The large deviation principle for certain series

Arcones, Miguel A.

ESAIM: Probability and Statistics, Tome 8 (2004), pp. 200-220

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.

Classification : 60F10
Keywords: large deviations, stochastic processes

[1] M.A. Arcones, The large deviation principle for stochastic processes I. Theor. Probab. Appl. 47 (2003) 567-583. | Zbl

[2] M.A. Arcones, The large deviation principle for stochastic processes. II. Theor. Probab. Appl. 48 (2004) 19-44. | Zbl

[3] J.R. Baxter and C.J. Naresh, An approximation condition for large deviations and some applications, in Convergence in ergodic theory and probability (Columbus, OH, 1993), de Gruyter, Berlin. Ohio State Univ. Math. Res. Inst. Publ. 5 (1996) 63-90. | Zbl

[4] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge, UK (1987). | Zbl | MR

[5] Y.S. Chow and H. Teicher, Probability Theory. Independence, Interchangeability, Martingales. Springer-Verlag, New York (1978). | Zbl | MR

[6] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Springer, New York (1998). | Zbl | MR

[7] J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, Inc., Boston, MA (1989). | Zbl | MR

[8] E.D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114 (1995) 303-309. | Zbl

[9] P. Hitczenko, S.J. Montgomery-Smith and K. Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables. Studia Math. 123 (1997) 15-42. | Zbl

[10] S. Kwapień and W.A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992). | Zbl | MR

[11] R. Latala, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails. Studia Math. 118 (1996) 301-304. | Zbl

[12] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, New York (1991). | Zbl | MR

[13] M. Ledoux, The Concentration of Measure Phenomenon. American Mathematical Society, Providence, Rhode Island (2001). | Zbl | MR

[14] J. Lynch and J. Sethuraman, Large deviations for processes with independent increments. Ann. Probab. 15 (1987) 610-627. | Zbl

[15] M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon. Geometric aspects of functional analysis (1989-90), Springer, Berlin. Lect. Notes Math. 1469 (1991) 94-124. | Zbl

[16] M. Talagrand, The supremum of some canonical processes. Amer. J. Math. 116 (1994) 283-325. | Zbl

[17] S.R.S. Varadhan, Asymptotic probabilities and differential equations. Comm. Pures App. Math. 19 (1966) 261-286. | Zbl