Some Bounds for the Expectations of Functions on Order Statistics and Their Applications

Let $$X_{1, N}\geqslant X_{2, N} \geqslant \cdots \geqslant X_{N , N}$$ X 1 , N ⩾ X 2 , N ⩾ ⋯ ⩾ X N , N be the order statistics of independent identically distributed random variables $$X_k$$ X k ( $$1\leqslant k \leqslant N$$ 1 ⩽ k ⩽ N ). For fixed natural K and a nonnegative bounded deterministic function $$G_N$$ G N on $$\mathbb {R}^N$$ R N satisfying mild conditions of Lebesgue’s measurability, we obtain the following bound for the expectations: $$\begin{aligned}&\mathbb {E}G_N \big (X_{1,N},X_{2,N},\ldots ,X_{K,N}, X_{K+1,N},\ldots , X_{N,N} \big ) \\&\quad \leqslant T \cdot \mathbb {E}G_N \big (X_{1,N}^{(1)},X_{1,N}^{(2)},\ldots ,X_{1,N}^{(K)}, X _{K+1,N},\ldots , X _{N,N} \big ) +\vartheta _T \end{aligned}$$ E G N ( X 1 , N , X 2 , N , … , X K , N , X K + 1 , N , … , X N , N ) ⩽ T · E G N ( X 1 , N ( 1 ) , X 1 , N ( 2 ) , … , X 1 , N ( K ) , X K + 1 , N , … , X N , N ) + ϑ T for any $$T \geqslant T_0(K)$$ T ⩾ T 0 ( K ) and any $$N \geqslant N_0(T)$$ N ⩾ N 0 ( T ) large enough; here constants $$\vartheta _T> 0$$ ϑ T > 0 tend to zero as T approaches infinity; $$X _{1,N}^{(i)}$$ X 1 , N ( i ) ( $$1\leqslant i\leqslant K$$ 1 ⩽ i ⩽ K ) are mutually independent copies of the maximum $$X_{1,N}$$ X 1 , N ; and each $$X _{1,N}^{(i)}$$ X 1 , N ( i ) is also independent of the sample $$\{X_k\}_{1 \leqslant k\leqslant N}$$ { X k } 1 ⩽ k ⩽ N . With $$G_N$$ G N as relevant indicator functions and $$N \rightarrow \infty $$ N → ∞ , these bounds are applied to study $$\mathrm{o}$$ o - and $$\mathrm{O}$$ O -type asymptotic properties of the following functions on order statistics: (Appl-1) the numbers of observations near the Kth extremes $$X_{K,N}$$ X K , N and (Appl-2) the sums of negative powers of spacings $$X_{K,N}-X_{i,N}$$ X K , N - X i , N ( $$K+1 \leqslant i \leqslant N$$ K + 1 ⩽ i ⩽ N ).