CAUCHY APPROXIMATION FOR SUMS OF INDEPENDENT RANDOM VARIABLES

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PII. S0161171203208206 http://ijmms.hindawi.com

CAUCHY APPROXIMATION FOR SUMS OF INDEPENDENT RANDOM VARIABLES

K. NEAMMANEE Received 6 August 2002

We use Stein’s method to ﬁnd a bound for Cauchy approximation. The random variables which are considered need to be independent.

2000 Mathematics Subject Classiﬁcation: 60F05, 60G50.

1. Introduction. In Stein’s work [19], the aim was to show convergence in distribution to the normal. His technique was novel. Stein’s technique was free from Fourier methods and relied instead on the elementary diﬀerential equation

f(w)−wf (w)=h(x)−Nh (w∈R), (1.1) whereh:R→Ris such that

_∞

−∞

h(x)e⁻^(1/2)x²dx <∞ (1.2) andNh=E(h(Z)), whereZ∼N(0,1).

Stein’s method was extended from normal distribution to the Poisson distribution by Chen [9]. Stein’s equation for Poisson with parameterλis

λf (w+1)−wf (w)=h(w)−Pλh

w∈Z⁺

, (1.3)

wherePλh=E(h(Z)),Z∼Poi(λ).

Since then, Stein’s method has found considerable applications in combina- torics, probability, and statistics. Recent literature pertaining to this method includes Arratia et al. [1,2], Baldi and Rinott [3], Barbour [4,5], Barbour et al. [6], Bolthausen and Götze [7], Chen [10,11], Goldstein and Reinert [12], Goldstein and Rinott [13], Götze [14], and Green [15]; the work of Holst and Janson [16]

gives an excellent account of this method. In this paper, we further develop the Stein technique to bound errors for a Cauchy approximation to the distribution ofW, the sum of independent random variables. In fact, there are some literatures (e.g., Boonyasombut and Shapiro [8], Neammanee [17], and Shapiro [18]) give a bound of Cauchy approximation in some kind of random variables.

But they used Fourier methods.

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This paper is organized as follows. Main results are stated inSection 2. Proof of main results is inSection 3, while an example is given inSection 4.

2. Main results. At the heart of Stein’s method lies a Stein equation. For example,

f(w)−wf (w)=g(w), w∈R,

λf (w+1)−wf (w)=g(w), w∈Z⁺ (2.1) are Stein equations for normal and Poisson distribution, respectively.

LetᏴ= {h:R→R|_∞

−∞(|h(x)|/(1+x²))dx <∞}, and for eachh∈Ᏼ, Cau(h)= 1

π _∞

−∞

h(x)

1+x²dx. (2.2)

The Stein equation for Cauchy distributionF F (x)= 1

π x

−∞

1

1+t²dt (2.3)

is

f(w)−2wf (w)

1+w² =h(w)−Cau(h). (2.4) It is easy to check that a solution of (2.4) isUh:R→Rdeﬁned by

Uh(w)=

1+w²^w

−∞

h(x)−Cau(h)

1+x² dx. (2.5)

Fixw0∈R, and choosehto be the indicator functionI_(−∞,w₀]which is deﬁned by

I_(−∞,w₀](w)=





1 ifw≤w0,

0 ifw > w0. (2.6) Letfw₀=UI_(−∞,w

0]. Then, by (2.2), (2.3), and (2.5), we see that fw₀(w)=



 π

1+w² F (w)

1−F w0

ifw≤w0, π

1+w² F

w0

1−F (w)

ifw≥w0. (2.7) The broad idea of Stein’s argument is as follows. First, for any w0∈R, a functionfw₀:R→Ris constructed to solve (2.4) whenhis the indicator func- tionI(−∞,w₀]. ReplacingwbyW, for any random variableW, it therefore follows that the diﬀerence betweenP (W≤w0)andF (w0)can be expressed as

E f_w₀(W )−2W fw₀(W ) 1+W²

. (2.8)

The main results are the following.

