Independent And Stationary Sequences Of Random Variables - Chapter 8 pps

Chapter 8CRAMER'S INTEGRAL THEOREM AND ITS REFINEMENT BY PETROV „ 1.. Statement of the theorem The first general result in the theory of large deviations was the integral theorem of Cram

Chapter 8

CRAMER'S INTEGRAL THEOREM AND ITS REFINEMENT BY PETROV

„ 1 Statement of the theorem

The first general result in the theory of large deviations was the integral theorem of Cramer [19], published in 1938, which has considerable computational and analytical usefulness It was refined and generalised

by Petrov [133] in 1954 In this chapter we discuss the work of Petrov, keeping for simplicity to the case of identically distributed variables X;

It should be remarked that the most natural method for proving integral theorems under Cramer's condition is the method of steepest descents, whose use for local theorems was described in the last chapter In the case in which the distributions of the X; are different such an approach encounters, however, considerable difficulty

Let the X, satisfy (7 1 1) and Cramer's condition (7 1 2), and let 2 (z) denote Cramer's series (7 2 20) Write

x -P(x)= ( tic)-2

1-00

e-2`zdt ,

Zn = (Xi + X2+ + X©) / ant ,

V (y) = P (X; < y) Then Petrov's refinement of Cramer's theorem, in the identically distri-buted case, has the following form

Theorem 8 1 1 For x > 1, x = o (n2 ), we have P(Z,, > X)

rx3

X

X

-exp+ n2

n2

1+0

n2 ,

P(Zn< - x)

=ex

p 3~

1+

x 0

\

n 2

n 2 )

n 2 ] 0(-X)

CRAMER'S INTEGRAL THEOREM ; ITS REFINEMENT BY PETROV Chap 8

„ 2 The introduction of auxiliary random variables Since

E(exp aJXXj) < oo ,

we may write, for Jhj <a,

00

R = R (h) =

f-00

ehl'dV (y)

(8 2 1) Let XX be independent random variables with the distribution function

When n=1 this follows trivially from (8 2 2) Suppose it is true for a particular value of n Then

P (x) = R-1

X

f -00 e"''d V (y) , (8 2 2) and write

(8 2 3) W©(x)=P(X1+ +X©<x),

W© X =P (X1+ +X©<x) Then

in=E(X,) = R-1

J

xe''XdV(x)=

-00

(8 2 4)

R'(h)

d log R,

-

=

R (h) dh and

(8 2 5) _

R©

R , 2 d2

62 V

log R (R

= (X;)

Write

(8 2 6)

F© (x) = P (XI + +X© < c0 x) , F© (x) = P (X1 + + X© - mn < 60 x)

We prove by induction on n the fundamental relation

(8 2 7) W© (x) = R"

T X

e - ' dW©(y) 00

8 2

INTRODUCTION OF AUXILIARY RANDOM VARIABLES co

W"+ 1 (x )

-

V (x - z) dW,,(z) _

f -o0

f

w

R-1

V(x-z)e-hwdWn(z) _

§~

x z

= Rn +1

J -00

dWn(z)e-hz

J - e -hsdV(~)

(( 8 2 8) 00

Making the substitution ~ =11- z, (8 2_8) becomes

J

Go

Jx

R "+1

f -00

dW" (z) e -hz

f -00

e-h(''-z)dn

V (h-Z) _

x

00

=Rn+1

J-e-h',d

V(~_z)dWn(z)=

cc

- cc

whence the induction succeeds, and (8 2.7) is proved

From (8 2.7) and (8.2.6) we have

F n (x) = Wn (xanl) ,

Fn (x) = Wn(mn + xan+) ,

so that

xan '/2

F (x) = Rn

f-00

e-h''dWn(y~)

Setting q= min + y6n2, we have

Letting x- oo,

1 = R n a - hmn

so that

e -h

(ax - mn'/z)/o'/z

Fn( x ) = Rn e -hmn'

exp (-h yan'/z) dFn

(y)

(8 2 11)

