Using the asymptotic expansion for the normal tail probability 1 (x) (x) x x and taking x > E(X), show, using

Using the asymptotic expansion for the normal tail probability 1 − Φ(x) ≃

ϕ(x)

x x → ∞

and taking x > E(X), show, using (6.14), that n

−1 log pr{Xn > x} → −K∗ (x)

as n → ∞, where X̄n is the average of n independent and identically distributed random variables. By retaining further terms in the expansion, find the rate of convergence to the entropy limit (6.5).