Optimality

Copulas, information, dependence and decoupling 203
4.1 and 5.7, we get that for all continuous functions fi : R→R (below, ri(x) are
polynomials corresponding to fi(x))
E
n�
fi(Xi) = E
i=1
= E
= E
The proof is complete.
n�
fi(ξi) +
i=1
n�
fi(ξi) +
i=1
n�
fi(ξi).
i=1
n�
�
c=2 1≤i1 ɛ)
≤ P(w(Um(ξt, . . . , ξt+m−1)) > (w(ɛ)∧w(−ɛ)))
≤ Ew(Um(ξt, ξt+1, . . . , ξt+m−1))/(w(ɛ)∧w(−ɛ))
= E (1 + Um(ξt, . . . , ξt+m−1))
× log(1 + Um(ξt, . . . , ξt+m−1)) /(w(ɛ)∧w(−ɛ))
= δt/(w(ɛ)∧w(−ɛ)).
If ɛ≥1, Chebyshev’s inequality and Um(ξt, . . . , ξt+m−1)≥−1 yield
(8.11) P(|Um(ξt, . . . , ξt+m−1)| > ɛ)≤ Ew(Um(ξt, . . . , ξt+m−1))
w(ɛ)
= δt/w(ɛ).
Similar to the above, by Chebyshev’s inequality and (7.3), for 0 < ɛ < 1,
P(|Um(ξt, . . . , ξt+m−1)| > ɛ)≤P(ψ(1+Um(ξt, . . . , ξt+m−1)) > (ψ(1+ɛ)∧ψ(1−ɛ))
(8.12)
≤ Eψ(1 + Um(ξt, . . . , ξt+m−1))/(ψ(1+ɛ)∧ψ(1−ɛ))
= D ψ
t /(ψ(1 + ɛ)∧ψ(1−ɛ)).