Independent And Stationary Sequences Of Random Variables - Chapter 11 doc

Classification of narrow zones by the function h We retain the notation of the last two chapters, and record here some newterminology.. Statement of the theoremsWe shall investigate the

Chapter 11

NARROW ZONES OF NORMAL ATTRACTION

„ 1 Classification of narrow zones by the function h

We retain the notation of the last two chapters, and record here some newterminology The narrow zones [0, A (n)] and [-A (n), 0], where A(n)

is continuous and increasing and A(n) = o (n b), will be described in „ 2

by means of a function h (x), non-decreasing and continuous in x>,2

It will turn out to be natural to distinguish three classes of possible tion h

func-Class I : This consists of functions h satisfying (for some ~o >0),(log x) 2+lo < h(x) <,X 2

(11 1 1)

If we write

h(x) = exp {H(log x)J ,then H(z) is required to be monotonic and differentiable, withH'(z) < 1, H'(z)-+0

(z *oc) ,

(11 1 2)H'(z) expH(z) > clzi +~'

(11 1 3)Class II : This consists of the non-decreasing continuous functions withP0(x) log x < h(x) < (log x)2 ,

h (x) = M(x) log x = N (log x) log x,

„ 2 Statement of the theorems

We shall investigate the narrow zones of local and integral normal vergence for variables of the class (d)defined in Chapter 9 In terms of thefunction h (x) we define A(n) by the equation

Theorem 11 2 2 The statement of Theorem 11 2 1 remains valid if theword "local" is replaced by "integral" throughout

If h belongs to Class II we define

A(n) _ {h(n)J4 = {M(n) log n}l- ,

(11 2 3)since, under the conditions defining this class, this differs only by a slowlyvarying function from that determined by (11 2 1)

Theorem 11 2 3 If h(x) belongs to Class II, the statement of Theorem

11 2 1 continues to hold

Theorem 11 2 4 If h(x) belongs to Class II, the statement of Theorem

11 2 2 continues to hold

Theorem 11 2 5 For h (x) in Class III and A (n) = {log n} the statementsofTheorems 11 2 1 and 11 2 2 continue to hold

„ 3 On the conditions imposed upon h(x)

We first comment on the conditions which the different classes, in cular Class I, impose on h (x) The inequality h (x) >,(log x) 2+~§ ensuresthat the zone is not too narrow ; it may be noted that, in the particularcase h (x) =(log x) 3 , ( 11 2.2) simply says that the Xj have finite third mo-ment On the other hand, h (x) < x+ implies that the zones are narrow inthe sense of Chapter 9 It should also be remarked that A (n) = n" cor-responds in (11 2 1) to h(x) =x4a/(2a+ 1), and 4a/(2a+ 1) < i for a <6 It isnatural to assume h to be monotonic and differentiable, so that H isalso If we also assume that h' is monotonic, this leads to (11 1 2) and, inview of the left-hand inequality of (11 1 1), to (11 1.3)

parti-„ 4 The necessity of (11 2 2) for Class I

Suppose that [0, A (n) p (n)] and [ -A (n) p (n), 0] are zones of normalattraction Then for n > n o we have

P (Z© > iA (n) p (n)) < exp(-16A (n)2 p (n) 2) ,

( 11 4 1)since A(n)-> oo as n-+ oo Suppose that (11 2 2) is not satisfied Then wecan find a sequence x m->oo such that either

P(X1 >x m ) > exp(-2h(x m )) ,

( 11 4 2)or

P(X1 < - m) > exp(-2h(-xm))

( 11 4 3)Suppose for instance that (11 4 2) holds For sufficiently large m choose

n so that

x m = an' A (n) p (n) + 0 ,

101 < 1

For sufficiently large ~, q,

h(cry) = exp[H(log gt + log ij)] < exp [H(log )] + o (log j) _

„ 5 The sufficiency of (11 2 2) for Class I

We now proceed to the proof that (11 2 2) is sufficient for [0, A(n)/p(n)]

and [ -A (n) / p (n), 0] to be zones of local normal attraction Indeed weshall prove more generally the sufficiency of