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Theorem2.1. LetX1, X2, . . . , Xnbe independent random variables withEXi

=0,EX_i²=σ_i², andE|Xi|⁴<∞. Then, P

W≤w0

−F w0

≤3 E

1−

n i=1

σ_i²+X_i² 1+W²

2

+4πmin





 n i=1

σ_i²,2 n

n i=1

σ_i² n i=1

EXi⁴





F w0

1−F w0

+C n i=1

EXi³,

(2.9)

whenW=X1+X2+···+Xn.

Corollary2.2. LetY1, Y2, . . . , Yn be identically independent random vari- ables with zero meansEY_i²=1/2andE|Yi|⁵<∞. Let Xi=Yi/√

nandW = X1+X2+···+Xn. Then,

P W≤w0

−F

w0< C

√4

n+Cmin 1 2,

2

EY_i⁴

F w0

1−F w0

. (2.10)

Throughout this paper,Cstands for an absolute constant with possibly dif- ferent values in diﬀerent places.

3. Proof of main results. Before we prove the main results, we need the following lemmas.

Lemma3.1. For any real numbersw0andw, (1) |fw₀(w)/(1+w²)| ≤π F (w0)(1−F (w0)) (2) |f_w₀(w)| ≤3

(3) |f_w₀(w)| ≤3+2π

(4) |(f_w₀(w)/(1+w²))| ≤6+2π (5) |(wfw₀(w)/(1+w²)²)| ≤3+5π. Proof. (1) follows directly from (2.7).

(2) Before we start the proof, we need the following inequalities:

−1

π ≤wF (w)≤0 forw≤0, (3.1) 0≤w

1−F (w)

≤ 1

π forw >0. (3.2)

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To show (3.1), we deﬁne g on (−∞,0] by g(w)= wF (w). Since g(w)= 2/π (1+w²)²>0,gis increasing. From this fact and the fact that

wlim→−∞g(w)= lim

w→−∞

1 π

w

1+w²+arctanw+π 2

=0, (3.3)

we haveg≥0. Hence,gis increasing and

−1 π = lim

t→−∞g(t)≤g(w)≤g(0)=0 (3.4) for anyw≤0. So (3.1) holds. To show (3.2), we can apply the same argument to the function ˜gon[0,∞) which is deﬁned by ˜g(w)=w(1−C(w)). Since fw₀(w)=f_−w₀(−w), it suﬃces to prove the lemma in the case wherew0≥0.

By (2.7), we have f_w₀(w)=



 1−F

w0

1+2π wF (w) ifw≤0, F

w0

−1+2π w

1−F (w) ifw≥w0

≤





1+2πwF (w) ifw≤0, 1+2πw

1−F (w) ifw≥w0

≤





3 ifw≤0, 3 ifw≥w0,

(3.5)

where we have used the fact that 0≤F (w)≤1 in the ﬁrst inequality and (3.1) and (3.2) in the second inequality. In the case where 0≤w ≤ w0, by monotonicity ofF and (3.2), we see that

0≤f_w₀(w)

= 1−F

w0

+2π 1−F

w0

wF (w)

≤1+2π

1−F (w) w≤3.

(3.6)

Hence, (2) follows from (3.5) and (3.6).

(3) follows immediately from (2) and the fact that f_w

0(w)= 2w 1+w²f_w

0(w)+2 1−w²

1+w²2fw₀(w). (3.7)

(4) and (5) follow from (2) and (3) and the facts that f_w₀(w)

1+w²

=f_w₀(w)

1+w² −2wf_w₀(w) 1+w²2 , wfw₀(w)

1+w²2

=wf_w

0(w)+fw₀(w)

1+w²2 −4w²fw₀(w) 1+w²3 .

(3.8)

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Lemma3.2. Let(W ,W )be an exchangeable pair of random variables, that is,

P (W∈B,W∈B) =P (W∈B, W∈B) (3.9)

for any Borel setsBandBonR, and there existsλ >0such that

E^WW=(1−λ)W , E|W−W|²<∞, (3.10) whereE^WWis the conditional expectation of Wwith respect toW. Then,

E

2W f (W ) 1+W² −1

λ(W−W ) f (W )

1+W²− f (W ) 1+W²

=0 (3.11)

for any functionf:R→R, for which there existsC >0such that for allw∈R, f (w)≤C

1+w²

. (3.12)

Moreover,

P

W≤w0

=C w0

+E

f_w₀(W )−1

λ(W−W )

fw₀(W )

1+W² −fw₀(W ) 1+W²

(3.13) for anyw0∈R.

