Independent And Stationary Sequences Of Random Variables - Chapter 20 ppsx

Thus we wouldwish to find necessary conditions and sufficient conditions, not too farapart, for the distribution function F„ x of the normalised sum of in-dependent, identically distribu

Chapter 20

SOME UNSOLVED PROBLEMS

In this chapter we list various unsolved problems and possible lines offurther research, classified according to the chapters from which theyarise They come from various sources, to which it would be difficult togive exact credit

Chapters 3-5(1) The problem of extending the results of © 3 4 to the case of conver-gence to a stable law of exponent a=2 (see [191], [21]) Thus we wouldwish to find necessary conditions and sufficient conditions, not too farapart, for the distribution function F„ (x) of the normalised sum of in-dependent, identically distributed random variables in the domain ofattraction of the stable law G, ,,(x) to satisfy

IF.(x)-G,,(x)I = 0(n -Y),

y >0 The method of © 3 4 does not work, since Theorem 1 6 1 is not applicable (2) What is the multi-dimensional analogue of Theorem 3 4 1?

(3) In the notation of © 3 5, write

zC„= sup sup

n2 IF,,(x)-0(x)I

X #3

Theorem 3 5 2 says that C,- *[(10)'+3]/6(27r)'- as n-+oo It would ofcourse be interesting to know C„ explicitly, but failing that, to find esti-mates of C,,, perhaps the second term of an asymptotic expansion

(4) Along the same lines as the last problem, setC* (x) = lim sup sup IF„ (x) -'P (x)I

It has been conjectured by Kolmogorov [82] that, for symmetric butions F,

(6) How is Prohorov's theorem (4 4 1) changed if convergence in

L 1 (- co, co) is replaced by convergence in L P (- oo, co) (1 < p < oo) ?(7) Again in the spirit of (1), extend the ~ results of © 5 3 to the case ofconvergence to stable laws

Chapters 6-14(1) In all the theorems of these chapters, the zones considered were ofwidth A(n)p(n) or A (n) / p (n), where p( n) is a function increasing arbi-trarily slowly to infinity Can the function p (n)be replaced by a constant,say 1? (Many results in this direction have been obtained by Nagaev )(2) The problem of uniform normal convergence has been studied fornarrow zones of general form, and for wide monomial zones The pro-blem remains of studying wide zones which are not monomial

(3) The derivation of asymptotic expansions in different zones of normalconvergence

(4) The analogues of (1)-(3) for convergence to Cramer's system of ing tails

limit-(5) The discovery of systems of limiting tails other than that of Cramer,and of their domains of attraction

(6) The establishment of sharp bounds for large deviations, and inparticular the computation of the best constant

(7) The discovery of wider classes of random variables, for whichintegral theorems are valid on the whole line

(8) As a particular case of (7), study the random variables with uous probability densities g(x) satisfying

contin-f

00 g(u)du =

1- F(") (x) = lim F~"~(- x) = 1 ?

(20 1 4)

"-0 J

2 ( )

limit-F1 and F2 need not coincide If a < 6, then (20 1 4) depends on the equality

of the first two moments, and does not even need the condition that thetails of F1 and F2 be identical Thus Kolmogorov's problem is solvedunder (20 1 5) In the absence of this condition the situation is obscure For variables of class (A) (© 14.1) it is easy to check that (20 1 4) is implied

by the equality of the first two moments of F1 and F2 in [0, c (log n)2],and also in [n? oo] ifF1(± x) ' F2(±x) Between these two zones

it will not hold unless the pseudomoments of F1 , F2 coincide, which is acondition on the distributions as a whole, and not merely on their tails (11) Extend the results of Chapters 9-14 to the case in which the variables

do not have the same distribution (Much has been done in this respect

by Petrov [128], [129] )(12) Deduce analogous results for Markov chains (Some results havebeen established by VA Statylyavichyus )

(13) Extend the results of Chapters 8-14 to random vectors (14) Investigate the large deviations of infinite-dimensional objects,such as the whole history of Markov chains

Chapter 15(1) Can Theorem 15 1 1 be proved in a neat analytical way using theapparatus of characteristic functions? (Kolmogorov [85] )

(2) If in Theorem 15 1 1 the uniform distance IF" - D"I is replaced by thevariational distance p ( F", D") (© 15 1), it is not known whether, as n-*oo,sup inf p ( F", D") > 0

Chap 20

SOME UNSOLVED PROBLEMS

393

Chapters 16-19

(1) A vast field of research is presented by the problem of characterisingstationary processes satisfying one or other of the conditions of weakdependence (See for example [87], [55], [56], [184] ) What conditions,for example, are laid upon the moments of a stationary non-Gaussianprocess by the strong mixing condition?

