PDE, shifted balls, sur-vey, volume of random sets.

3 Recent news

The recent preprints are: Aurzada and Lifshits [26], Jiange Li and Madiman[179], Lototsky [215].

The recent publications are: Ai [2], Aurzada and Kramm [22], Devulder [94],Friedland, Giladi, and Guedon [119], Yueyun Hu [145], Kobayashi [163], Lee,Peres, and Smart [177], Lototsky and Moers [216], Rozovsky [283, 284].

Also added or updated: Griffin [141].

4 Corrections and updating of the list

Everybody is encouraged to send the corrections and new references in thearea to mikhail@lifshits.org.Thanks in advance !

2

References

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[2] X. Ai, A note on Karhunen–Loeve expansions for the demeaned stationaryOrnstein-Uhlenbeck process, Statist. & Probab. Letters 117 (2016), 113–117.

[3] X. Ai, W.V. Li, and G. Liu, Karhunen–Loeve expansion for the detrendedBrownian motion, Statist. & Probab. Letters 82 (2012), 1235–1241.

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[14] F. Aurzada and Ch. Baumgarten, Survival probabilities for weighted randomwalks, ALEA 8 (2011), 235–258.

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[16] F. Aurzada and S. Dereich, Small deviations of general Levy processes, Ann.Probab. 37 (2009), 2066–2092, http://arxiv.org/abs/0805.1330.

[17] , Universality of the asymptotics of the one-sided exit problem forintegrated processes, Ann. Inst. H. Poincare 49 (2013), 236–251.

[18] F. Aurzada, S. Dereich, and M.A. Lifshits, Persistence probabilities for anintegrated random walk bridge, Probab. Math. Statist. 34 (2014), 1–22,http://arxiv.org/abs/1205.2895.

[19] F. Aurzada, F. Gao, Th. Kuhn, W.V. Li, and Shao Q.-M., Small deviationsfor a family of smooth Gaussian processes, Theor. Probab. Appl. 26 (2013),153–168, http://arxiv.org/abs/1009.5580.

[20] F. Aurzada, I.A Ibragimov, M.A. Lifshits, and van Zanten H.,Small deviations of smooth stationary Gaussian processes, Theor.Probab. Appl. 53 (2008), 697–707 (English), 788–798 (Russian),www.arXiv.org/abs/0803.4238.

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[22] , The first passage time problem over a moving boundary for asymp-totically stable Levy processes, J. Theor. Probab. 29 (2016), 737–760,http://arxiv.org/abs/1305.1203.

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[24] F. Aurzada and M. Lifshits, Small deviation probability via chaining, Stoch.Proc. Appl. 118 (2008), 2344–2368, www.arXiv.org/abs/0706.2720.

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[26] , Small deviations of sums of correlated stationary Gaussian se-quences, http://arxiv.org/abs/1606.01072, 2016.

[27] F. Aurzada, M.A. Lifshits, and W. Linde, Small deviations of stable processesand entropy of associated random operators, Bernoulli 15 (2009), 1305–1334,www.arXiv.org/abs/0804.1883.

[28] F. Aurzada and Th. Simon, Small deviations for stable convolution processes,ESAIM. Probability & Statistics 11 (2007), 327–343.

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[29] F. Aurzada and Th. Simon, Persistence probabilities & exponents, Levy Mat-ters, Lecture Notes in Math., vol. 2149, Springer, 2015, pp. 183–221.

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[43] Ph. Berthet and M.A. Lifshits, Some exact rates in the functional law of theiterated logarithm, Ann. Inst. H.Poincare 38 (2002), no. 6, 811–824.

[44] Ph. Berthet and Z. Shi, Small ball estimates for Brownian motion under aweighted sup-norm, Studia Sci. Math. Hung. 36 (2001), no. 1-2, 275–289.

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[54] E. Bongiorno and A. Goia, A clustering method for Hilbert functional databased on the small ball probability, http://arxiv.org/abs/1501.04308, 2015.

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[59] M. Bramson and D. Griffeath, Capture problems for coupled random walks,Random Walks, Brownian Motion and Interacting Particle Systems (R. Dur-rett and H. Kesten, eds.), Birkhauser, Boston, 2001, pp. 153–188.

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[61] J.C. Bronski, Small ball constants and tight eigenvalue asymptotics for frac-tional Brownian motions, J. Theoret. Probab. 16 (2003), no. 1, 87–100.

[62] P. Caithamer, Large and small deviations of a string driven by a two-parameter Gaussian noise whitein time, J. Appl. Probab. 40 (2003), 946–960.

[63] R.H. Cameron and W.T. Martin, The Wiener measure of Hilbert neighbor-hoods in the space of real continuous functions, J. Math. Phys. 23 (1944),195–209.

[64] M Capitaine, Onsager-Machlup functional for some smooth norms onWiener space, Probab. Theor. Relat. Fields 102 (1995), no. 2, 189–202.

[65] A. Carbery and J. Wright, Distributional and Lq norm inequalities for poly-nomials over convex bodies in Rn, Math. Research Letters 8 (2001), 233–248.

