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

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 → ∞ (dependent on a n = o ( n ) ), define S n = ∑ i = 1 n Y i , T n = a n − 1 z n − 1 ∑ k = 1 n S k k , V n = a n z n − 2 ∑ i = 1 n 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 sums of i.i.d. random variables, J. Math. Anal. Appl. 334 (2007) 1246–1259].