(2) In Theorem 18 2.3, can the uniform mixing condition be replaced bythe strong mixing condition?

(3) Conjecture : If a sequenceXj stationary in the strict sense is

uniform-ly mixing and satisfies

E(X < co ,

Jim

V E Xi = co ,n-oo

j=1

then it satisfies the central limit theorem, (cf Theorems 18 5 1, 18 5 2) (4) Can Theorem 18 5 3 be refined in the following way? Let a stationarysequence XX be strongly mixing with mixing coefficient a (n), and let

E IX;I2+a< co for some S > 0 Can one give a number a = a (8) such thatthe central limit theorem holds whenever a (n) = o (n - a), and for which anexample of a sequence not satisfying the central limit theorem can befound for any a (n) with lim sup a (n) na > 0, it being always assumed ofcourse that V(E XX) ca?

(5) How far from the best possible condition is (1) of Theorem 18 6 1?For instance, suppose that Xi are independent variables taking two valueswith constant probabilities p, q, and that

Yj = Y( , X;-1, X;, Xj+1, ) What is the precise order of magnitude of the quantity

y(n) = E{Yo - E(YoI X-n, , Xn)} 2

which ensures that Y satisfies the central limit theorem? In the context of

© 19 3, how can this condition be expressed in terms off e L2(0, 1)

(6) How are the conditions for the central limit theorem to hold forMarkov chains related to the conditions of weak dependence? For example,letf (Xj) be the stationary sequence derived from a homogeneous Markovchain XX (© 19 1) For this sequence to satisfy the central limit theorem, is

it enough that it be regular? If not, is it enough that it be strongly mixing?

and a>0 This theorem is due to Karamata [62] For its proof we require twolemmas

Appendix 1

SLOWLY VARYING FUNCTIONS

A positive function h (x), defined for x > 0, is said to be slowly varying if,for all t > 0,

h(tx)lim

Clearly (A1 2) is equivalent to the assertion that

q (x) = xah (x) ,where h (x) is a slowly varying function

Theorem A1 1 A slowly varying function h(x) which is integrable onany finite interval may be represented in the form

x-* 00

lira E(X)=0,

X_ 00

App I

SLOWLY VARYING FUNCTIONS

Lemma A1 1 lim

rx

= 1 ,X-,, a (x)

so that a (x) is also slowly varying It is also bounded, so thatlim {a(rx) -a(x)} = 0

X- oe

If H (x) = f o h(t) dt, then (A 1 6) maybe writtena(x)

= dx log H (x) ,

(x)

( )

as x-+ oo, or equivalently,Jim r

a (xt) dt = log r X-ccf1

Since log r = f i t-1dt, (A1 9) shows that, as x-~ oo,

App 1

SLOWLY VARYING FUNCTIONS

to invoke such properties, it is necessary that h (n) be not merely slowlyvarying, but also have a slowly varying extension h(x) defined for allx>0

An example of a slowly varying function h(x) which does not have such

an extension is

h (n) = (number of simple divisors of n) + (log n)?

(1) lim hhxX t) = 1 , (t > 0) X--~ co

( )

(2) For all s > 0,lim x¢h (x) = co ,X- oo

=

k-, ao 2k-<t-<2k+1 h(2 )

Appendix 2

THEOREMS ON FOURIER TRANSFORMS

For p > 0, we denote by LP the collection of functionsf(x)for which

Theorem A2 3 Let F (x) be the Fourier transform off (x) E Lp( 1 < p < 2) If

also f (x) E L q and f' (x) E L p , then xF (x) E L 2 and - ixF (x) is the Fourier transform off' (x)

of f' (x), and the right-hand side to - ixF (x) This theorem is a slight generalisation ofTheorem 68 of [180]

Appendix 3

A THEOREM ON CONVERGENCE OF CONDITIONALEXPECTATIONS

Theorem A3.1 Let X be a random variable with E I X I P <oo

and let Tin be a a-algebra of events for each integer n, with

n- - oo

lim EIE(X I 9JIn) - E(X 19n0,,)IP=O

(A3 1)n-oo

In particular, if X is measurable with respect to JJk~, then

lim EIE(X I 9JIn)-XIP = 0

(A3 2)

n-+ oo

The proof may be found in © 7 1 of [31]

