Moderate deviations for two sample t-statistics

Let X1,. . . , Xn1 be a random sample from a population with mean µ1 and variance, and X1,. . . , Xn1 be a random sample from another population with mean µ2 and variance independent of Xi, 1 ≤ i ≤ n1. Consider the two sample t-statistic. This paper shows that ln P (T ≥ x) ~ -x²/2 for any x: = x (n1, n2) satisfying x → ∞, x = o (n1 + n2) 1/2 as n1, n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, in addition, E|X1|3 < ∞, E|Y1|3 < ∞, then holds uniformly in x ∈ (O, o ( (n1 + n2) 1/6) )