Limit Theorems for Lower-Upper Extreme Values from Two-Dimensional Distribution Function

Document Type : Original Article

Author

Dep. of Mathematics, Faculty of Science, Zagazig University, Egypt

Abstract

The limiting distribution of the random vector
(V̅_(k,k^':n) - b̅_n) / a̅_n = (X_(1,k:n) - b₁ₙ) / a₁ₙ, (X_(2,n-k^'+1:n) - b₂ₙ) / a₂ₙ
 
k and k' being constants, are investigated, necessary and sufficient conditions for waking convergence of the distribution of the above vector are obtained. The conditions under which the components of the vector (V̅_(k,k^':n) - b̅_n) / a̅_n  are asymptotically independent are also obtained. Some cases are examined when the number of observations is a random variable.

Keywords