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Section: New Results

Large Deviations Inequalities

Participant : Xiequan Fan.

Let (ξi)i=1,...,n be a sequence of independent and centered random variables satisfying Bernstein's condition, for a constant ε>0,

|𝔼ξik|12k!εk-2𝔼ξi2,forallk2andalli=1,...,n. (12)

Denote by

Sn=i=1nξiandσ2=i=1n𝔼ξi2. (13)

The well-known Bernstein inequality (1946) states that, for all x>0,

(Sn>xσ)infλ0𝔼eλ(Sn-xσ). (14)

In the i.i.d. case, Cramér (1938) has established a large deviation expansion under the condition 𝔼e|ξ1|<. For all 0x=on, one has

(Sn>xσ)1-Φ(x)=ex3nλxn1+O1+xn,n, (15)

where λ(·)=c1+c2xn+... is the Cramér series and the values c1,c2,... depend on the distribution of ξ1.

Bahadur-Rao (1960) proved the following sharp large deviations similar to (15 ). Assume Cramér's condition. Then, for given y>0, there is a constant cy depending on the distribution of ξ1 and y such that

Snn>y=infλ0𝔼eλ(Sn-yn)σyty2πn1+Ocyn,n, (16)

where ty, σy and cy depend on the distribution of ξ1 and y.

We present an improvement on Bernstein's inequality. In particular, we establish a sharp large deviation expansion similar to the classical results of Cramér and Bahadur-Rao. The following theorem is our main result.

Theorem 0.1 Assume Bernstein's condition. Then, for all 0x<112σε,

(Sn>xσ)=infλ0𝔼eλ(Sn-xσ)Fx,εσ, (17)

where 2πM(x) is the Mills ratio, the function

Fx,εσ=M(x)+28θR4xε/σεσ (18)

with

R(t)=(1-t+6t2)3(1-3t)3/2(1-t)7,0t<13, (19)

and |θ|1. In particular, in the i.i.d. case, for all 0x=o(n),n,

(Sn>xσ)-M(x)infλ0𝔼eλ(Sn-xσ)=O1ninfλ0𝔼eλ(Sn-xσ) (20)

and thus

(Sn>xσ)M(x)infλ0𝔼eλ(Sn-xσ)=1+o1. (21)