Estimations with the expectation form

It is shown in the above that the normal approximation error is related to the auxiliary functionf_h. Hence, we are interested in some bound estimations concerning the func-tion f_h. In this subsection, we shall give estimations based on the expectation form of f_h whereh is the indicator function and the call function. The method used here was presented in Stein [77]. The more general case was studied in Barbour [3] to get higher order estimations. In the next subsection, we propose a new method based on the integral form of f_h.

For any real number α let I_α be the indicator function I_α(x) = 11_{x≤α}, and let C_α = (x−α)₊ be the “Call” function. We first recall the inequality concerning the normal distribution functions, which can be found in Stein [77] and Chen and Shao [18].

Proposition 3.3.2 Denote by φσ(x) the density function and Nσ(x) the cumulative distribution function of the central normal distribution N(0, σ²), then

(1− Nσ(x)< ^σ2φ_x^σ(x), x >0

Nσ(x)< ^σ2φ_|x|^σ(x), x <0. (3.27) Proof. We first consider the case wherex > 0. Notice that φ⁰_σ(x) = −_σ^x2φ_σ(x), then direct calculation gives that

1− Nσ(x) =− Z _∞

x

σ²

t dφ_σ(t)< σ² x φ_σ(x).

The case wherex <0 is similar. 2

For technical reason, we introduce the following notation: for any function h such that hφ_σ is integrable on (−∞,−x)∪(x,+∞) for any x >0, let fe_h,σ be the function defined overR\ {0}by

fe_h,σ(x) = ( ₁

σ²φσ(x)

R_∞

x h(t)φ_σ(t)dt, x >0

−_σ²_φ¹_σ(x)Rx

−∞h(t)φ_σ(t)dt, x <0. (3.28)

We writefe_h instead offe_h,σ when there is no ambiguity. We also give the expectation form offe_h:

fe_h(x) = (^√

2π σ E

h(Z+x)e^−Zxσ211_{Z>0}

, x >0

√2π σ E

h(Z+x)e^−Zxσ211_{Z<0}

, x <0 (3.29) where Z ∼ N(0, σ²). Notice that in general, fe_h can not be extended as a contin-uous function on R. if E[|h(Z)|] < +∞, we have fe_h(0−) = −^√_σ^2πE[h(Z)11_{Z<0}] and fe_h(0+) = ^√_σ^2πE[h(Z)11_{Z>0}]. The two limits are equal if and only if Φ_σ(h) = E[h(Z)] = 0. Furthermore, if Φ_σ(h) = 0, thenfe_h coincides with the solutionf_h of the Stein’s equation.

We introduce the definition set E of fe_h as below and we only study h∈ E in the following. LetE be the set of functions g defined on R\ {0} taking values on Rsuch that g is locally of finite variation and has finite number of jump points and that g satisfies R

|g(x)|φ_σ(x)11_{|x|>a}dx < ∞ for any a > 0. It’s evident that (3.28) is well defined for anyh∈E. In fact, the above condition specifies the regularity of functions we are interested in and we exclude the “irregular” functions which are not contained inE.

Proposition 3.3.3 We have following properties of fe_h for any x∈R\ {0}: 1. f_h(x) =fe¯h(x) where ¯h=h−Φ_σ(h);

2. fe_h(−x) =−fe_h(−t)(x);

3. fe_h is one solution of the following equation

xfe_h(x)−σ²fe_h⁰(x) =h(x). (3.30) Proof. 1) and 2) are directly by definition. For 3), it is easy to verify that fe_h defined by (3.28) is one solution of the differential equation (3.30). 2

Remark 3.3.4 We call (3.30) the decentralized Stein’s equation. It is useful to intro-ducefe_h since by taking derivatives, we shall often work with non-centralized functions.

Moreover, by 1) of Proposition 3.3.3, the properties off_h can be deduced directly.

We first give some simple properties of the function fe_h.

Proposition 3.3.5 1) Let h₁ and h₂ be two functions and a₁ and a₂ be two real numbers. Then fe_a1h1+a2h2 =a1fe_h1 +a2fe_h2.

2) If|h(x)| ≤g(x), then |fe_h| ≤ |fe_g|.