[66] U. Cetin, On certain integral functionals of squared Bessel processes,Stochastics 87 (2015), no. 6, 1033–1060, http://arxiv.org/abs/1209.4919.

[67] X. Chen, Moderate and small deviations for the ranges of one-dimensionalrandom walks, J. Theoret. Probab. 19 (2006), no. 3, 721–739.

[68] X. Chen, J. Kuelbs, and W.V. Li, A functional LIL for symmetric stableprocesses, Ann. Probab. 28 (2000), no. 1, 258–277.

[69] X. Chen and W.V. Li, Quadratic functionals and small ball probabilities ofthe m-fold integrated Brownian motion, Ann. Probab. 31 (2003), 1052–1077.

[70] , Small deviation estimates for some additive processes, Proc.Conf. High Dimensional Probability. III. Progress in Probability, vol. 55,Birkhauser, 2003, pp. 225–238.

[71] X. Chen, W.V. Li, J. Rosinski, and Q.M. Shao, Large deviations for lo-cal times and intersection local times of fractional brownian motions andriemann-liouville processes, Ann. Probab. 39 (2011), 729–778.

[72] W. Chu, Small value probabilities for supercritical multitype branching pro-cesses with immigration, Statist. & Probab. Letters 93 (2014), 87–95.

[73] W. Chu, W.V. Li, and Y-X. Ren, Small value probabilities for supercrit-ical branching processes with immigration, Bernoulli 14 (2014), 377–393,http://arxiv.org/abs/1301.6840.

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[74] K.L. Chung, On the maximal partial sums of independent random variables,Trans. Amer. Math. Soc. 64 (1948), 205–233.

[75] Z. Ciesielski and S.J. Taylor, First passage times and sojourn times for Brow-nian motion in space and the exact Hausdorff measure of the sample path,Trans. Amer. Math. Soc. 103 (1962), 434–450.

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[84] R. Davis and S.I. Resnick, Extremes of moving averages of random variableswith finite endpoint, Ann. Probab. 19 (1991), 312–328.

[85] P. Deheuvels and M.A. Lifshits, Probabilities of hitting of shifted small ballsby centered Poisson processes, J. Math. Sci. 118 (2003), no. 6, 5541–5554.

[86] P. Deheuvels and G. Martynov, Karhunen-Loeve expansions for weightedWiener processes and Brownian bridges via Bessel functions, Proc. Conf.High Dimensional Probability. III. Progress in Probability, vol. 55,Birkhauser, 2003, pp. 57–93.

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[87] A. Dembo, J. Ding, and F. Gao, Persistence of iterated partial sums, Ann.Inst. H.Poincare 49 (2013), 873–884, http://arxiv.org/abs/1205.5596.

[88] A. Dembo and F. Gao, Persistence of iterated partial sums,http://arxiv.org/abs/1101.5743, 2011.

[89] A. Dembo, E. Mayer-Wolf, and O. Zeitouni, Exact behavior of Gaussianseminorms, Statist. & Probab. Letters (1995), 275–280.

[90] D. Denisov and V. Wachtel, Exit times for integrated random walks, Ann.Inst. H.Poincare 51 (2015), 167–193, http://arxiv.org/abs/1207.2270.

[91] S. Dereich, Small ball probabilities around random centers of Gaussian mea-sures and applications to quantization, J. Theor. Probab. 16 (2003), no. 2,427–449.

[92] S. Dereich and M.A. Lifshits, Probabilities of randomly centered small ballsand quantization in banach spaces, Ann. Probab. 33 (2005), 1397–1421,arXiv: math.pr/0402220.

[93] S. Dereich and M. Scheutzow, High-resolution quantization and entropy cod-ing for fractional Brownian motion, Electronic J. Probab. 11 (2006), 700–722, arXiv: math.pr/0504480.

[94] A. Devulder, Persistence of some additive functionals of Sinai’swalk, Ann. Inst. H. Poincare Probab. Statist. 52 (2016), 1076–1105,http://arxiv.org/abs/1402.2267.

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[101] , Estimates for the small ball probabilities of the fractional Browniansheet, J. Theor. Probab. 13 (2000), no. 2, 357–382, Erratum: ibid, vol 14(2001) 607.

[102] T. Dunker, T. Kuhn, M.A. Lifshits, and W. Linde, Metric entropy of inte-gration operators and small ball probabilities for the Brownian sheet, C.R.Acad. Sci. Paris, Ser. I 326 (1998), 347–352.

[103] , Metric entropy of integration operators and small ball probabilitiesfor the Brownian sheet, J. Approximation Theory 101 (1999), 63–77.

[104] T. Dunker, W.V. Li, and W. Linde, Small ball probabilities for integrals ofweighted Brownian motion, Statist. & Probab. Letters 46 (2000), 211–216.

[105] T. Dunker, M.A. Lifshits, and W. Linde, Small deviations of sums of in-dependent variables, High Dimensional Probability, Progress in Probability,vol. 43, Birkhauser, Basel, 1998, pp. 59–74.

