Cauchy approximation for sums of independent random variables

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PII. S0161171203208206 http://ijmms.hindawi.com © Hindawi Publishing Corp.

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 equa-tion

f(w)−wf (w)=h(x)−Nh (w∈R), (1.1)

whereh:R→Ris such that

_∞

−∞

_h(x)_e−(1/2)x2

dx <∞ (1.2)

andNh=E(h(Z)), whereZ∼N(0,1).

Stein’s method was extended from normal distribution to the Poisson dis-tribution 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).

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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)=_π1 _∞

−∞

h(x)

1+x2dx. (2.2)

The Stein equation for Cauchy distributionF

F (x)=_π1 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(−∞,w₀]. Then, by (2.2), (2.3), and (2.5), we see that

fw0(w)=

  

π1+w2_{F (w)}₁₋_F_w 0

ifw≤w0,

π1+w2_F_w 0

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 fw0(W )−

2W fw0(W ) 1+W2

. (2.8)

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

=0,EXi2=σi2, andE|Xi|4<∞. Then,

PW≤w0

−Fw0

≤3

E

1−

n

i=1

σ2 i+Xi2 1+W2

2

+4πmin

    

n

i=1

σ_i2,2

n

i=1

σ_i2

n

i=1

EXi 4

    F

w0

1−Fw0

+C

n

i=1

EXi3,

(2.9)

whenW=X1+X2+···+Xn.

Corollary2.2. LetY1, Y2, . . . , Yn be identically independent random vari-ables with zero meansEY2

i =1/2andE|Yi|5<∞. Let Xi=Yi/√nandW =

X1+X2+···+Xn. Then,

_P W≤w0

−Fw0<

C

4

√

n+Cmin

1 2,

2

EYi4

Fw0

1−Fw0

.

(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+w

2₎_{| ≤}_{π F (w}

0)(1−F (w0)) (2) |f_w0 (w)| ≤3

(3) |fw0(w)| ≤3+2π (4) |(fw0(w)/(1+w

2₎₎_{| ≤}₆₊₂_π (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:

−_π1 ≤wF (w)≤0 forw≤0, (3.1)

(4)

To show (3.1), we deﬁne g on (−∞,0] by g(w)= wF (w). Since g(w)=

2/π (1+w2₎2_>_0,_g_{is increasing. From this fact and the fact that}

lim w→−∞g

_(w)₌ _lim w→−∞

1

π

_w

1+w2+arctanw+

π

2

=0, (3.3)

we haveg≥0. Hence,gis increasing and

−_π1 = 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

fw0(w)=

  

1−Fw0

1+2π wF (w) ifw≤0, _F

w0

−1+2π w1−F (w) ifw≥w0

≤   

1+2πwF (w) ifw≤0,

1+2πw1−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≤fw0(w)

=1−Fw0

+2π1−Fw0

wF (w) ≤1+2π1−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 w0(w)+

21−w2

1+w22fw0(w). (3.7)

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

fw0(w)

1+w2

=fw0(w) 1+w2 −

2wfw0(w)

1+w22 ,

wfw0(w)

1+w22

=wfw0(w)+fw0(w)

1+w22 − 4w2_f

w0(w)

1+w23 .

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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

EW_W₌₍₁₋_{λ)W ,} _E_|_W₋_W_|2_<_∞_, _(3.10)

whereEW_W_{is the conditional expectation of} _W_{with respect to}_W_{. Then,}

E

₂_{W 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)≤C1+w2_. _(3.12)

Moreover,

PW≤w0=Cw0+E

fw0(W )− 1

λ(W−W ) _f

w0(W ) 1+W2 −

fw0(W ) 1+W2

(3.13)

for anyw0∈R.

