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(1)

### O n t h e r e m a i n d e r t e r m in the C L T

1. Introduction and results

Let {X,} be a sequence of iid random variables with distribution function F and suchthat

and

Let

and define

### a. = s~p I~[Sn <_-- Xf~]--~(X) I

Where ~b ( . ) is the distribution function of the standard normal variable.

The well-known Berry--Ess6en theorem states that if ElXlla=/~a<~, then

### A,<=Cflan -~12,

where C is a generic constant. Among the many extensions and generalizations, Ess~en (1969) proved that if

then

### A,~Con -~/~.

This result extends the Berry--Ess6en theorem by relaxing the condition of finite thir'd moment,

On the other hand, Paulauskas (1969) showed that if

= f

### txl.la(F(x)_~(x)) I <0o

then An<=Cmax (7, 7a/4)n -a/~, while later on, Zolotorov (1971), obtains an exten- sion of Paulauskas' theorem assuming the finiteness of the third absolute difference moment. Precisely, if ~ a = f

### Ixl21F(x)-~(x)ldx<oo,

then A,_~Cmax (ca,

### e~l~n-ll=

see Zolotorov (1971). Since )r Paulauskas' result follows from that of Zoloto- roy. Then the natural question to ask is whether we can extend the result of Zoloto- rov (1971) by assuming the finiteness of the truncated absolute moments. This, in fact, is the main goal of this note. Let

### (1.1) ~,=su>p{a fl,~,~ x'lF(x)-~(x)ldx+2z fl,,,>=lxllF(x)-~(x)lclx }.

Then the following result is the main theorem of this note,

(2)

166 lbrahim A. Ahmad Theorem 1.

### I f y < ~ , then

A n ~ C m a x (V, y1/4)n-l/2.

Note that since ~ < 0 , then Essden's result follows f r o m the theorem, also note that if x a < co then y < co a n d thus Z o l o t o r o v ' s result follows f r o m the theorem 9

2. P r o o f of "theorem 1 We distinguish between two cases; ? => 1 and y < 1.

By Ess6en inequality

### (see

Feller (1966), p. 512), ( 2 . 1 )

First assume that y => 1.

1 r - 1 d 24

## [ {')1"

where T > 0 , is an arbitrary constant a n d f n ( t ) = f n with

### f(t)

the charac- teristic function o f X 1. F r o m n o w on we shall denote all generic constants (which m a y be different) by C. The following lemmas are necessary for the proof.

L e m m a 2.1.

C > 0 ,

(2.2)

(2.3)

Note that

= e " x

### = (it) f ? . [e u x - 1 - itx]

[F(x) - # (x)] dx

9 t 2

### + (-it) (L~J>l,l-' x[F(x)- ~(x)] dx) (1 + Oe),

(3)

On the remainder term in the CLT 167

by application of Taylor expansion and integration by parts with 1011 and 102]< 1.

Thus (2.4)

Lemma 2.2.

(2.5)

(2.6)

Ixl>jt1-1 Ixi

051tlaT.

]t1~_(1/2C7)

### and for some constant

C > 0 , Ifft)l <=

### Ce -'=/a.

By Lemma 2.1 and the assumption we have

9 ]tla7)

### <= e-t=/2(1 + Ce 1/8~

]tlaT) = e-"/2(1 + C "7" It] a)

e - t Z / 2 e C l l t [ a r = e - ( t 2 / 2 ) O - 2 c l t l . r ) ~ e - t 2 / 4

Lemma2.3.

C>O,

(2.7)

Note that

t f t

### f ( t--~-}-e-t~/z" <-C. 7. tt[an-a/'

while Thus by Lemma2.1, ~

### )]<=l/n/2CT, f(-~nn)]<=e -'~/".

Hence using these results,

(2.8) t It[ a ,-1 4,

Now, choosing

by Lemma 2.2 for

jt z

2 n

(2.9)

### <= C. y. [tl3n-1/2e -t~/s.

in (2.1) we get using Lemma 2.3,

### A. <= (-d "-r

(4)

168 Ibrahim A. Ahmad : On the remainder term in the CLT

we get that f o r this choice

Thus f o r

### ttl~ 1/7112C7 TM

lowing modification o f L e m m a 2.3, (2.10)

Hence in (2.1) let (2.11)

N e x t assume that 7 < 1 , thus 7~14~7 a n d it follows f r o m L e m m a 2.1 that

### [ f( t ) - e-t=/2[ <_ C7~/4

It }a. I f we p u t the a s s u m p t i o n t h a t It I -<- I/2C71/4 in L e m m a 2.2 we get the fol-

a n d

a n d thus

<

### lS'f,,'--=/-e-':'=

[tl-ldt-i-C'7 n-1'2

KVBJ

< C~t/~n-l/=+

### CTn-tl~+ CTnl/~ <= C7~14n-~/~.

'Thus the t h e o r e m is completely proved.

References

1. ESSEEN, C. G. (1969). On the remainder term in the central limit theorem.

### Arkiv fgr Mat.,

8, 7--15.

2. PAULAUSKAS, V., (1969). A refinement of the Lyapounoff's theorem.

### Litov. Mat. Sb.,

9, 323-- 328.

3. ZOLOTOROV, V. M., (1971). Estimations of difference of distributions in Levy's metric.

### Steklov Math. Inst.,

112, 224--231.

Department of Mathematical Sciences University of Petroleum

and Minerals

Dharan, Saudi Arabia

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