We note that the left-hand side of (A3 1) is finite for all n,since by Jensen'sinequality [31], for any a-algebra 91,

EIE(XI9)IP<E{E(IXIPI 91 )} = EIXI P

In case p=2, the point of view of © 16 3 gives (A3 1) a simple geometricmeaning IfHn is the subspace of L 2 (Q)consisting of the random variablesmeasurable with respect to Di n , and Pn is the projection operator onto

H, then it is easy to verify that

E (X 1 9JIn) = PnX

Chapter 1

©© 1-3 : The development of probability theory on the basis of the cepts of measure theory, including the definition of conditional expec-tation and conditional probability, comes from Kolmogorov [76] Theorem 1 3 1 is cited without proof in [82]

con-© 4 : Characteristic functions were first used to prove limit theorems inprobability theory by Lyapunov [106] Their basic properties werestudied by Levy [93]

© 5 : Theorem 1 5 1 comes from Levy [93] The remaining theorems ofthis section are due to Esseen [33], [34]

© 6 : Theorem 1 6 1 is well known For much more general results, seethe book by Linnik [102], from which our proof is taken

© 7 : Infinitely divisible distributions were first studied by de Finetti [38] Formula (1 7 1) was discovered by Levy [94], but in the case of finite vari-ance had been earlier obtained by Kolmogorov [78] Theorem 1 7 2comes from Khinchin [70], Theorem 1 7 3 from Gnedenko [39]

of Linnik [99], Zolotarev [188] and Medygessy [109]

© 4 : Theorems 2 4 2 and 2 4 5 were proved by Bergstrom [6], Theorem2.4 1 by Bergstrom [6] and Pollard [135], Theorem 2 4 3 by Skorokhod[174] The asymptotic expansions of Theorems 2 4 4 and 2.4 6 are new,the leading terms having been obtained by Skorokhod [174] and Linnik[99] respectively

© 6 : The domain of attraction of the normal law was studied by Khinchin[69] and Levy [95] The domains of attraction of stable laws with ex-ponent a =A2 were investigated by Gnedenko [40] and Doeblin [30] Theorems 2 6 1 and 2 6 2 are reformulations of the results of these authors Theorem 2 6 3 is due to Sakovich [165], Theorem 2 6 4 was proved for

a = 2 by Khinchin [69] and for a:A2 by Gnedenko Theorem 2 6 5 is new

Chapter 3

© 3 : Esseen [33] For more restrictive conditions see Cramer [23] The case of convergence to stable laws with a :A 2 was investigated byCramer [21] and Zolotarev [191]

© 4 : A new result

© 5 : The first estimate of F„ - 0, in the spirit of Theorem 3 5 1, was tained by Lyapunov [101], the final result by Esseen [33] Theorem 3 5 2was proved by Esseen [35], the supplementary results by Rogozin [152]

vari-© 3 : Gnedenko [44], [45] Extensions to non-identical summands bySmith [179] and Petrov [122]

© 4 : Prokhorov [138] In the work of Sirazhdinov and Mamatov [172]bounds were obtained for I Ip - 0I I for the case of normal convergence

© 5 : Theorems 4 5 1 and 4 5 3 are new, Theorems 4 5 2 and 4.5 4 due toEsseen [33] For the non-identical case see- Petrov [125]

Chapter 7

© 5 : Bernstein [7] and Richter [149] introduced refinements of the equalities (7 5 3) (In Richter's work there is an easily corrected error : informula (1) the expression

) X

Chapter 9Linnik [100], [101], [104]

Chapter 10Petrov [130] Other local theorems for large deviations have been ob-tained by Richter [146] and Nagaev [120]

Chapters 11, 12These chapters largely describe the results of [101]

Chapter 13Petrov [131]

Chapter 14Linnik [104]

Chapter 15

©© 1-4 : The basic theorem of this chapter is due to Kolmogorov [85],whose paper also describes the history of this problem Concentration

404

NOTES

functions were introduced by Levy Theorem 15 2.1 is an amplificationdue to Rogozin [153] of a result of Kolmogorov [84] Lemma 15 3 5comes from Prokhorov [140] Meshalkin [111] has shown that

inf sup IFn -DI >, Cn- 3(log n) -4

© 3 : The geometrical interpretation comes from Kolmogorov [79], [80]

