Precise asymptotics in the deviation probability series of self-normalized sums

Let X,X1,X2,…X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means. Set Sn=X1+⋯+XnSn=X1+⋯+Xn and Wn2=X12+⋯+Xn2. In the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self-normalized sums, Sn/WnSn/Wn. For positive functions g(x)g(x), ϕ(x)ϕ(x), α(x)α(x) and κ(x)κ(x), we obtain the precise asymptotics for the following deviation probabilities of self-normalized sums:α(ϵ)∑n=1∞g(ϕ(n))ϕ′(n)E[(Sn/Wn)2I(|Sn|⩾Wn(ϵϕ(n)+κ(n)))]. The results given can be considered as the generalization of that in the complete moment convergence, law of iterated logarithm and large deviation for self-normalized sums.