Proof. DeﬁneF:R²→Rby

F (w,w) =(w−w)f (w)

1+w²+ f (w) 1+w²

. (3.14)

Then,F is antisymmetric, that is,F (w,w) = −F (w, w). By Stein [20, pages 9–10], we haveEF (W ,W )=0, which implies that

0=E(W−W ) f (W )

1+W²+ f (W ) 1+W²

=E(W−W ) 2f (W )

1+W²+f (W )

1+W²− f (W ) 1+W²

=2E

E^WW−Wf (W )

1+W²+E(W−W )f (W )

1+W²− f (W ) 1+W²

= −λE

2W f (W ) 1+W²

+E(W−W ) f (W )

1+W²− f (W ) 1+W²

=E

2W f (W ) 1+W² −1

λ(W−W ) f (W )

1+W²− f (W ) 1+W²

.

(3.15)

Then, (3.11) holds and (3.13) follows from (3.11) and (2.4) whenh=I(−∞,w₀].

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Lemma3.3. Let(W ,W )be an exchangeable pair of random variables such that

E^WW=(1−λ)W , E|W−W|²<∞ (3.16) withλ >0. Then, for anyw0∈R,

P

W≤w0

=C w0

+Ef_w₀(W )

1−1

λE^W(W−W )² 1+W²

+2

λ

E(W−W )²W fw₀(W ) 1+W²2

+1 λ

_∞

−∞E(W−W )

w−W+W 2

× I

w≤W

−I(w≤W )

f_w₀(w) 1+w²

dw

−2 λ

_∞

−∞EW−W

w−W+W 2

× I

w≤W

−I(w≤W )

wfw₀(w) 1+w²2

dw.

(3.17)

Proof. Letw0∈R. ForW <W, we see that fw₀(W )

1+W² −fw₀(W )

1+W² −(W−W )f_w₀(W )

1+W² +2(W−W )W fw₀(W ) 1+W²2

= W

W

fw₀(w) 1+w²

−f_w₀(W )

1+W² +2W fw₀(W ) 1+W²2

dw

= W

W

f_w₀(w)

1+w² −2wfw₀(w)

1+w²2 −f_w₀(W )

1+W² +2W fw₀(W ) 1+W²2

dw

= W

W

w W

f_w₀(y) 1+y²

dy dw−2 W

W

w W

yfw₀(y) 1+y²2

dy dw

= W

W

W y

f_w₀(y) 1+y²

dw dy−2 W

W

W y

yfw₀(y) 1+y²2

dw dy

= W

W

(W−y)

f_w₀(y) 1+y²

dy−2

W W

(W−y)

yfw₀(y) 1+y²2

dy,

(3.18)

and by the same argument we can show that fw₀(W )

1+W² −fw₀(W )

1+W² −(W−W )f_w

0(W )

1+W² +2(W−W )W fw₀(W ) 1+W²2

= W

W(w−W )

f_w₀(w) 1+w²

dw−2 W

W(w−W )

wfw₀(w) 1+w²2

dw

(3.19)

forW < W.

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So, fw₀(W )

1+W² −fw₀(W )

1+W² −(W−W )f_w

0(W )

1+W² +2(W−W )W fw₀(W ) 1+W²2

= _∞

−∞(W−w)

I(w≤W ) −I(w≤W )

f_w₀(w) 1+w²

dw

−2 _∞

−∞(W−w)

I(w≤W )−I(w≤W )

wfw₀(w) 1+w²2

dw.