3) If |h(x)| ≤g(x) and if ^g(x)_|x| is decreasing whenx >0 and is increasing when x <0, then |fe_h(x)| ≤^g(x)_x .

Proof. 1) and 2) are evident by definition.

For 3), we first study the case where x > 0. By definition of fe_h, we know that

|fe_h(x)| ≤ _σ2φ¹σ(x)

R_∞

x g(t)φσ(t)dt. Notice that φ⁰_σ(x) =−_σ^x2φσ(x), then

|fe_h(x)| ≤ − 1 φ_σ(x)

Z _∞

x

g(t)

t dφ_σ(t).

Since ^g(x)_x is decreasing, we get the inequality. Whenx <0, the proof is similar. 2

Corollary 3.3.6 We have |xfe₁(x)| ≤1.

Proof. It is a direct consequence of Proposition 3.3.5 applied to h=g= 1. 2 The following two propositions allow us to estimate the derivatives of f_h in the expectation form. The argument is based on the fact that the polynomial functions increase slower than the exponential functions.

Proposition 3.3.7 Let Z ∼N(0, σ²). Then for any non-negative integersl, m satis-fyingl≤m and for any x >0,

E

11_{Z>0}x^lZ^me^−Zxσ2

≤ 1 2

lσ² e

l

E

|Z|^m−l

. (3.31)

Where, by convention, 0⁰ = 1.

Proof. Consider the functionf(y) =y^le^−σy2, it attains the maximum value aty=lσ², then|f(y)| ≤(^lσ_e²)^l. Then the lemma follows immediately. 2

Proposition 3.3.8 Let Z∼N(0, σ²). Then for any x >0 we have xE[11_{Z>0}e^−Zxσ2]≤ σ

√2π. (3.32)

Proof. If suffices to observe thatxE[11_{Z>0}e^−Zxσ2] = √^σ

2πxfe₁(x). Then applying

Corol-lary 3.3.6 gives (3.32). 2

Remark 3.3.9 For the case x <0 in (3.31) and (3.32), we can obtain by symmetry the following inequalities for any integers 0≤l≤m:

E[11_{Z<0}|x|^l|Z|^me^−Zxσ2]≤ 1 2

lσ² e

l

E[|Z|^m−l]

and E[11_{Z<0}|x|e^−Zxσ2]≤ ^√σ_2π.

Corollary 3.3.10 For any x∈R, |fe₁(x)| ≤ ^√_2σ^2π. Proof. Since lim

x→0+fe₁(x) = √

2π/2σ, lim

x→0−fe₁(x) =−√

2π/2σ, and lim

|x|→+∞

fe₁(x) = 0, we only need to prove that fe₁ is decreasing when x >0 and whenx <0 respectively.

In fact, by Corollary 3.3.6, we have fe₁⁰(x) = _σ¹2(xfe₁(x)−1) ≤ 0 for any x > 0 and fe₁⁰(x) =−_σ¹2(1−xfe1(x))≤0 for any x <0. 2 By Proposition 3.3.5 2), we can give the upper bound of fe_h andfe_h⁰ for all bounded functionsh by Corollary 3.3.6 and Corollary 3.3.10 as below.

Proposition 3.3.11 Let h be a bounded function on R and letc₀ =khk, then 1) |fe_h(x)| ≤√

2πc₀/2σ, 2) |fe_h⁰(x)| ≤2c₀/σ²,

Proof. 1) is direct by Proposition 3.3.5 since|h(x)| ≤c0 for any x∈R.

2) By Stein’s equation, fe_h⁰(x) = _σ¹2(xfe_h(x)−h(x)). Notice that for any x∈R,

|xfe_h(x)| ≤ |c₀xfe₁(x)| ≤c₀, |h(x)| ≤c₀.

So |fe_h⁰(x)| ≤2c0/σ². 2

The indicator function satisfies the boundedness condition withc₀ = 1. So Propo-sition 3.3.11 applies directly. Let I_α(x) = 11_{x≤α}. We now give the estimation for xf_I⁰_α. For technique reasons, we consider a small lag of β where 0 ≤ β ≤ 1. Then 0≤ |I_α−β| ≤1 and we shall see that the bound is uniform onβ.

