Independent And Stationary Sequences Of Random Variables - Chapter 13 pdf

Chapter 13MONOMIAL ZONES OF INTEGRAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS © 1.. Formulation This chapter is devoted to Petrov's theorems on integral convergence, whose local a

Chapter 13

MONOMIAL ZONES OF INTEGRAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS

© 1 Formulation

This chapter is devoted to Petrov's theorems on integral convergence, whose local analogues have been proved in Chapter 10 We keep the notation of that chapter, but do not restrict the variables Xjto belong to the class (d) The basic results are the following analogues of Theorem

10 1 1

Theorem 13 1 1 Let p (n) + oo be an arbitrary increasing function, and suppose that, for some a < 2,

Here Ar5] (z) is the truncated Cramer series, and s in the integer defined by (10 1 6)

Theorem 13 1 2 If,for all x in 0 < x < n' p (n), all n >, n 0 , and positive con-stants n o, a o we have

P(Zn>x) < e -§ o"2 , P(Z"< -x) < e-aox2 , (13 1 4) then (13 1 1) necessarily holds, and the conclusion of Theorem 13 1 1 applies

Then, uniformly in 0 < x < na/p (n),

3

X

P (Z,, > x) ti {1- 0 (x) } exp x~ AI5]

3 x P(Z„< -x) - 0(-x) exp

{ - AN ~- n

x

(13 1 3)

1 3 2

THE PROBABILITY OF A LARGE DEVIATION

The deduction of (13 1 1) from (13 1 4) is exactly like that given in „ 8 2 Thus no truncated power series 7i[s] (z) other than that of Cramer can possibly appear in formulae of the type (13 1 2), (13 1 3) Further, in the collective Theorem 13 1 1 the role of the linear functionals a j , is played

by the moments of Xj

„ 2 An upper bound for the probability of a large derivation For the sequel, we shall need inequalities like (12 2.3) and (12 2.10) for the probability of large deviations We cannot however use the inequalities already proved, since these depended on the vanishing of the first (s + 3) cumulants, which is not here assumed

Suppose then that X 1 , X2 , are independent and identically distributed, with

E(Xj)=0,

V(Xj)=6 2 ,

and suppose that (13 1 1) is satisfied We prove that, for any monotonic function p ( n) -3oo there exist positive constants c1 and c2 such that P(Sn>n"+Z/p(n)) < c 1 exp[ - c2n 2a /p(n)2]

( 13 2 1) for all sufficiently large n, where

n

Sn = y Xj j= 1

To prove this assertion, we consider the normalised sum

Z n = Sn /6n- ,

(13 2 2) and a modified random variable

Zn = Zn+ Yn /6n 2 ,

( 13 2 3) where Yn is a random variable, independent of the Xj ,and having a normal distribution with mean 0 and variance n 2" Thus

E(2n) =0,

V(Zn )= 1+6 -2 n 2 " -1

( 13 2 4) The distribution functions of Znand2 n will be denoted by Fn(x) and Fn (x)

respectively, and their characteristic functions byf n (t) andfn (t), so that

n2"-1 t

2

fn (t) = fn (t) eXp

-

X62

(13 2 5) 245

246 CRAMER'S SYSTEM OF LIMITING TAILS Chap 1 3

The random variable 2n has a continuous distribution, with density

1 a Pn( W ) = - / e-itx ,,(t)dt

00

which is everywhere continuous Integrating from x 1 to x2 , (where x l <x 2 will be chosen later),

F"(X2) -F.(X 1)

1 2?t 1 27L

op e- itx2-e- 1.1x1

.fn(t)dt = It

00

roo e-iuan'/zx2-e-iuvn l /zxl

1

Zn2±uz

()"d

e_

vuu,

-1.u

-00

where v (u) is the common characteristic function of the variables X 3

If e is a fixed positive number, then

_ 1 E e u '/2x2 a iuon'/2xt

Fn(x2) - Fn(x1) = -~

27r _E -1.u

+B a - E•" 2a , ( 13 2 6) where as usual e denotes a positive constant By virtue of (2 6 35) there exists i > 0 such that