(3.20)

ByLemma 3.2, we have P

W≤w0

=C w0

+E

f_w₀(W )−1 λ

f_w₀(W )(W−W )² 1+W² +1

λ

f_w₀(W )(W−W )² 1+W² +2

λ

(W−W )²W fw₀(W ) 1+W²2 −2

λ

(W−W )²W fw₀(W ) 1+W²2

−1

λ(W−W )

fw0(W )

1+W² −fw0(W ) 1+W²

=C w0

+Ef_w

0(W )−1

λEE^Wf_w₀(W )(W−W )² 1+W² +2

λ

E(W−W )²W fw₀(W ) 1+W²2 −1

λE(W−W )

×

fw₀(W )

1+W² −fw₀(W )

1+W² −(W−W )f_w₀(W )

1+W² +2(W−W )W fw₀(W ) 1+W²2

=C(w0)+E

f_w₀(W ) 1−1

λE^W(W−W )² 1+W²

+2 λ

E(W−W )²W fw₀(W ) 1+W²2 −1

λE(W−W )

×

fw₀(W )

1+W² −fw₀(W )

1+W² −(W−W )f_w₀(W )

1+W² +2(W−W )W fw₀(W ) 1+W²2

=C w0

+E

f_w₀(W ) 1−1

λE^W(W−W )² 1+W²

+2

λ

E(W−W )²W fw₀(W ) 1+W²2

−1

λE(W−W ) _∞

−∞(W−w)

I(w≤W ) −I(w≤W )

f_w₀(w) 1+w²

dw

+2

λE(W−W ) _∞

−∞(W−w)

I(w≤W ) −I(w≤W )

wfw₀(w) 1+w²2

dw, (3.21) where we have used (3.20) in the last equality.

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For ﬁxedw, we deﬁneF:R²→Rby

F (x,x) =(x−x) x−x

2

I(w≤x) −I(w≤x) . (3.22)

Then, F is antisymmetric. Since W and W are exchangeable,EF (W ,W ) =0.

Thus,

E(W−W )(w−W )

I(w≤W ) −I(w≤W )

=E(W−W )

w−W+W

2 +W−W 2

I(w≤W ) −I(w≤W )

=E(W−W )

w−W+W 2

I(w≤W )−I(w≤W ) −EF (W ,W )

=E(W−W )

w−W+W 2

I(w≤W )−I(w≤W ) .

(3.23)

By (3.21) and (3.23), the lemma is proved.

Proof ofTheorem2.1. Let X1, X2, . . . , Xn be independent random variables andW=X1+X2+···+Xn. In order to prove the theorem, we introduce additional random variablesI, X1,X2, . . . ,Xn, andW deﬁned in the following way. The random variablesI, X1, X2, . . . , Xn, X1,X2, . . . ,Xn are independent,I is uniformly distributed over the index set{1,2, . . . , n}, eachXihas the same distribution as the correspondingXiandW=W+(XI−XI). Then,(W ,W )is an exchangeable pair. We note that

E^WW=W+E^WXI−E^WXI=W−1 n

n i=1

Xi=

1−1 n

W ,

E|W−W|²=E XI−XI²= 1 n

n i=1

E Xi−Xi²= 2 n

n i=1

σ_i².

(3.24)

Then, the assumptions ofLemma 3.3are satisﬁed withλ=1/n. Moreover, we know that

E|W−W|³=E XI−XI³= 1 n

n i=1

E Xi−Xi³≤ 8 n

n i=1

EXi³, (3.25) E|W−W|⁴=E XI−XI⁴= 1

n n i=1

E Xi−Xi⁴≤16 n

n i=1

EXi⁴. (3.26)

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To prove the theorem, letw0∈R. ByLemma 3.3, we obtain P

