Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance ${\sigma }_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance ${\sigma }_2^2$ independent of {Xi,1 ≤ i ≤ n1}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. 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 $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.