The small ball asymptotics in Hilbertian norm for the Kac–Kiefer–Wolfowitz processes

A. I. Nazarov   ; Yu. P. Petrova

Teoriâ veroâtnostej i ee primeneniâ, Tome 60 (2015) no. 3, pp. 482-505

Bibliographie

[1] Li W. V., Shao Q. M., “Gaussian processes: inequalities, small ball probabilities and applications”, Stochastic Processes: Theory and Methods, Amsterdam, North-Holland, 2001, 533–597 | MR | Zbl

[2] Lifshits M. A., “Asymptotic behavior of small ball probabilities”, Probab. Theory and Math. Statist., Proc. VII International Vilnius Conference (1998), VSP/TEV, Vilnius, 1999, 453–468

[3] Lifshits M. A., Bibliography on small deviation probabilities, 2014

[4] Sytaya G. N., “O nekotorykh asimptoticheskikh predstavleniyakh gaussovskoi mery v gilbertovom prostranstve”, Teoriya sluchainykh protsessov, 2 (1974), 93–104

[5] Zolotarev V. M., “Gaussian measure asymptotic in $L_2$ on a set of centered spheres with radii tending to zero”, 12th Europ. Meeting of Statisticians (Varna, 1979), 254

[6] Hoffmann–Jœrgensen J., Shepp L. A., Dudley R. M., “On the lower tail of Gaussian seminorms”, Ann. Probab., 7:2 (1979), 310–342 | MR

[7] Ibragimov I. A., “O veroyatnosti popadaniya gaussova vektora so znacheniyami v gilbertovom prostranstve v sferu malogo radiusa”, Zap. nauchn. sem. POMI, 1979, 85, 75–93 | MR | Zbl

[8] Li. W. V., “Comparison results for the lower tail of Gaussian seminorms”, J. Theoret. Probab., 5:1 (1992), 1–31 | DOI | MR

[9] Dunker T., Lifshits M. A., Linde W., “Small deviation probabilities of sums of independent random variables”, Progr. Probab., 43 (1998), 59–74 | MR | Zbl

[10] Gao F., Hanning J., Torcaso F., “Comparison theorems for small deviations of random series”, Electron. J. Probab., 8:21 (2003), 1–17 | MR

[11] Nazarov A. I., Nikitin Ya. Y., “Exact $L_2$-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems”, Prob. Theory Related Fields, 129:4 (2004), 469–494 | DOI | MR | Zbl

[12] Nazarov A. I., “Exact small ball asymptotics of Gaussian processes and the spectrum of boundary value problems”, J. Theor. Probab., 22:3 (2009), 640–665 | DOI | MR | Zbl

[13] Birkhoff G., “On the asymptotic character of the solutions of certain linear differential equations containing a parameter”, Trans. Amer. Math. Soc., 9:2 (1908), 219–231 | DOI | MR

[14] Birkhoff G., “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Amer. Math. Soc., 9:4 (1908), 373–395 | DOI | MR | Zbl

[15] Tamarkin J., “Sur quelques points de la theorie des equations defferentielles lineaires ordinaires et sur la generalisation de la serie de Fourier”, Rendiconti del Circolo Matematico di Palermo, 34:1 (1912), 345–382 | DOI | Zbl

[16] Tamarkin J., “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions”, Math.Ż., 27:1 (1928), 1–54 | DOI | MR

[17] Shkalikov A. A., “Kraevye zadachi dlya obyknovennykh diffenetsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Tr. seminara im. I. G. Petrovskogo, 9, Izd-vo Mosk. un-ta, M., 1983, 190–229 | MR

[18] Nazarov A. I., Nelineinye zadachi v teorii funktsii, Problemy matematicheskogo analiza, 26, Pod red. N. N. Uraltsevoi, Tamara Rozhkovskaya, Novosibirsk

[19] Nazarov A. I., Pusev R. S., “Tochnaya asimptotika malykh uklonenii v $L_2$-norme s vesom dlya nekotorykh gaussovskikh protsessov”, Zap. nauchn. sem. POMI, 364 (2009), 166–199 | MR | Zbl

[20] Nazarov A. I., Pusev R. S., “Teoremy sravneniya dlya veroyatnostei malykh uklonenii vesovykh $L_2$-norm grinovskikh gaussovskikh protsessov”, Algebra i analiz, 25:3 (2013), 131–146 | MR

[21] Nikitin Ya. Yu., Kharinskii P. A., “Tochnaya asimptotika malykh uklonenii v $L_2$-norme dlya odnogo klassa gaussovskikh protsessov”, Zap. nauchn. sem. POMI, 311 (2004), 214–221 | MR | Zbl

[22] Deheuvels P., Martynov G., “Karhunen–Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions”, High Dimensional Probability, III (Sandjberg, 2002), 55, Birkhäuser, Basel, 2003, 57–93 | DOI | MR | Zbl

[23] Gao F., Hanning J., Lee T.-Y., Torcaso F., “Laplace transforms via Hadamard factorization”, Electron. J. Probab., 8:13 (2003), 1–20 | MR

[24] Nazarov A. I., “Ob odnom semeistve preobrazovanii gaussovskikh sluchainykh funktsii”, Teoriya veroyatn. i ee primen., 54:2 (2009), 209–225 | DOI | MR | Zbl

[25] Vladimirov A. A., Sheipak I. A., “O zadache Neimana dlya uravneniya Shturma–Liuvillya s samopodobnym vesom kantorovskogo tipa”, Funktsionalnyi analiz i ego prilozheniya, 47:4 (2013), 18–29 | DOI | MR | Zbl

[26] Kac M., Kiefer J., Wolfowitz J., “On tests of normality and other tests of goodness of fit based on distance methods”, Ann. Math. Stattist., 26:2 (1955), 189–211 | DOI | MR | Zbl

[27] Gikhman I. I., “O nekotorykh predelnykh teoremakh dlya uslovnykh raspredelenii i o svyazannykh s nimi zadachakh matematicheskoi statistiki”, Ukr. matem. zhurn., 5:4 (1953) | Zbl

[28] Gikhman I. I., “Protsessy Markova v zadachakh matematicheskoi statistiki”, Ukr. matem. zhurn., 6 (1954), 28–36

[29] Durbin J., “Weak convergence of the sample distribution function when parameters are estimated”, Ann. Statist., 1:2 (1973), 279–290 | DOI | MR | Zbl

[30] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Uchebnoe posobie, Izd-vo Leningradskogo un-ta, Leningrad, 1980 | MR

[31] Seneta E., Pravilno menyayuschiesya funktsii, Per. s angl., Nauka, M., 1985 | MR

[32] Blair J. M., Edwards C. A., Johnson J. H., “Rational Chebychev approximations for the inverse of the error function”, Math. Comput., 30:136 (1976), 827–830 | DOI | MR | Zbl