RUS  ENG
Полная версия
ЖУРНАЛЫ // Записки научных семинаров ПОМИ

Зап. научн. сем. ПОМИ, 2003, том 298, страницы 5–21 (Mi znsl1148)

Exact small ball constants for some Gaussian processes under $L^2$-norm
L. Beghin, Ya. Yu. Nikitin, E. Orsingher

Литература

1. L. Beghin, Ya. Nikitin, E. Orsingher, Exact small ball constants for some Gaussian processes under the $L^2-$norm, Technical Report № 4. Dept. of Statistics, Probability and Applied Statistics, Rome University “La Sapienza”, January 2002
2. X. Chen, W. V. Li, “Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion”, Ann.Probab., 31 (2003), 1052–1077  crossref  mathscinet  zmath
3. C. Donati-Martin, M. Yor, “Fubini's theorem for double Wiener integrals and the variance of the Brownian path”, Ann. Inst. H. Poincaré, 27 (1991), 181–200  mathscinet  zmath
4. R. M. Dudley, “On the lower tails of Gaussian seminorms”, Ann. Prob., 7:2 (1979), 319–342  crossref  mathscinet  zmath
5. T. Dunker, M. A. Lifshits, W. Linde, “Small deviations of sums of independent variables”, Proc. Conf. High Dimensional Probability, Ser. Progress in Probability, 43, Birkhäuser, 1998, 59–74  mathscinet  zmath
6. F. Gao, J. Hannig, T.-Y. Lee, F. Torcaso, “Laplace transforms via Hadamard Factorization with applications to small Ball probabilities”, Electronic J. of Probability, 8 (2003), 13  mathscinet  zmath
7. F. Gao, J. Hannig, T.-Y. Lee, F. Torcaso, “Exact $L^2$ small balls of Gaussian processes”, J. Theor.Probab. (to appear)
8. F. Gao, J. Hannig, F. Torcaso, “Integrated Brownian motions and exact $L_2$-small balls”, Ann.Prob., 31:3 (2003), 1320–1337  crossref  mathscinet  zmath
9. I. S. Gradshteyn, I. M. Ryzhik, Tables of integrals, sums, series and products, 5th ed., Nauka, Moscow, 1971
10. N. Henze, Ya. Yu. Nikitin, “A new approach to goodness-of-fit testing based on the integrated empirical process”, Journ. Nonpar. Statist., 12 (2000), 391–416  crossref  mathscinet  zmath
11. N. Henze, Ya. Yu. Nikitin, “Watson-type goodness-of-fit tests based on the integrated empirical process”, Math. Meth. Stat., 11 (2002), 183–202  mathscinet  zmath
12. I. A. Ibragimov, “On the probability that a Gaussian vector with values in a Hilbert space hits a sphere of small radius”, J. Sov. Math., 20 (1982), 2164–2174  crossref
13. M. Kac, “On deviations between theoretical and empirical distribution”, Proc. Nat. Acad. Sci. USA, 35 (1949), 252–257  crossref  mathscinet  zmath  adsnasa
14. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, 7-nd ed., Akademische Verlagsges. Geest and Portig, 1961
15. D. Khoshnevisan, Z. Shi, “Chung's law for integrated Brownian motion”, Trans. Amer. Math. Soc., 350:10 (1998), 4253–4264  crossref  mathscinet  zmath
16. A. Lachal, “Study of some new integrated statistics: computation of Bahadur efficiency, relation with non-standard boundary value problems”, Mathematical Methods of Statistics, 10:1 (2001), 73–104  mathscinet  zmath
17. W. V. Li, “Comparison results for the lower tail of Gaussian seminorms”, Journ. Theor. Prob., 5:1 (1992), 1–31  crossref  mathscinet
18. W. V. Li, Q. M. Shao, “Gaussian processes: Inequalities, Small Ball Probabilities and Applications”, Stochastic Processes: Theory and Methods, Handbook of Statistics, 19, eds. C. R. Rao, D. Shanbhag, 2001, 533–597  mathscinet  zmath
19. M. A. Lifshits, Gaussian Random Functions, Kluwer, 1995  mathscinet  zmath
20. M. A. Lifshits, “Asymptotic behavior of small ball probabilities”, Prob. Theory and Math. Stat., eds. B.Grigelionis et al., VSP/TEV, 1999, 453–468
21. A. I. Nazarov, Ya. Yu. Nikitin, Exact small ball behavior of integrated Gaussian processes under $L_2-$norm and spectral asymptotics of boundary value problems, Preprint. Studi Statistici № 70. Istituto di Metodi Quantitativi, Universita Bocconi, Milan, Febbraio, 2003
22. A. I. Nazarov, “On the sharp constant in the small ball asymptotics of some Gaussian processes under $L_2$-norm”, Nonlinear equations and mathematical analysis, Problems of Math. Anal., 26, T. Rozhkovskaya, Novosibirsk, 2003, 179–214  mathscinet
23. G. R. Shorack, J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York, 1986  mathscinet
24. G. N. Sytaya, “On some asymptotic representations of the Gaussian measure in a Hilbert space”, Theory of Stochastic Processes, 2, 1974, 93–104
25. G. S. Watson, “Goodness-of-fit tests on a circle”, Biometrika, 48 (1961), 109–114  mathscinet  zmath
26. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1996  mathscinet  zmath
27. V. M. Zolotarev, “Asymptotic behavior of Gaussian measure in $l_2$”, Journ. Soviet Math., 24 (1986), 2330–2334  crossref


© МИАН, 2025