Moderate deviations for two sample t-statistics

Let X 1 ,...,X n 1 be a random sample from a population with mean µ 1 and variance , and X 1 ,...,X n 1 be a random sample from another population with mean µ 2 and variance independent of { X i ,1 ≤ i ≤ n 1 }. Consider the two sample t-statistic . This paper shows that ln P(T ≥ x) ~ -x ²/2 for any x := x(n 1 ,n 2 ) satisfying x → ∞, x = o(n 1 + n 2 ) 1/2 as n 1 ,n 2 → ∞ provided 0 c 1 ≤ n 1 /n 2 ≤ c 2 E|X 1 | 3 , E|Y 1 | 3 , then holds uniformly in x ∈ (O,o((n 1 + n 2 ) 1/6 ))