© 4 : Khinchin [65]

© 5 : Theorem 16 5.1 was proved by Cramer [20], although equation(16 5 1), without Z(,~), was known to Kolmogorov [79] For proofs ofTheorem 16 5 1 not using the spectral theory of unitary operators, see[31], [187]

©© 6, 7 : Kolmogorov [80]

Chapter 17

© 1 : Regular processes were studied by Vinokourov [183] ; Lemma 17 1 1

is due to Wold [185] The idea of linear regularity, and condition (17.1 7),come from Kolmogorov [80] Theorem 17.1 2 is a very special case oftheorems on the spectrum of K-systems and K-flows [86], [171]

© 2 : The strong mixing condition was introduced by Rosenblatt [158],the uniform mixing condition by Ibragimov [152] The results of thissection come from [184] and [57]

© 3 : Further information on the spectral densities of strongly mixingprocesses may be found in [87], [55], [56], [163]

Chapter 18

© 1 : An alternative technique to that of Bernstein for proving limittheorems is Markov's method of moments, which has been applied tostationary processes by Leonov and Shiryaev [88], [89], [91], [92] Theorems 18 1 1 and 18 1 2 are due to Ibragimov [184]

405

©© 2, 3 : Theorems 18 2 2 and 18 3.2 are from Leonov [90] Theorems

18 2 3 and 18 3 3 are new Equation (18 2 7) comes from Robinson [151]

©© 4-7 : Mainly the results of Ibragimov [52], [51] Related tions not confined to stationary processes may be found in the work ofVolkonskii and Rozanov [184] and in that of Rozanov [160], [161],[162] Theorem 18 4 2 comes from [184] The first limit theorems forstrongly mixing processes were proved by Rosenblatt [158] A variantunder a condition weaker than strong mixing has been proved by Sinai[169] Estimates for the rate of convergence may be found in [177] The method for deducing the analogous results in continuous time is due

investiga-to Kolmogorov [77] Results similar investiga-to those of © 5 were obtained byCiucu [14], [15], see also [16]

Chapter 19

© 1 : The central limit theorem for finite Markov chains was proved byMarkov himself [108] Theorem 19 1 2 comes from Nagaev [117],whose method of proof differs from ours In [118] and [119] the condi-tions of that theorem are further relaxed The most complete results oninhomogeneous Markov chains were found by Dobrushin [27] andStatulevicius [176] In [110] Meshalkin has enumerated all possiblelimit distributions for sums of random variables defined on a finite homo-geneous Markov chain

© 2 : m-dependent random variables were first studied by Hoeffding andRobbins [51] Theorem 19 2 1 is due to Diananda [25], [26]

© 3 : The results of this section, which are due to Ibragimov [53], areamplifications of theorems of Kac [60] Leonov [89] has investigated thedistribution of values of sums of the form Ef (A kt), where f is defined on

an n-dimensional cube, and the integral matrix A has no eigenvalueswhich are roots of unity

© 4 : These profound results in the metric theory of continued fractionswere obtained by Khinchin [66], [67] The first part of Theorem 19 4 1 is

by Ryll-Nardzewski [164], the second by Ibragimov [53] Theorem

19 4 2 is due to Ryll-Nardzewski [164], but weaker variants were known

to Khinchin The central limit theorem for continued fractions was firstproved by Doeblin [29], Theorem 19 4.3 by Ibragimov [153] In [54] acentral limit theorem was proved for the denominators qn (t)

The metric theory of more general number systems has been studied byRenyi [144] and by Rokhlin [155]

© 5 : Rosenblatt [158]

SOME CONTRIBUTIONS OF RECENT YEARS

I A IBRAGIMOV, V V PETROV

The present chapter is a review of contributions published in the yearsbetween the appearance of the original (Russian language) version of thisbook and the present translation (1965-1970) Its authors have notattempted to review all such contributions pertaining to the book'ssubject matter ; consideration is essentially given to those works which

to a certain extent extend or develop the results of preceding chapters,such as those solving the problems of Chapter 20 Thus proofs are eitherwholly omitted, or only touched on The referencing of this chapter isself-contained ; all references given are to the complementary referencelist at the conclusion of the chapter