202 NARROW ZONES OF NORMAL ATTRACTION

Chap 11

local normal attraction, since [0, A (n)/ p ( n)] and [ -A(n) / p( n), 0] arenarrower Indeed, if we write A P (n) = A (n) Y (n), the arguments of „ 4applied to (11 5 1) and (11 2 2) show that

g n

11 6

INVESTIGATION OF THE FUNDAMENTAL INTEGRAL

2 0 3

„ 6 Investigation of the fundamental integral

In the notation of Chapter 9, we have the equation (cf (9.3.7), (9 4.5))

n2f

n_°

p n (W) =

0(t)nexp(-itxn4)dt+B exp(-c 1 e 2 X§'~n~) (11 6 1)27c _n - °

In order to use the method described in Chapter 9, we need to estimate4) ()(t) in I ti < n- ° We have

J

00

( 11 6 2)

Moreover,lim xh'(x) = oc ,

204 NARROW ZONES OF NORMAL ATTRACTION Chap I I

For x>,x1,h(x)-q log

i = B exp {Bq+q log Q o -h(Q 0)} exp(2q) 1+ ~§+B =

which proves the

h(Qo(q)) - log q

be non-decreasing, since the equation

(11 7 1)

Q0(q) h'(Q0 (q))implies that

L(q) =q-1{q-1h(Qo(q))-1} • (11 7 3)

Thus L(q) is non-decreasing in q From (11 6 9) we have0(q) (t§)

4pcgc(to) tq =

B exp(B - C1 q - qL (q) + qL (q)) =q!

log Q 0 (m) = H -1 {2XP (n)}

(11 7 11)Then

exp H {log Q0 (m) } = I {Qo (m) } = e 2 '' ("1

( 11 7 12)Substituting this in (11 17 10) and taking account of (11 5 8), we through(11 7 10) into the form

B exp[m(B-p 2 (n)+2XP (n)-logm-m -1 exp2XP (n))]=

= B exp[m (B -p2 (n)-exp 2XP (n) -log m) + 2XP (n) -log m]

We now show that

so thatexp [H(log Q0 (m))] H'(log Q0 (m)) = mThus, from (11 7 11),

11 8

We remark that, because of (11 5 6),exp (2Xp(n)) = eB n1 - 2°

„ 8 Investigation of K(t)From (11 7 10),

tmsup K (m)(t)

'Y r = K(r)(0) = B exp (Br + r log Q0 (r) - h (Q 0 (r)))

( 11 8 3)Since

2X,(n) >, H (log n/p (n)) = log h (n4/ p (n)) ,

we haveexp (2Xp (n)) > h (n4/ p( n)) > [log(n4/p (n))]2+ '§ ,

208

NARROW ZONES OF NORMAL ATTRACTION

M

xh' (x) > (log x)1 + ~' ,

so that, from (11 6 4),{log Qo(r)}1+~' < r,

proving (11 8 12) Thus (11 8 9) gives, in this range,

11 9

MORE INVESTIGATION OF K(t)

= B exp [r (B + r

0 48 log n)] _

= B exp [(-0 47 log n)r] = Bn-0 ' 47 r

(11 8 13)Summing these expressions over C 3 < r,< (log n)' +'~' gives

Bn_2

( 11 8 14)Finally, consider the range

(log n)1+4'<r<m

Because L(q) is monotonic, (11 8 9) does not exceed

Bexp[r{B+logQ o (m)-m-lh(Q 0 (m))-logm-° 1 logn}] (11 8 15)

For r,> m, 1 tI ,< n "`,

Comparing this with (11 7 10) and (11 7 13), this becomes

B exp [r {B - (2m)-1 exp(2Xp (n)) - p 2 (n) +(On log n} ] (11 8 16)From the definition of con, this is

210 NARROW ZONES OF NORMAL ATTRACTION

~; c ;

tr

00

r_

exp {nK3(t) } = 1 + E Xr1r + B exp (- c 4 exp 2XP (n))

r=3 r and therefore

e-~2 Xr

~rd~

r 3

+ B exp { -C6 exp [2XP (n)] } (11 9 7)

(11 9 8)for which the methods of „ 9 5 give the estimate

B exp [-n4 1-2 °] exp[r{B+2 log r - (2 -1u1) log n}]

