Asymptotic behaviour of near-maxima of Gaussian sequences

Let $$(X_1,X_2,\ldots ,X_n)$$ ( X 1 , X 2 , … , X n ) be a Gaussian random vector with a common correlation coefficient $$\rho _n,\,0\le \rho _n>1$$ ρ n , 0 ≤ ρ n > 1 , and let $$M_n= \max (X_1,\ldots , X_n),\,n\ge 1$$ M n = max ( X 1 , … , X n ) , n ≥ 1 . For any given $$a>0$$ a > 0 , define $$T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)$$ T n ( a ) = j , 1 ≤ j ≤ n , X j ∈ ( M n - a , M n ] , K n ( a ) = # T n ( a ) and $$S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1$$ S n ( a ) = ∑ j ∈ T n ( a ) X j , n ≥ 1 . In this paper, we obtain the limit distributions of $$(K_n(a))$$ ( K n ( a ) ) and $$(S_n(a))$$ ( S n ( a ) ) , under the assumption that $$\rho _n\rightarrow \rho $$ ρ n → ρ as $$n\rightarrow \infty ,$$ n → ∞ , for some $$\rho \in [0,1)$$ ρ ∈ [ 0 , 1 ) . Copyright Springer-Verlag Berlin Heidelberg 2014