Proposition 3.3.12 For any real number β∈[0,1],

|xfe_I⁰_α−β(x)| ≤

√2π 2σe +|α|

σ². Proof. First we consider the case x >0, by definition,

fe_Iα−β(x) =

√2π

σ E[11_{Z>0}(I_α(x+Z)−β)e^−Zxσ2].

Then

fe_I⁰_α−β(x) =−

√2π

σ³ E[11_{Z>0}(I_α(x+Z)−β)Ze^−Zxσ2] + 11_{x≤α} 1 σ²e^x

2−α2 2σ2 .

Using Proposition 3.3.7 withl=m= 1 and the fact thatkIα−βk ≤1 we get

xE[11_{Z>0}(I_α(x+Z)−β)Ze^−Zxσ2]≤ σ² 2e, and

x11_{x≤α} 1 σ²e^x

2−α2 2σ2 ≤ |α|

σ². So combining the two terms, we have

|xfe_I⁰_α−β(x)| ≤

√2π 2σe +|α|

σ².

By a similar calculation forx <0, we get the proposition. 2 We can now resume the estimations of fIα by using feIα for the indicator function.

Corollary 3.3.13 Let Iα(x) = 11_{x≤α}. Then kf_Iαk ≤

√2π

2σ , kf_I⁰_αk ≤ 2 σ², and

|xf_I⁰_α| ≤

√2π 2σe +|α|

σ².

Proof. By Proposition 3.3.3, f_Iα = feI¯α where ¯I_α = I_α−P(Z ≤ α) = I_α − Nσ(α).

Since |I¯_α| ≤ 1, we can apply Proposition 3.3.11 to obtain the first two inequalities.

Proposition 3.3.12 implies the third one. 2

We have presented above a quite formal way to estimate fe_h and its derivatives and the method is easy to apply. However, the bound estimation we get is not always optimal. Stein [77] gave the following estimations for the indicator function:

0<|fIα(x)| ≤min√ 2π 4σ , 1

|x|

, |f_I⁰_α| ≤ 1 σ².

Compared to the constants obtained by Stein, those in Corollary 3.3.13 are twice larger.

In the following, we consider the functions with bounded derivatives. The increasing speed of these functions are at most linear. The call function satisfies this property.

Proposition 3.3.14 Let h be an absolutely continuous function on R.

1) Let c₁ =|h(0)| and suppose thatc₀=kh⁰k<+∞, then

|fe_h(x)| ≤

√2πc₁

2σ + 2c₀, |fe_h⁰(x)| ≤

√2πc₀ σ

1 + 1

2e

+ c₁ σ².

2) If, in addition, c2 =khk<+∞, then it suffices to prove the inequalities for x >0.

1) As|h(x)| ≤c₁+c₀|x|, we have

|fe_h(x)| ≤

√2π

σ E[11_{Z>0}(c₁+c₀Z+c₀x)e^−Zxσ2].

Then inequalities (3.31) and (3.32) yield

|fe_h(x)| ≤

2π. Taking the derivative, fe_h⁰(x) =

√2π

σ E[11_{Z>0}h⁰(Z+x)e^−Zxσ2]−

√2π

σ³ E[11_{Z>0}Zh(Z+x)e^−Zxσ2]. (3.33) Notice that the first term isfe_h0(x) withh⁰ being bounded and the second term can be estimated as above. So 2) If, in addition, h itself is bounded by c₂, we apply the above method by using (3.33) to get

|xfe_h⁰(x)| ≤c₀+

√2πc₂ 2σe .

3) By (3.33),

By the linearity of fe_h with respect toh, we know that fe_h⁰0 =fe_g⁰₁+

XN

i=1

_ife_I⁰

µi−βi. So Proposition 3.3.12 and 2) imply that

|xfe_h⁰0(x)| ≤ |xfe_g⁰₁(x)|+

The other two terms are estimated by (3.31) and (3.32) as above,

For the call function, we apply directly the above Proposition.

Corollary 3.3.15 Let C_k= (x−k)⁺, then

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