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

Let {Y,Yi;i≥1}{Y,Yi;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 zn→∞zn→∞ (dependent on an=o(n)an=o(n)), define Sn=∑i=1nYi,Tn=an−1zn−1∑k=1nSkk,Vn=anzn−2∑i=1nYi2. Then we show that (Tn,Vn) 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 sums of i.i.d. random variables, J. Math. Anal. Appl. 334 (2007) 1246–1259].