IV (u)i" exp(-2u 2 6 2n)

in but < E 1 Therefore (for any monotonic p 1 (n)-+ oo) we have, in the range

na 2/p1 (n) Jul C E1

the inequality

Iv(u)I" < exp [-62n 2a /p1(n) 2 ]

Taking E=E1 in (13 2 6), we therefore have Fn(x2) - Fn(x1) =

1 e -iuan1/zx2 -e- iuan'/2xt _

e -2n 2au2

v (u)" du +

2 7~ J ul <-na-'/z/pi(n) -1.u

+B exp[-82n 2a/p1(n)2] ( 13 2 7)

As in Chapters 9 and 10, we introduce the function

K (t) = log v (t) ,

13 2

and obtain from (13 2 7),

THE PROBABILITY OF A LARGE DEVIATION

Fn(x2)Fn(xl)

-f

2n

-~

Jug _<<na/2/pl(n)

m

x exp -2n 2 au 2 +n Z 'ur du+Bexp[- E3n2alpl(n) 2 ]

r=2

r-where 1//r = K( r )(0) and m = In2a/p1(n)2 ] If _

P

KP(Z) = ZQ' 2 Z 2 + ll/r Zr/r!

r=3 where

d2 = 6 2+n 2a- 1 this becomes

e -ioun 1/zx2

-e-ioun `/2x1

- iu

1 (in•C -1/2/pj(n) e-zon e /zx2-e-zone/2x1 Fn( X 2) Fn( X l)

-

X 2nl -lna

1/2Ip1(n)

Z

x

247

(13 2 8)

(13 2 9)

xexp(nKN(z)) exp n E ~ r ir dz +

N+1 r +B exp[-e3n 2a /p1(n) 2 ] , (13 2 10) where N is a large positive constant to be determined later

We now apply the method of steepest descent, setting

na

x 1

x1 =

up (n)

, t =

nT ,

so that t -).O as n co Form the equation

(13 2 11)

dz {KN (z)-zta} = 0

(13 2 12)

For sufficiently small t, this equation has a unique real root z o with the same sign as t and tending to zero as t >0 In (13 2 10) we deform the con-tour of integration to the three sides of a rectangle passing through z o ,

to give IF.(x2) - Fn(x1) = I1+12 +13 +B exp IF; 3n 2a /pl(n) 2 ] ,

( 13 2 13)

2 4 8 where

I1

-and

1 z+ins - '/2/nj(n)

l E n (z, t)d- , 27CI .Izn-ina - '/2/P (n)

zo+ina - '/2/P1(n)

1 2 = - 1 En (Z, t)dZ ,

2761 in" - '/z/p j (n)

1 zo- in a- '/2/P1(n)

I3 = E n (z, t) dz ,

27zi -in•C-'/2/pl(n)

1 -e (X2-x1)vn'/2 / m

E n (z, t) = exp [n (K N (z) - zt6)] exp n E i z

z

r N+1 Along the line z = z 0 + iv we have

CRAMER'S SYSTEM OF LIMITING TAILS Chap 13

N KN ) (Zo) (i

v

)r 1~

r=2~

and substituting iv = v {nKN (z 0 ) }2 we get I1 = J,(27r)-1 {nKN (zo) } 2 exp {n(KN(zo) - zo ta) }

KN (Z) - Zta = K N (Z0) - Z 0 to + Y

where

J1 = exp -2w2 + Qr( Z O)( iW )r x

r

~S2 =3 r!nor-1

lW 1-exp -(x 2 -x 1 )an 2 z0+

[nKN(ZO)] 12

z o + iw / [nKN (z 0 )]

m

x exp

f

n (z 0 + iw/ [n1 (z 0 )]?)r dw , (13 2 15)

r = N + 1 r ° Qr( Z 0) KN ) ( Z O) [1 (ZO)] 2r

and 0 is the interval

,

jwj \ n•`[KN(zo)]z/pl(n)}

Now require that p 1 (n) and N should satisfy

lim p(n)1p, = oo , lim p 1 (n) = oo , ( 13 2 16)

N >, 7+2a

(13 2 14)