Proof. DeﬁneF:R2_→_R_by

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

=2EEWW−Wf (W )

1+W2+E(W−W )

_{f (}_{W )}

1+W2−

f (W )

1+W2

= −λE

₂_{W f (W )}

1+W2

+E(W−W ) _{f (}_{W )}

1+W2−

f (W )

1+W2

=E

₂_{W f (W )}

1+W2 − 1

λ(W−W ) _{f (}_{W )}

1+W2−

f (W )

1+W2

.

(3.15)

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

EW_W₌₍₁₋_{λ)W ,} _E_|_W₋_W_|2_<_∞ _(3.16)

withλ >0. Then, for anyw0∈R,

PW≤w0

=Cw0

+Efw0(W )

1−1_λEW(W−W )2 1+W2

+2_λE(W−W )

2_{W f} w0(W )

1+W22

+_λ1 ∞

−∞E(W−W )

w−W+₂W

×Iw≤W−I(w≤W )

fw0(w) 1+w2

dw

−_λ2 _∞

−∞EW−W

w−W+₂W

×Iw≤W−I(w≤W )

wfw0(w)

1+w22

dw.

(3.17)

Proof. Letw0∈R. ForW <W, we see that

fw0(W ) 1+W2 −

(W−W )fw0(W ) 1+W2 +

2(W−W )W fw0(W )

1+W22

= W

W

_f

w0(w) 1+w2

−fw0(W ) 1+W2 +

2W fw0(W )

1+W22

dw = W W fw0(w)

1+w2 −

2wfw0(w)

1+w22 −

fw0(W ) 1+W2 +

2W fw0(W )

1+W22

dw = W W w W _f w0(y) 1+y2

dy dw−2

W

w

W

yfw0(y)

1+y22

dy dw = W W W y _f w0(y) 1+y2

dw dy−2

W

y

yfw0(y)

1+y22

dw dy

= W

W

(W−y) _f

w0(y) 1+y2

dy−2

W

(W−y)

yfw0(y)

1+y22

dy,

(3.18)

and by the same argument we can show that

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)

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So,

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 )

fw0(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

PW≤w0

=Cw0

+E

fw0(W )− 1

λ

fw0(W )(W−W ) 2

1+W2 + 1

λ

fw0(W )(W−W ) 2

1+W2

+2_λ(W−W )

2_{W f} w0(W )

1+W22 − 2

λ

(W−W )2_{W f} w0(W )

1+W22

−1

λ(W−W ) _f

w0(W ) 1+W2 −

fw0(W ) 1+W2

=Cw0+Efw0(W )− 1

λEE

Wfw0(W )(W−W ) 2

1+W2

+2 λ

E(W−W )2_{W f} w0(W )

1+W22 − 1

λE(W−W )

×

fw0(W ) 1+W2 −

(W−W )fw0(W ) 1+W2 +

2(W−W )W fw0(W )

1+W22

=C(w0)+E

fw0(W ) 1− 1

λE

W(W−W )2 1+W2

+2_λE(W−W )

2_{W f} w0(W )

1+W22 − 1

λE(W−W )

×

fw0(W ) 1+W2 −

(W−W )fw0(W ) 1+W2 +

2(W−W )W fw0(W )

1+W22

=Cw0

+E

fw0(W ) 1− 1

λE

W(W−W )2 1+W2

+2_λE(W−W )

2_{W f} w0(W )

1+W22

−1_λE(W−W ) ∞

−∞(W−w)

I(w≤W ) −I(w≤W )

fw0(w) 1+w2

dw

+2_λE(W−W ) ∞

−∞(W−w)

I(w≤W ) −I(w≤W )

wfw0(w)

1+w22

dw,

(3.21)

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For ﬁxedw, we deﬁneF:R2_→_R_by

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+₂W+W−₂WI(w≤W ) −I(w≤W )

=E(W−W )

w−W+₂WI(w≤W )−I(w≤W ) −EF (W ,W )

=E(W−W )

w−W+₂WI(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 vari-ables 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

EWW=W+EWXI−EWXI=W− 1

n

i=1

Xi=

1−_n1

W ,

E|W−W|2=E XI−XI2= 1

n

i=1

E Xi−Xi2= 2

n

i=1

σi2.