=B exp(-n 1-24 u) exp[r{B+2log m-(2-,u 1 ) log n}] (11 9 9)Since

Bexp(-4n 1-2 "`)= B exp{-c 4 exp(2XP(n))}

(11 9 10)

11 10

COMPLETION OF THE PROOF OF THEOREM 11 2 1

A similar analysis can be made of

„ 10 Completion of the proof of Theorem 11 2.1

We now investigate, for 3 < r,<- m, the integral

00

J

e-~~2e-`~x~rd =Hr10)(x)e-2x2

- co(cf. „ 10 6) The sum of the terms in (11 9 11) with 3 < r <C3 isBe-2x2

p log q - p (1- 2°) log n = B - v (n) ,

2 1 2

NARROW ZONES OF NORMAL ATTRACTION

log q < -2L(1 -2°) log n = (2-°) log n

(11 10 8)(remarking that (1-2°) log n-* oo) Then (11 10 5) has the estimate

sup {B exp [-q(1 -2p) log p 1 (n)]} + B exp [-qp(2-°) log n] _

P

=B exp [-gp3(n)] (11 10 9)Summing over 3< q< p 2 (n), we have the estimate

B/p4(n)

(11 10 10)This proves that [0, n2-1"1p1 (n)] and [- nZ- R/p 1 (n) ] are zones of localnormal attraction, and completes the proof of Theorem 11 2 1

„ 11 The corresponding integral theorem

Consider first the monomial zones [0, n"] and [-n", 0], where a< 6is aconstant It is not difficult to go from this to the general narrow zone

We introduce auxiliary normal variables Y© with zero mean and variance

n 2 a, which therefore have characteristic functions

exp (-2n2a t 2)and set

Z;, = (S,,+ Y©)n - Let

we show that, if this is a zone of normal attraction for Z©, then it is a zone

of normal attraction for Zn We haveY©EN(0, n"),

P(n S©>x) i P(n S©+n-+Y©>x+n2«-Z In -ZI1,©I<n2a-2)and 2a-2<3-2<0

Suppose that, under (11 11 2),

00

-j

P(n - S©+n-2 Y©>x)

P(n -2 S©>x) (1+0(1))~

x

~ +nz '/z

e-2"2 du+o(1) f~e-2"2du

a

2< n-'< n-ESx -1

(11 11 9)

P(n-ZSn >x) < (1+o(1))(2ir) 2

„ 12 Calculation of the auxiliary limit distribution

Take 1 < x1 < n"/ p (n) and x 2 = n K, where K is a positive constant to bechosen later, and write

1-Fn (x) = P(X i + +Xn + Yn >xn2)

(11 12 1)The event on the right-hand side implies that at least one of the events

XX>2x2n-2, Yn > 2x2 n-2 occurs, and by (9.2 1), (11 11 3) and (11 11 9),for sufficiently large K, its probability is bounded by

We have

i

oo

e-onl/2itx2-e-an'/2itx1Fn(x2) -Fn('X1)

4~n(t) = exp (-2n 2 « t2) = B exp(- Zn 2"EO) ,

(11 11 13)

i n

exp [-?(n+n2a)t2] exp n

= 27r n

°

exp

C - i(n + n 2a) t 2 ] (1 + j xr~ r x-n-

r=3

x t -1 {exp(-an+itx2)-exp(-an'itx1)}dt+B exp(-c 7 n2a)

(11 12 7)Setting = tn2 , this becomes

Fn (x2)-Fn(xl) =

i

= 2

~~ exp[-i(1+n-2°) 2] 1 + Z r n2r

x

00

2 1 6

then

F©(x2)-F©(x1)= I

~exp(-zv 2 ) ( 1 + M

yrlr 1+P r 27r ~o- 00

As in „„ 5 6, we must first estimate

NARROW ZONES OF NORMAL ATTRACTION

( 11 13 6)

11 14

COMPLETION OF THE PROOF OF THEOREM 11 2 2

But

Jxl n'`-ZI < 1/P(n)

so that (11 13 6) isB

the sum in (11 13 2) is, for 1 < x< n"/ p (n) and fixed t,

B exp [Bq+q(1-2p)-2t-1)(a log n-log p(n))+

-pq log q-pq log p-(1-2p) q log q-q(1-2p) log (1-2p)+

+q log q+2t log q+2t log (1-2p)]