- 00

00

J -00

exp (-hy6n z) dFn (y) ,

(8 2 10)

"

f

cc

1-Fn (x)=R e-hmn

exp(-hy6nZ)dFn (y)

(8 2 12)

(ax-Fn )/a'/a

1 73

CRAMER'S INTEGRAL THEOREM ; ITS REFINEMENT BY PETROV Chap 8

„ 3 Proof of the theorem Since

log R = log ~~ e"''dV (y) _ y Y'' by ,

(8 3 1)

00

y12 v!

where yy are the cumulants of the X3 (7 2 6), we have

d

00

Fn = - log log R = z i i hv_ l ,

(8 3 2)

v=2 (v)•

Y2 = 2 _2=

dh2

2

00

log R = E v

Y -2)! by-2 > 0

(8 3 3)

v=2 (

In the notation of „ 7 2, logR=K(h),

so that the factor multiplying the integral (8 2.12) is exp {n (K (h) - hK' (h)) }

( 8.3 4)

We now choose h to be the solution of the saddle point equation

where i=x/n 2 =o(1) According to (7 2 18) and (7 2 19) there holds, for sufficiently large n, the equation

K (h) - hK' (h) = K (h) - aT = -lz 2 + i 3 (z) ,

( 8 3 6) where A (z) is Cramer's series (7.2.20) Moreover, m= K' (h), so that by (8 3 5), ax -mn? = 0 Substituting (8 3 6) into (8 2 12), we therefore have

f

00

1-F©(x)=exp{n(-42+t3A(i))}

exp(-h6:n1y)dF©(y) (8 3.7)

0

We therefore have only to examine the integral in (8.3 7)

00 exp (- hini y) dF© (y)

0

Now F, is the distribution of the normalised sum

(8 3 8)

8 3

PROOF OF THE THEOREM

175

so that we can use the theorems of „ 3 5 From (8 3 3),

a = 6 + O (h) ,

(8 3 9)

and

Fn (y)

(y) + Qn (y) ,

Q, ,(y) = Bn - 2 ,

( 8 3 10)

so that

~co

exp(-hdn-Iy)dFP(y)=(2n)-J

00

exp(-hin#y-iy 2

)dy-0

0

-Q n (0)+h6n* Jo exp(-hdnly)Q n (y)dy

( 8 3 11)

The last integral in (8 3 11) is

o

Bn - hQnz

J exp(-hin y) dy = Bn - ,

(8 3 12)

0

and Q n (0) = Bn - , so that it remains only to estimate the integral

-

ex

hen

dy •

(8 3 13

00

(27r) -f

f o

p(

-

? y f )

) Now (8 3 2) and (8 3 5) imply that

hQnz = h6n++ 0(h 2 n+) ,

mn Z =C2

hn 4 + O (h 2 0) ,

so that

han 4 = mn4 o ' + O (h 2 n 4 )

( 8 3 14)

Substituting (8 3 14) into (8 3 13), we have the expression

00

(2n)-2

J exp(-mn 4 6 -1 y) exp(Bh2n4y) eXp(- 2y 2 )dy =

0

h- 1n1/2

exp(-mn 4 6 -1 y) exp(-Zy 2 )dy(1+0(h))+

0

+B exp(-2n2 h -2) _

f

00

=(2rr)_2

exp(-Fnn4 6 - 'y)exp(-2y2 )dy(1+0(h)) (8 3 15)

0

=exp (m e n/26 2 )(1-0(mn4 /a))(1 +O(h))

( 8 3 16)

CRAMER'S INTEGRAL THEOREM ; ITS REFINEMENT BY PETROV Chap 8

According to (8 3 4),

K'(h)/u = M-/c = ,r,

so that Ten/2a2=1ni2. Using this and (8 3 13), and substituting (8 3 16) into (8 3 7), we find that

1-F©(x)

= exp {ni3 ?,(i)} (1 +0(h))

(8 3 17) 1-'P(x)

Since h = 0 (i) = 0 (X/n2'2), the theorem follows