Moderate deviation principles for trajectories of sums of independent Banach space valued random variables

Hu, Yijun; Lee, Tzong-Yow

doi:10.1090/S0002-9947-01-02893-8

Moderate deviation principles for trajectories of sums of independent Banach space valued random variables
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by Yijun Hu and Tzong-Yow Lee PDF

Trans. Amer. Math. Soc. 355 (2003), 3047-3064 Request permission

Abstract:

Let $\{X_n\}$ be a sequence of i.i.d. random vectors with values in a separable Banach space. Moderate deviation principles for trajectories of sums of $\{X_n\}$ are proved, which generalize related results of Borovkov and Mogulskii (1980) and Deshayes and Picard (1979). As an application, functional laws of the iterated logarithm are given. The paper also contains concluding remarks, with examples, on extending results for partial sums to corresponding ones for trajectory setting.

References

A. de Acosta, Upper bounds for large deviations of dependent random vectors, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 4, 551–565. MR 791911, DOI 10.1007/BF00532666
A. de Acosta, Moderate deviations and associated Laplace approximations for sums of independent random vectors, Trans. Amer. Math. Soc. 329 (1992), no. 1, 357–375. MR 1046015, DOI 10.1090/S0002-9947-1992-1046015-4
A. de Acosta, Large deviations for vector-valued Lévy processes, Stochastic Process. Appl. 51 (1994), no. 1, 75–115. MR 1288284, DOI 10.1016/0304-4149(94)90020-5
A. de Acosta, Moderate deviations for empirical measures of Markov chains: lower bounds, Ann. Probab. 25 (1997), no. 1, 259–284. MR 1428509, DOI 10.1214/aop/1024404288
A. de Acosta and Xia Chen, Moderate deviations for empirical measures of Markov chains: upper bounds, J. Theoret. Probab. 11 (1998), no. 4, 1075–1110. MR 1660920, DOI 10.1023/A:1022673000778
Aloisio Araujo and Evarist Giné, The central limit theorem for real and Banach valued random variables, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 576407
Arcones, M.A.: The large deviation principle for stochastic processes, Preprint, 2000.
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
E. Bolthausen, Laplace approximations for sums of independent random vectors, Probab. Theory Relat. Fields 72 (1986), no. 2, 305–318. MR 836280, DOI 10.1007/BF00699109
L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
Chen, Xia: On the lower bound of the moderate deviations of i.i.d. random variables in a Banach space, Chinese Ann. Math. 11A, 621-629(1990). (in Chinese).
Chen, Xia: Probabilities of moderate deviations for $\mathbf {B}$-valued independent random vectors, Chinese J. Contemporary Math. 11, 381-393(1990).
Xia Chen, Moderate deviations of independent random vectors in a Banach space, Chinese J. Appl. Probab. Statist. 7 (1991), no. 1, 24–32 (Chinese, with English and Chinese summaries). MR 1204537
Amir Dembo and Tim Zajic, Large deviations: from empirical mean and measure to partial sums process, Stochastic Process. Appl. 57 (1995), no. 2, 191–224. MR 1334969, DOI 10.1016/0304-4149(94)00048-X
Amir Dembo and Tim Zajic, Uniform large and moderate deviations for functional empirical processes, Stochastic Process. Appl. 67 (1997), no. 2, 195–211. MR 1449831, DOI 10.1016/S0304-4149(97)00006-9
Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Jones and Bartlett Publishers, Boston, MA, 1993. MR 1202429
Deshayes, J., Picard, D.: Grandes et moyennes déviations pour le marches aleatoires, Astérisque 68, 53-71 (1979).
Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR 997938
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
V. Goodman, J. Kuelbs, and J. Zinn, Some results on the LIL in Banach space with applications to weighted empirical processes, Ann. Probab. 9 (1981), no. 5, 713–752. MR 628870, DOI 10.1214/aop/1176994305
Yi Jun Hu, Large deviations of trajectories of sums of series of a class of dependent random variables, Chinese Ann. Math. Ser. A 19 (1998), no. 2, 197–210 (Chinese, with Chinese summary); English transl., Chinese J. Contemp. Math. 19 (1998), no. 2, 137–153. MR 1651622
Jiang, Tiefeng: Moderate deviations for double arrays with applications to moving average processes, Gaussian processes and Cesaro’s sums, Preprint (1999).
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Michel Ledoux, Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 2, 267–280 (French, with English summary). MR 1162575
Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
James Lynch and Jayaram Sethuraman, Large deviations for processes with independent increments, Ann. Probab. 15 (1987), no. 2, 610–627. MR 885133
Oswald Teichmüller, Erreichbare Randpunkte, Deutsche Math. 4 (1939), 455–461 (German). MR 49
Howard Levi, On the values assumed by polynomials, Bull. Amer. Math. Soc. 45 (1939), 570–575. MR 54, DOI 10.1090/S0002-9904-1939-07038-9
Paul H. Schuette, Large deviations for trajectories of sums of independent random variables, J. Theoret. Probab. 7 (1994), no. 1, 3–45. MR 1256390, DOI 10.1007/BF02213359
S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19 (1966), 261–286. MR 203230, DOI 10.1002/cpa.3160190303
Wu, Liming: Some general methods of large deviations and applications, Preprint (1991), in $\langle \langle$Habilitation a diriger des recherches $\rangle \rangle$, Laboratoire de Probabilités de Paris VI, 1993.