W≤w0

−C w0

≤sup

w∈R

f_w₀(w)E

1−nE^W(W−W )² 1+W²

+2n

E(W−W )²W fw₀(W ) 1+W²2

+nsup

w∈R

f_w₀(w) 1+w²

E

_∞

−∞|W−W|

w−W+W 2

×I(w≤W ) −I(w≤W ) dw +2nsup

w∈R

wfw₀(w) 1+w²2

E

_∞

−∞|W−W|

w−W+W 2

×I(w≤W ) −I(w≤W ) dw

≤sup

w∈R

f_w₀(w)E

1−nE^W(W−W )² 1+W²

+2n

E(W−W )²W fw₀(W ) 1+W²2

+

nsup

w∈R

f_w₀(w) 1+w²

+2nsup

w∈R

wfw₀(w) 1+w²2

E

× W∨W

W∧W |W−W|

w−W+W 2

dw

≤sup

w∈R

f_w₀(w) E

1−nE^W(W−W )² 1+W²

2

+2n

E(W−W )²W fw₀(W ) 1+W²2

+

n 2sup

f_w

0(w) 1+w²

+nsup

wfw₀(w) 1+w²2

E|W−W|³

≤3 E

1−nE^W(W−W )² 1+W²

2

+2n

E(W−W )²W fw₀(W ) 1+W²2

+6n(π+1)E|W−W|³

≤3 E

1−nE^W(W−W )² 1+W²

2

+2n

E(W−W )²W fw₀(W ) 1+W²2

+C

n i=1

EXi³,

(3.27) where the fourth inequality comes from (4) and (5) ofLemma 3.1and the last inequality comes from (3.25). SinceXi andXiare independent and have the same distribution,

E^W(W−W )²=E^W XI−XI

2

= 1 n

n i=1

Xi−Xi

2

= 1 n

_n

i=1

σ_i²+ n i=1

X²_i

. (3.28)

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Hence,

E

1−nE^W(W−W )² 1+W²

2

=E

1−nE^W(W−W )² 1+W²

2

=E

1− n i=1

σ_i²+X_i² 1+W²

2

.

(3.29)

Next, we will give a bound of 2nE(W−W )²(W fw₀(W )/(1+W²)²).

FromLemma 3.1(1),

2nE(W−W )²W fw₀(W ) 1+W²2

≤2π F w0

1−F w0

ⁿ

i=1

E Xi−Xi²

=4π F w0

1−F w0ⁿ

i=1

σ_i²,

2nE(W−W )²W fw₀(W ) 1+W²2

≤2nπ F w0

1−F w0

E|W−W|²|W|

≤2nπ F w0

1−F w0

E XI−XI⁴ EW²

=8π F w0

1−F

w0n n i=1

σ_i² n i=1

EXi⁴. (3.30)

Hence,

2nE(W−W )²W fw₀(W ) 1+W²2

≤4πmin





 n i=1

σ_i²,2 n

n i=1

σ_i² n i=1

EXi⁴



F w0

1−F w0

.

(3.31)

This completes the proof.

4. Proof ofCorollary 2.2. Using Taylor’s formula, we see that 1

1+W²=1−W²+CW³ for some|C|<1, 1

1+W²2=1−2W²+CW³ for some|C|<1.

(4.1)

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Hence, E

1 1+W²

≤1 2+ C

√n, E 1

1+W²2

≤ C

√n, E

!n i=1X_i² 1+W²

=E _n

i=1

X_i²

−E _n

i=1

X_i²

W²+C1E _n

i=1

X_i²

W³

≤1 4+ C

√n, E

!n i=1X_i² 1+W²2≤ C

√n, E !ⁿ

i=1X_i² 1+W²

2

≤C n,

(4.2)

which implies that

E

1− 1 1+W²

1 2+

n i=1

X_i² 2

=1−E 1

1+W²

−2E !ⁿ

i=1X²_i 1+W²

+1

4E 1

1+W²2

+E !n

i=1X_i² 1+W²2

+E

!ⁿ

i=1X_i² 1+W²

2

≤ C

√n.

(4.3)

Clearly, that

C n i=1

EXi³≤ C

√n, 4πmin





 n i=1

σ_i²,2 n

n i=1

σ_i² n i=1

EX_i⁴





F w0

1−F w0

≤Cmin 1 2,

2

EY_i⁴

F w0

1−F w0

.

(4.4)

Hence, by (4.3) and (4.4), the example is proved.

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K. Neammanee: Department of Mathematics, Faculty of Science, Chulalongkorn Uni- versity, Bangkok 10330, Thailand

E-mail address:[email protected]