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[113] , Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some Gaussian diffusion processes in the Lp-norm,2 ≤ p ≤ ∞, Problems of Information Transmission 44 (2008), no. 2, 138–155, (Russian version: Problemy Peredachi Informazii, 2008, vol. 44, no. 2,75-96.).

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[117] K. Fleischmann and V. Wachtel, Lower deviation probabilities for supercrit-ical Galton-Watson process, Ann. Inst. H. Poincare 43 (2007), 233–255.

[118] , On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Boettcher case, Ann. Inst. H. Poincare 45 (2009),201–225.

[119] O. Friedland, O. Giladi, and O. Guedon, Small ball esti-mates for quasi-norms, J. Theor. Probab. 29 (2016), 1624–1643,http://arxiv.org/abs/1410.0780.

[120] A.N. Frolov, On probabilities of small deviations for compound Cox processes,J. Math. Sci. 145 (2007), no. 2, 4931–4937.

[121] , Limit theorems for small deviation probabilities of some iteratedstochastic processes, J. Math. Sci. 188 (2013), no. 6, 761–768, Russian ver-sion: Zap. Nauchn. Semin. POMI, 2011, 396, 218-232.

[122] , Small deviations of iterated processes in the spaceof trajectories, Central Eur. J. Math. 11 (2013), 2089–2098,http://arxiv.org/abs/1208.6148.

[123] T. Fujita and S. Kotani, The Onsager-Machlup function for diffusion pro-cesses, J. Math. Kyoto Univ. 22 (1982), 131–153.

[124] F. Gao, Entropy estimate for k-monotone functions via small ball probabilityof integrated Brownian motions, Electron. J. Probab. 13 (2008), 121–130.

[125] F. Gao, J. Hannig, Lee T.-Y., and F. Torcaso, Laplace transforms viaHadamard factorization with applications to small ball probabilities, Elec-tronic J. Probab. 8 (2003), no. 13, 1–20.

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[126] , Exact L2-small balls of Gaussian processes, J.Theoret. Probab. 17(2004), no. 2, 503–520.

[127] F. Gao, J. Hannig, and F. Torcaso, Comparison theorems for small deviationsof random series, Electronic J. Probab. 8 (2003), no. 21, 1–17.

[128] , Integrated Brownian motions and exact L2-small balls, Ann. Probab.31 (2003), no. 3, 1320–1337.

[129] F. Gao and W.V. Li, Small ball probabilities for the Slepian Gaussian fields,Trans. Amer. Math. Soc. 359 (2005), no. 3, 1339–1350.

[130] , Log-level comparison for small deviation probabilities, J. Theoret.Probab. 19 (2006), no. 3, 535–556, see also J. Theoret. Probab., 2007, v.20,no 1, 1-23.

[131] F. Gao, W.V. Li, and J.A. Wellner, How many Laplace transforms of prob-ability measures are there?, Proc. Amer. Math. Soc. 138 (2010), 4331–4334,www.arxiv.org/abs/1001.0200.

[132] F. Gao, Z. Liu, and X. Yang, Conditional persistence of Gaussianrandom walks, Electronic Comm. Probab. 19 (2014), no. 70, 1–9,www.arxiv.org/abs/1001.0200.

[133] F. Gao and X. Yang, Upper tail probabilities of integrated Brow-nian motions, Sci. China Math. 58 (2015), no. 5, 1091–1100,http://arxiv.org/abs/1410.4936.

[134] V.A. Gasanenko, The probability of duration of a Wiener process for movingboundaries on a large time interval, Stochastic Systems and Their Applica-tions. Collect. Sci. Works, Kiev, 1990, (in Russian), pp. 17–23.

[135] , On certain exact relations for sojourn probabilities of a Wienerprocess, Ukr. Math. J. 51 (1999), no. 6, 942–947.

[136] , A total expansion of functional of exit time from a small ball fordiffusion process, Isatistic, J. Turkish Statist. Assoc. 3 (2000), no. 3, 83–91.

[137] , Small deviation of Brownian motion with large drift, Statist. &Probab. Letters 80 (2010), no. 21-22, 1618–1622.

[138] S. Gengembre, Probabilites de petites deviations pour les processus station-naires gaussiens, Preprint, 2003.

[139] N.L. Gorn and M.A. Lifshits, Chung’s law and Csaki function, J. Theor.Probab. 12 (1998), 399–420.

[140] P. S. Griffin, Probability estimates for the small deviations of d-dimensionalrandom walk, Ann. Probab. 11 (1983), 939–952.

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[141] P.S. Griffin, Probability estimates for the small deviations of d-dimensionalrandom walk, Ann. Probab. 11 (1983), 939–952.

[142] K. Grill, A liminf result in Strassen’s law of iterated logarithm,Probab.Theory Relat. Fields 89 (1991), 149–157.

[143] , Some remarks on Chung’s law, Period. Math. Hung. 41 (2000),no. 1-2, 167–173.

[144] J. Hoffman-Jørgensen, Bounds for the Gaussian measure of a small ball ina Hilbert space, Aarhus University Preprints, N 18, 1976.

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