(3.24)

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

E|W−W|3₌E _X

I−XI3= 1

n

i=1

E Xi−Xi3≤ 8

n

i=1

EXi3, (3.25)

E|W−W|4₌E _X

I−XI4= 1

n

i=1

E Xi−Xi4≤ 16

n

i=1

(9)

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

PW≤w0

−Cw0

≤sup w∈R

fw0(w)E

1−nEW

(W−W )2 1+W2

+2n

E(W−W )2_{W f} w0(W )

1+W22

+nsup w∈R

fw0(w) 1+w2

E

∞

−∞|W−W|

w−W+₂W

×I(w≤W ) −I(w≤W ) dw

+2nsup w∈R

wfw0(w)

1+w22

E

∞

−∞|W−W|

w−W+₂W

×I(w≤W ) −I(w≤W ) dw

≤sup w∈R

_f w0(w)E

1−nEW(W−W )

2

1+W2

+2nE(W−W )

2_{W f} w0(W )

1+W22

+ nsup w∈R

fw0(w) 1+w2

+2nwsup∈R

wfw0(w)

1+w22

E × W∨W

W∧W |

W−W|w−W+W

2

dw

≤sup w∈R

fw0(w)

E

1−nEW(W−W )2 1+W2

2

+2nE(W−W )

2_{W f} w0(W )

1+W22

+ n 2sup f_w 0(w) 1+w2

+nsup

wfw0(w)

1+w22

E|W−W|3

≤3 E

1−nEW(W−W )2 1+W2

2

+2nE(W−W )

2_{W f} w0(W )

1+W22

+6n(π+1)E|W−W|3

≤3 E

1−nEW(W−W )2 1+W2

2

+2nE(W−W )

2_{W f} w0(W )

1+W22

+C n i=1

EXi 3

,

(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,

EW₍_W₋_{W )}2₌_EW _X I−XI

2

= 1 n

n

i=1

Xi−Xi

2

= 1 n

n

i=1

σ2 i +

n

i=1

X2 i

(10)

Hence,

E

1−nEW(W−W )2 1+W2

2

=E

1−nEW(W−W )2 1+W2

2

=E

1−

n

i=1

σ2 i +Xi2 1+W2

2

.

(3.29)

Next, we will give a bound of 2nE(W−W )2_{(W f}

w0(W )/(1+W2)2). FromLemma 3.1(1),

2nE(W−W )2

W fw0(W )

1+W22

≤2π Fw0

1−Fw0

n

i=1

E Xi−Xi 2

=4π Fw01−Fw0 n

i=1

σ2 i,

2nE(W−W )2

W fw0(W )

1+W22

≤2nπ Fw0

1−Fw0

E|W−W|2_|_W_|

≤2nπ Fw0

1−Fw0

_E

XI−XI 4

EW2

=8π Fw01−Fw0

n

i=1

σ2 i

n

i=1

EXi4.

(3.30)

Hence,

2nE(W−W )2

W fw0(W )

1+W22

≤4πmin

    

n

i=1

σi2,2

_nn

i=1

σi2 n

i=1

EXi4

    F

w01−Fw0.

(3.31)

This completes the proof.

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

1

1+W2=1−W

2₊_CW3 _{for some}_|_C_|_<₁_,

1

1+W22=1−2W

(11)

Hence,

E

₁

1+W2

≤1₂+√C n, E

1

1+W22

≤√C

n,

E !n

i=1Xi2 1+W2

=E

n

i=1

X2 i

−E

n

i=1

X2 i

W2₊_C

1E

n

i=1

X2 i

W3

≤1₄+√C n,

E !n

i=1Xi2

1+W22≤

C √

n, E

!n i=1Xi2 1+W2

2

≤C n,

(4.2)

which implies that

E

1−₁₊1_W2

1 2+

n

i=1

Xi2

2

=1−E

₁

1+W2

−2E

!n i=1X2i 1+W2

+1₄E

1

1+W22

+E !n

i=1Xi2

1+W22

+E

!n i=1Xi2 1+W2

2

≤√C n.