13 2

THE PROBABILITY OF A LARGE DEVIATION Then on 0,

Izo+iw/[nKN(z0)]2

I < Cn a-1 /pl(n)

for sufficiently large n (the positive constant C may be different in different formulae), so that, for r < m = [n"'/p,(n)2] and w c Q we have

I ~ r

l W

r r

zo + [nKN (ZO)]

C exp fr C +

1- 2a

log r -(2 - a) log n -log p1 (n)]

4a

1 ~

C/pl (n)84r

Hence

r

lW

r

r = [ log n] r ! ZO + [ nKN (Z0)]z

for sufficiently large n It is not difficult to see that, under (13 2.17), a similar bound obtains for

[log n]

so that for w cQ and n sufficiently large, we have eX

m

Z

iw

r

1 + T

~

r

1

=

p

n r=N+l r!

0

+ [ nK N( Z 0)] a

with

ITI < Cn -1

Using this fact, we easily find that JJ1I < CnC

for all sufficiently large n

For the root z o of (13 2 12) we have

t

1'/ 3

2 ZO=

G - 264 t + ,

KN(z0)-ZOta= - Zt 2

(1+Bn 2x-1

)+t 3 1(t),

where

G2

'~~ 2 03

04 -3

~(t) = 6G

3 +

2466

t+ +

is a series converging for small t, in which, by taking N sufficiently large,

we can make arbitrarily many of its terms agree with those of the Cramer

C n

249

2 50 CRAMER'S SYSTEM OF LIMITING TAILS Chap 13

series ;.(t) Hence from (13 2 14),

h = J

1 , exp{n[-zt2 (1+Bn 2 a -1)+t 3 ; (t)]}(1+0(1)), (13 2 18) 2nun'

and thus for sufficiently large n, taking (13 2 11) into account,

(h(< Ce-4nt2 = C exp [-n2a / 4a2p(n) 2 ] The integrals I2 and I3 are estimated by the methods used for the similar integrals in „ 10 4, to give the bounds

(IS( < C exp [- E5 85n2,/P1(n)2], (s=2,3) (13 2 20) Combining (13 2 13), (13 2.18) and (13 2 20), we obtain

F§(x2)-F§(x1) C exp[-E6n2a/p(n)2] , (13.2.21) for all finite x 2> x l and all sufficiently large n

We now set x 2= np, where p> (2a+1)/4a+3 , and

Sn =S§+Yn Then 3n> np implies that one of the events

`rJ

i np-2' ~rn np-2 occurs, and by (13 1 1) P(Xj >, np -2 ) < C exp{-(np-2)4a/(2a+1)}

and P(Yn >' np -2 ) < e -n ,

so that 1-Fn (x2)= P{S n >1 x 2on 1 -} < C(n+1)e -n (13.2 22) Adding (13 2 21) and (13 2 22), we therefore have

1-Fn(x1) < C exp [-E6n2alp(n) 2] (13 2 23)

It remains only to replace Fn by Fn in (13 2 23) We have

1 -Fn(x1) = P(S n +Yn >, x1Qn#) >, P(S n >,x 1un-1)P(Yn >,0)=

=2P(Sn i x 16nl),

(13 2 19)

13 3

INVESTIGATION OF THE BASIC FORMULA

2 5 1

so that (13 2 23) gives 1-Fn(x1) < C exp [-96n2a/p(n)2]

Since x1= n"/ up (n) this is the inequality (13 2 1) which we set out to prove Replacing Xi by - X

., we also have the inequality P(Sn < -n"+z/p(n)) < cl exp [c2n2"/P(n)2]

Moreover, the argument used in „ 12 2 shows that, for any n1 <n, P(IS§1I > n"+?/P(n) < c3 exp [-c4n2a/P(n)2]

„ 3 Investigation of the basic formula

Having established the basic inequalities, we can now proceed as in

„„ 12 3-7 As in „ 12 3, we write XL = Xi +,J where d i are independent with

diEN(0, n-10)

(13 3 1) Keeping the notation of Chapter 12 and using (13 2 1) we proceed to the formula (cf (12 7 1))

