Optimality

198 V. H. de la Peña, R. Ibragimov and S. Sharakhmetov
Theorem 7.5. The following inequalities hold:
P (h(X1, . . . , Xn)>x)≤P(h(ξ1, . . . , ξn) > x) + φX1,...,Xn (P(h(ξ1, . . . , ξn) > x)) 1
2
,
P (h(X1, . . . , Xn) > x)≤ � 1 + φ 2 X1,...,Xn
� 1/2 (P(h(ξ1, . . . , ξn) > x)) 1/2 ,
P(h(X1, . . . , Xn) > x)≤(e−1)P(h(ξ1, . . . , ξn) > x) + δX1,...,Xn,
P(h(X1, . . . , Xn) > x)≤
x∈R, where ψ(x) =|x| q/(q−1) − 1.
8. Appendix: Proofs
�
1 + D ψ
� 1 (1− q
X1,...,Xn
)
(P(h(ξ1, . . . , ξn) > x)) 1
q , q > 1,
Proof of Theorem 2.1. Let us first prove the necessity part of the theorem. Denote
T(x1, . . . , xn) =
� x1
−∞
···
� xn
−∞
Let k∈{1, . . . , n}, xk∈ R. Let us show that
(8.1)
(1 + Un(t1, . . . , tn))
T(∞, . . . ,∞, xk,∞, . . . ,∞) = Fk(xk),
n�
dFi(ti).
xk∈ R, k = 1, . . . , n. It suffices to consider the case k = 1. We have
T(x1,∞, . . . ,∞)
� x1 � ∞ � ∞
= . . . (1 + Un(t1, . . . , tn))
−∞
�
−∞
��
−∞
�
n
= F1(x1) +
n�
= F1(x1) + Σ ′′ .
�
c=2 1≤i1