(4.3)

Clearly, that

C

n

i=1

EXi 3

≤√C n,

4πmin

    

n

i=1

σ_i2,2

n

i=1

σ_i2

n

i=1

EX_i4     F

w0

1−Fw0

≤Cmin 1 2,

2

EY4

i

Fw0

1−Fw0

.

(4.4)

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

References

[1] R. Arratia, L. Goldstein, and L. Gordon,Two moments suﬃce for Poisson approx-imations: the Chen-Stein method, Ann. Probab.17(1989), no. 1, 9–25. [2] ,Poisson approximation and the Chen-Stein method, Statist. Sci.5(1990),

no. 4, 403–434.

[3] P. Baldi and Y. Rinott,On normal approximations of distributions in terms of dependency graphs, Ann. Probab.17(1989), no. 4, 1646–1650.

[4] A. D. Barbour,Stein’s method and Poisson process convergence, J. Appl. Probab.

25A(1988), 175–184.

(12)

[6] A. D. Barbour, L. H. Y. Chen, and W.-L. Loh,Compound Poisson approximation for nonnegative random variables via Stein’s method, Ann. Probab.20(1992), no. 4, 1843–1866.

[7] E. Bolthausen and F. Götze,The rate of convergence for multivariate sampling statistics, Ann. Statist.21(1993), no. 4, 1692–1710.

[8] V. Boonyasombut and J. M. Shapiro,The accuracy of inﬁnitely divisible approxi-mations to sums of independent variables with application to stable laws, Ann. Math. Statist.41(1970), 237–250.

[9] L. H. Y. Chen,Poisson approximation for dependent trials, Ann. Probab.3(1975), no. 3, 534–545.

[10] ,Stein’s method: some perspectives with applications, Probability Towards 2000 (New York, 1995) (L. Accardi and C. C. Heyde, eds.), Lecture Notes in Statist., vol. 128, Springer, New York, 1998, pp. 97–122.

[11] ,Non-uniform bounds in probability approximations using Stein’s method, Probability and Statistical Model with Applications: A Volume in Honor of Thephilos Cacoullous (Ch. A. Charalambides, M. V. Koutras, and N. Bal-akrisshnan, eds.), Chapman&Hall/CRC Press, Florida, 2000, pp. 3–14. [12] L. Goldstein and G. Reinert,Stein’s method and the zero bias transformation with

application to simple random sampling, Ann. Appl. Probab.7(1997), no. 4, 935–952.

[13] L. Goldstein and Y. Rinott,Multivariate normal approximations by Stein’s method and size bias couplings, J. Appl. Probab.33(1996), no. 1, 1–17.

[14] F. Götze,On the rate of convergence in the multivariate CLT, Ann. Probab.19

(1991), no. 2, 724–739.

[15] T. A. Green,Asymptotic enumeration of generalized Latin rectangles, J. Combin. Theory Ser. A51(1989), no. 2, 149–160.

[16] L. Holst and S. Janson, Poisson approximation using the Stein-Chen method and coupling: number of exceedances of Gaussian random variables, Ann. Probab.18(1990), no. 2, 713–723.

[17] K. Neammanee,On the rate of convergence of distribution functions of sums of reciprocals of logarithm of random variables to the Cauchy distribution, to appear.

[18] J. M. Shapiro,On the rate of convergence of distribution functions of sums of re-ciprocals of random variables to the Cauchy distribution, Houston J. Math.

4(1978), no. 3, 439–445.

[19] C. Stein,A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory (California), Univ. California Press, 1972, pp. 583–602.

[20] ,Approximate Computation of Expectations, Institute of Mathematical Sta-tistics Lecture Notes—Monograph Series, vol. 7, Institute of Mathematical Statistics, California, 1986.

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