Self-normalized Cramér type moderate deviation theorem for Gaussian approximation

Berry–Esseen type bounds for Gaussian approximation of standardized sums have been extensively studied under exponential type moment conditions. In this paper, a Cramér type moderate deviation theorem is established for self-normalized Gaussian approximation under finite moment conditions. More specifically, let X1,X2,…,Xn be i.i.d. Rp-valued random vectors with zero means. Let Sn,j=∑ i=1nXij and Vn,j2=∑i=1nXij2. We show that if the correlation matrix of X1 is Ip and the third moment of X1 is finite, then P(max 1≤j≤pSn,j/Vn,j>x) P(max1≤j≤pZj>x)→1 uniformly for 0≤x≤o(n1/6) and for all p≥1, where Z1,…,Zp are independent standard normal random variables. Similar result is also established for large x when X1 has a general correlation matrix. The proof is based on a new Cramér type moderate deviation theorem for the minimum of several self-normalized sums. As an application, we propose a high dimensional one-sample t-test that allows for an exponential growth of p without requiring the commonly used sub-Gaussian assumption.