On chapter 3

In recent years a great deal of work has been devoted to estimating theremainder term of the central limit theorem M Katz [52] has obtainedthe following generalization of the Berry-Esseen estimate (Theorem 3.5 1) Let X1, , Xn be independently and identically distributed randomvariables with zero mean and positive variance Q 2 LetE (X i g (Xl )) < oo,for a non-negative even function g (x) with the properties that g (x) andx/g (x) are non-decreasing in the region x > 0 and lim g (x) = + oo Put

X-+ w n

< ~2 6n~ E(X1g(X1))g(

V M Zolotarev [13] has noted that, in this last estimate, one may put

C = 0.82

V V Petrov'[33], L V Osipov [25], L V Osipov and V V Petrov [26],

W Feller [47], and others, have studied the generalization of the Esseen estimate to non-identically distributed independent randomvariables

Berry-Considerable progress has been made in respect of the subject of uniform estimates of the remainder term in the central limit theorem,which appear as essential refinements of the uniform estimates We firstnote the following result of S F Kolodiazhniy [18], of interest beyondthe central limit theorem alone, and relevant to a theorem of Esseen(Theorem 3 6 1) Let F(x) be an arbitrary distribution function, with finiteabsolute moment of order p > 0. Put J = sup IF (x) - O(x) 1 If 0 < d < e - 1,

As indicated in [18], this estimate is optimal in a certain sense The following important refinement of the Berry-Esseen estimate is due

to S V Nagaev [24] Let X1 , X2 , , Xn be independent and identicallydistributed random variables such that E(X1) =0,E(X2)=U2

> 0, E IX1 1 3= /33< co Then

I Fn(x ) -P(x)I < a3

n (1

#+ Ix13)

for all x Here C is an absolute constant A generalization of this result

to non-identically distributed random variables was obtained by A alis [1] Non-uniform and uniform estimates of the remainder in thecentral limit theorem without assumptions concerning the existence ofmoments of the random variables under investigation may be found inthe paper of L V Osipov and V V Petrov [26]

Biki-We now pass onto an account of some recent results concerning totic expansions in the central limit theorem Let X 1 , X2, be a sequence

asymp-SOME CONTRIBUTIONS OF RECENT YEARS

407

j yJKdV(y)+

~y~ ~vn 1 (1+ixi)

+a-K- 1n2K-1(1+IxI-K-1

Here 8=a2(12 E IX113)-1 and c(K) is a positive constant, depending only

on K The function P,(-0) is the same as in Theorem 3 3 3*) This result

uniformly with respect to x

* _ n the paper of V V Petrov [31] there are explicit formulae for the functions P v (- 0)

=0 ( n-4K+ 1)

In the case K=3, A Bikialis [1] has shown that the preceding relationstill holds if Cramer's condition (C) is replaced by the weaker requirementthat the distribution of X1 be non-lattice

Let us now consider a sequence of independent random variables

X 1 , X2 , having the same lattice distribution, on the possible values

a + mh (m = 0, ± 1, + 2, ) where h is the (maximum) lattice distance forthe distribution Let E (X 1 ) = 0, E (X 1) = a 2 >0, and E I X 1 1'< oo for some

r > 3 Then, as shown by L V Osipov [27], there exists a positive functionc(u), such that

e(u) =0 and[r]-2

h

v F,,(x)-Hnr(x)

•• sin 2n lx S2"+ 1(x) 2

~ 1 (27rl)2"+1

(K =1,2, )

Of some interest are upper and lower estimates of the remainder term

in asymptotic expansions, having the same order Let X 1 , X2 , be asequence of independent and identically distributed random variableswith E(X1)=0, E(X2)= a 2 > 0 and E 1X 1 1"< co for some integer K ->-3

We putV(X) =P(X1 <x), /l n , v = 6 -v n -iv+1J

(1+lxl)r

I A IBRAGIMOV, V V PETROV

n K+ ILn,K+ 1 I + Ln,K + 2 9 Y"n,li -

Let XI , X2 , be a sequence of independent random variables with thesame distribution function V(x), zero mean and finite positive variance

Q2 Let § I = 0, § 2 = Q2, §3, §4, be a specified numerical sequence inwhich the numbers § 3 , §4, may be arbitrarily chosen Let QK(x),x=1, 2, be polynomials with coefficients expressed in terms of §3, ,YK+2, in the same manner as the coefficients of the classical polynomials

