• Aucun résultat trouvé

~,=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,


Academic year: 2022

Partager "~,=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,"

En savoir plus ( Page)

Texte intégral


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

Ibrahim A. Ahmad

1. Introduction and results

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





S , = z ~ = I X k,

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

o = :up { f,x,=, xadF(x) + zf,.,>= x=dF(x)} < ~'"


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


then A,_~Cmax (ca,


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,


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


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

First assume that y => 1.

1 r - 1 d 24

& < = - g f _ r l ] , ( t ) - e - ' V = l t t + - ~ T ,

[ {')1"

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


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.

For all real t and

C > 0 ,

I f ( t ) - e - m 2 l <= CIt}~7.




Note that

f(t) -- e - ta/2

= e " x

= f ~ . [eUX- l - - i t x - ~ ] d[F(x)-g~(x,]

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

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

+ f jx>,,t_l[- itx-O,.itxltF(x)- (x)l dx}

9 t 2

= Ot) ~fl~,~l,l_,x'[F(x)-~(x)]dx(1-01)

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


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.




If(t)-e-"/21 <= OaltI 8 (3 f ix,~_l,t_lx2iF(x)-~(x)l dx +04t2(2 f

Ixl>jt1-1 Ixi

IF(x)-~(x)l dx) <=


Let 7<00. Then for all


and for some constant

C > 0 , Ifft)l <=

Ce -'=/a.

By Lemma 2.1 and the assumption we have

If(O[ <= If(t)-e-"/~t +e -''/~

"<= CltlaT+e -t*/2 = e-t'/2(l +e"/2C

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


For some


and all Itl<=l/n/2CT, we have



Note that

f" (-~-~)--e -''/~ <= C7[t[an-1/2e-,'/8.

t f t

.~j=of e ,'s12, .

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,

- e - <= C . 7 . 7 ~ Zs=oe

(Itl -,,t"-q

~ C , 7 . [ ~ n ) Zj=o e t ,, )

Now, choosing


by Lemma 2.2 for

jt z

2 n


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

in (2.1) we get using Lemma 2.3,

C. 7 r r t2 e-"/s dt + C. 7/t/~ <= C. 7" n-i/~

A. <= (-d "-r


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

we get that f o r this choice

[f(t)l<-_e -t=/~.

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-

[f" (-~) ,e-t~/21<= @l/4it [an-1/~ e -t~js.


a n d


a n d thus

A, <= ~1 --e-'=12 ]tl-t dt




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

rl t~=rol


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

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

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


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.

Received 79----06--09

Ibrahim A. Ahmad

Department of Mathematical Sciences University of Petroleum

and Minerals

Dharan, Saudi Arabia


Documents relatifs

The longterm goal of this project is to move away from the current model of storing similar and/or duplicate network information in multiple databases and to move to a

Let us recall that a space is locally compact if each neighbourhood of a point x contains a compact neighbourhood. (Notice that the product of two locally compact spaces is

For Hosts with word lengths 16, 32, 48, etc., bits long, this string is necessarily in the last word received from the Imp.. For Hosts with word lengths which are not a multiple of

Definition 3.31 (Test preorder). In what follows, we use the notions of lift and weight of a path, introduced in Sect. The first part is obvious. Hence, φ µ is not observable, and

When it is asked to describe an algorithm, it has to be clearly and carefully done: input, output, initialization, loops, conditions, tests, etc.. Exercise 1 [ Solving

If W is a non-affine irreducible finitely generated Coxeter group, then W embeds as a discrete, C-Zariski dense subgroup of a complex simple Lie group with trivial centre, namely

Our goal is to reprove the key results in Davenport’s Multiplicative Number Theory and Bombieri’s Le Grand Crible using only “pretentious methods.” The past semester I taught

Note: This examination consists of ten questions on one page. a) Considering two arbitrary vectors a and b in three-dimensional Cartesian space, how can the angle between

No calculators are to be used on this exam. Note: This examination consists of ten questions on one page. Give the first three terms only. a) What is an orthogonal matrix?

These equations govern the evolution of slightly compressible uids when thermal phenomena are taken into account. The Boussinesq hypothesis

Tracer la courbe représentative et donner les coordonnées du sommet et l'équation de.. l'axe

As stated in the beginning, the real purpose of these notes is not to simply repeat the standard calculation of the derivative of the logarithm function, but to replace the

Les contributions scientifiques de cette thèse peuvent être résumées comme suit: (1) le sujet de cette thèse, qui est une combinaison de réaction et de scalaire passif

The symbol denotes implication, the arrow pointing from the premiss to the conclusion; and the double-headed arrow ~ means 'implies and is implied by'... Let ~

These binomial theorems are evidently generalizations of corresponding theo- rems by NSrlund t (in the case where the intervals of differencing are identical).

In the mid-1980s the second author observed that for a connected reductive complex algebraic group G the singular support of any character sheaf on G is contained in a fixed

B., The Cauchy problem with polynomial growth conditions for partial differential operators with constant

John8 College, Cambridge, England. Ahnqvist &amp; Wiksells

However, the proof of Theorem 2.7 we present here has the virtue that it applies to other situations where no CAT(0) geometry is available: for instance, the same argument applies,

We will show that one can always find a suitable prototypical surface in Model B, and two switch moves among the four introduced in Proposition 5.1, such that the prototypes of the

By 1., the group homomorphism from S n into {±1} which maps σ onto det u σ −1 coincides with the signature on the set of transpositions, because a transposition is its own

The Galois section is good if it universally kills Chern classes of line bundles, a property that is directly linked to lifting to the maximal cuspidally central quotient π cc 1 (U,

Determine the symmetric bilinear form φ associated to the quadratic form q and write its matrix A with respect to the canonical basis4. Determine the kernel of the bilinear