The moderate deviation principle for self-normalized sums of sums of i.i.d. random variables

摘要

Let {Y, Y-i; i >= 1} be a sequence of nondegenerate, independent and identically distributed random variables with zero mean, which is in the domain of attraction of the normal law. For a Suitably defined sequence z(n) -> infinity (dependent on a(n) = o(n)), define S-n = Sigma(n)(i=1) Y-i, T-n = a(n)(-1) z(n)(-1) Sigma(n)(k=1) Sk/k, V-n = a(n)z(n)(-2) Sigma(n)(i=1) Y-i(2). Then we show that (T-n, V-n) satisfies the partial large deviation principle (PLDP) introduced by Dembo and Shao [A. Dembo, Q.M. Shao, Self-normalized moderate deviations and lils, Stochastic Process. Appl. 75 (1998) 51-65; A. Dembo, Q.M. Shao, Self-normalized large deviations in vector space, in: Eberlein, Hahn, Talagrand (Eds.), Proceedings of the Obervolfach meeting, High-dimensional Probability, in: Progress in probability, vol. 43, 1998, pp. 28-32]. The corresponding moderate deviation principle follows. The Central Limit theorem has been recently obtained by Pang, Lin and Hwang [T.X. Pang, Z.Y. Lin. K.S. Hwang, Asymptotics for self-normalized random products Of SLIMS of i.i.d. random variables,j. Math. Anal. Appl. 334 (2007)1246-1259].