Q K (x) = (27r)' ex2 / 2PK( - 0) are expressed in terms of the cumulants

Y3,

YK+2 (see e g [31]) A sequence of numbers /31, # 2 , is

construct-denotes that 0< Jim inf a„/b„<, Jim sup a„/b„< oo

SOME CONTRIBUTIONS OF RECENT YEARS 411

ed as follows : & is defined in terms of § l , uK in the same manner asmoments are expressed in terms of cumulants, i e N 1 = § 1, N 2 = §2 +J U2,F'3+ 3§1 §2+ §1 We then have the following theorem, due to I A Ibragimov [15]

For the relation

n = n

n j=1

(2 ) s 1K=1, 2, to hold uniformly with respect to x, it is necessary (and fordistribution V(x) satisfying Cramer's condition (C), sufficient) that thefollowing conditions be satisfied

1) the absolute moments up to order K+ 1 of the distribution V (x) arefinite, and

L-00 xmdvx= ()

2)J

conver-= E(X2)< oo, and F

n(x) the distribution function

of the normed sum

<co,(0<5<1) ; E{X1 log (1+IX1I)}<oo, (6 =0)

If X1 , , Xn are independent random variables each with the normal

Let X1 , X2 , , Xn be independent random variables with the same bution function V (x), with E (X 1 ) = 0, E(X2) = 1

distri- We introduce the domoments

P n_#

jXj <x)-0(x) j=1

-V V Petrov [32] has obtained the following refinements of Theorems4.5 2 and 4 5 4-which are due respectively to B V Gnedenko and Esseen -without auxiliary conditions

If, for some n=n o the random variable Z n has an absolutely continuousdistribution with bounded density pn (x), then there exists a function ~ (n)

independent of x such that lim n,x ~( n) =0 and

n (1 + Jx~ )

If the random variable X 1 may only take values of the form a+Nh

(N=0, +1, ±2, ) where h is the maximal lattice distance, and a some fixed number, then there exists a function b (n) independent of N such that lim 5 (n) = 0 and

for all N, where

/2

V=1

5 (n)

n iK-1 (1 + IxI K )

O(x) = ( 27i:)-+e-x2/2, P"(-O) = dPv( -o)

Local limit theorems for sums of independent non-identically distributed random variables have been investigated by : V V Petrov [35] ; V A Statuliavicius [40] ; A A Mitalauskas and V A Statuliavicius [19] ;

N G Gamkrelidze [4], [5] ; D A Moskvin, L P Postnikova, and A A Yudin [21] ; and V L Pipiras and V A Statuliavicius [37]

-00

converges if and only if E {Xi log (1 + 1X11)} < oo Discarding the ment of finite variance of the random variable X 1 , Heyde has shown that

require-if the distribution function of X 1 , V (x),belongs to the domain of attraction

of the normal law, and further the condition

00

Fn(x) - P (x), which take into account the dependence of this difference

on nand x For example, from the results of L V Osipov [28] and V V Petrov [32] on asymptotic expansions in limit theorems, cited above, wearrive at the following conclusions Let X1, X2 , be a sequence of in-dependently and identically distributed random variables with zero meanand finite moment E 1X1 IK for some integer x,> 3 If the distribution ofthe random variable X1 satisfies Cramer's condition (C), then

K-2IIFn-(PIIP=

Pfor arbitrary p>,1 If the random variable (un 2 ) -1 E ;=1Xj, where a 2=

SOME CONTRIBUTIONS OF RECENT YEARS

415

E(X i), has for some n = no an absolutely continuous distribution withbounded density p„ (x), then

K~2 P

P"(-0)

+o (n fK+1IIPn-0II) = "/2

1 n

Pfor any p> 1 Here 0(x) = (27r)-+e- x2 / 2 ,

00 iIuI1 P =

Let F (x) be the distribution function of a random variable with zeroexpectation, positive variance 2 and finite moments of all orders Let

YK be the cumulant of order x of the distribution F(x) V A Statuliavicius[59] has obtained relations of Cramer type for {1-F (xa)} / { 1-0(x)}

and IF(- xu)/'P (-x)} in the interval 1 < x < bd,where

x Ho-2 (K-2>-1 ,A== Q inf

x33 IYKI /

and H and 6 are certain positive constants His results imply Theorem

8 4.1 (a refinement of Cramer's theorem), if for F(x) we take the tion function of the normed sums of independent random variables eachwith the same distribution function V(x), satisfying the condition00

distribu-eh xdV < oo, jhi < A, for some A >0 (Cramer's condition (A))

The paper [59] also contains information on the estimation of constants

~-in rema~-inder terms of Cramer-type relations