00

1 -F,,(x) = Rne hmn

e-haul/2vdFn(v)+0Pn3 ,

(13 3.2) f (x - mn'/z )/o'/a where

R =

e l (hy) p (y) dy ,

(13 3 3) 00

and pn3 is defined in „ 12 6 Since the case a<-! has already been investigated, we may suppose that

x > n 6 - E for any e > 0 Thus we may take

From (12 8 10), (12 8 18) and (12 8 12) we have

logR=

K

hp yp-+ OC7n-9=LKI(h)+6C,n-9, (13 3 5)

Fn =

p=2

P~

(13.3 6) K

hp-1

E yp

+ OC7 n-8 = mIKI (h)+OC7n-8 , p=2

(P-1)

2 5 2 CRAMER'S SYSTEM OF LIMITING TAILS Chap 13

K

hp -2

6 2 = 1 7p + 9C, n - ' _ a 2[K]( h) + 0C, n - ' , (13 3 7)

p=2 (P-2)1 where the superscript [K] denotes a truncated power series in h, and the

Yp are the cumulants of X; ; in particular

(cf (12 9 2), the notation of Chapter 12 being retained) From (13 3 2) we have

1-Fn(x)=Rne-hmn e-han '/zvdF'n(v)+8Pn3

0

From (13 3 4), h>n- -ZE

If z' = z - 9C, n - 6 , we easily find (cf (8 3 6)),

LK](h)-h

d L

K1 (h) = -'-z'2+z'3~[2K](z')+Bn-3 ,

(13 3 10)

In view of (13 3 5}-(13 3 8) the factor multiplying the integral in (13 3 10) may be written

exp n(L[K](h) - h dh L[K] (h) + 8C, n - 6 (13 3 11)

Now write z = n - x = m and choose h as the solution of the saddle point equation

Fn -z = d L K] (h)-z+6C,n-6 = 0 (13 3 12)

(13 3 13)

where )± K3 is a truncated Cramer series By virtue of the definition of z',

LK](h)-h

dd L

K J(h)= - 1 22 2 +z 3 ) "(z)+Bn -3 (13 3 14)

Y2 = 1+n-2o Thus

m[K] = dh L[K]

(h) + 9C, n - 9 (13 3 8)

Following „ 8 3, we take x = FRO, where

1 < x < n"/P 7 (n) (13 3 9)

1 3 4

COMPLETION OF THE PROOF

2 5 3

Substituting this shows that (13 3.11) is equal to

exp[(-?nr 2 +i 3 A[2K1 (r))+Bn -2 ]

( 13 3.15)

„ 4 Completion of the proof Inserting (13 3 15) into (13 3 10), we have

f

00

1-Fn(x)=exp(-2ni2+ni3At~K](i)+Bn-2)

e-n•"1/z•dFn(v)+

0

+ epn3 (13 4 1) For the computation of the integral we follow the argument of „ 12 9 to give

J0

-e-hvn 1/zv dFn ( v ) = (27r)-'x-1

(1 + 9C11 h)

(13 4.2)

0

Since r=n-2 x, the substitution of (13 4.2) into (13 4.1) gives

3 1-Fn(x) _ (2~)-2 x-1 e -2x2 exp x ~,[2KI x

x

n 2

n2

x (1+Bn -2 )(1+Bh)+6p n3 , (13 4 3)

or, because of (13 3.9) and (12 10.5),

3

1- Fn (x) = {1- 0 (x) } exp x~ ~= K~ x x

n2

n2

x (1+Bn-2)(1+Bh)(1+Bx-1) (13 4 4) But x > n2 and h = Bn- 2 x = Bn"- 2/ p 7(n), so that for the values of x described in (13 3 9), (13 4 4) gives

1-Fn(x) ^, eXp x 3 ~I2K] x )

( 13 4 5) 1-O(x)

n 2

n 2, Since Xi = XX + d we find without difficulty that (13 4 5) holds for the original variables also In view of the definition of K, we can replace

[2K] by (s], to obtain (13 1 2) Finally (13 1 3) is derived by replacing X3

by -Xi

The theorems of this chapter of course contain those of Chapter 12 on normal attraction as a special case