(11 14 1)

The term corresponding to t is thus

B exp[t(log q - 2a log n + 2 log p (n))] _

= B exp [t (log m-2a log n+2 log p(n))]

(11 14 3)

In thislog m = 2a log n -log p o (n) +0,

101 < 1

If we take p o (n) so thatlog p o (n) > 2 log p (n) ,

A similar argument for [- nap (n), -1] completes the proof of Theorem

11 2 2 in the special case of monomial zones We now proceed to thegeneral case

„ 15 The general case of narrow zones

We now follow the argument of „„ 11-14 to prove Theorem 11 2.2 Let

XXbe random variables with E (Xi) = 0, V (X;) = 1, and suppose that(11 2 2)

is satisfied, where h (x) is a function of Class I We begin by following „ 3 ;determine ° from (11 3 6) and set

(11 14 5)

(11 14 6)

x,

(11 15 3)For 1 < x 1 <n"/ p6(n), Q = q - 2s = q (1- 2p), we easily find (for example

by the method of steepest descents) the estimate (Q >, 1),

Now let x 1 < m 2 Then (11 5 4) is estimated asBQxQ-1 e - 2x2 e2Qlo9Q

to give an error term of

B exp [q(B+ 1-2p)z log m- p log q-(1-2p)log q+

+log q-p log p-(1-2p) log(1-2p)-Z log n+pl log n)] =

= B exp[q { B +(2 -p) log m + p log q - p log p +

which shows that [1, m 2] is a zone of normal attraction This completesthe proof of Theorem 11 2 2

„ 16 The transition to Theorems 11 2 3-5The remaining theorems refer to the "very narrow" zones Functions ofClass III satisfy

3 log x < h(x) < M log x,

(11 16.1)and (11 2 2) implies the existence of third moments, but not that ofmoments of all orders In this case it is possible [4] to establish by classi-cal methods that [0, (log n)2/ p(n)] and [ - (log n)-'/p (n), 0] are zones oflocal normal attraction for variables in (d), and of integral normalattraction in general, and that [0, (log n)l p (n) ] and [ - (log n)4p (n), 0]

will not be so unless all the moments exist These assertions, which prise Theorem 11 2 5, can also be proved by the arguments describedbelow

com-We shall, however, confine ourselves to functions h (x) of Class II, i e those with

(11 16 7)Let Q (q) be the solution of the equation

h(x) = (q+4) log x,

(11 16 8)M(x) = q+4

J Qq)

and

f

00exp(q log x -h (x))dx = BQ (q) exp [q log Q (q)] _

0

= B exp [(q+ 1) log Q(q)]

221

(11 16 10)

( 11 16 12)Following „ 6, we find that

K(q)(0) = B exp [(q+ 1) logM -1 (q+4)] ,

(11 16 13)and

sup

°

tmK(' )(t) -

X (n) = log A (n) = 0 (log log n)

( 11 17 2)Let r =10 - 6, and choose m by the condition

log M -1 (m+4) = lcu log n+B

(11 17 3)Thus

= B exp [log M -1 (m + 4) - °m log n] (11 16 14)

(11 17 1)

11 17

To study the entire function exp (nK3(t)) we set

224

NARROW ZONES OF NORMAL ATTRACTION

Now

andlog r-(e-° 1 ) log n<e log m-u 1 ) log n< -4 log n ,

( 11 18 5)because of (11 17 4) Therefore (11 18 3) is

11 18

COMPLETION OF THE PROOF

225

from (11 18 6) ; the sum of the remaining terms with 3 < r < C 3 will then be

Here log r<log m=B log log n (see (11 17 4)), so that (11 18 9) is

Bd0' -°')

( 11 18 10)Summing over C3< r < m gives an error

Bn-2 ,

( 11 18 11)

so that (11 18 6) givespn(x)=(27r)-'e-zx2(1+o(1))

(11 18 12)This proves Theorem 11 2 3

The corresponding integral Theorem 11 2 4 is proved exactly as in „ 15,the rough estimates derived in „„ 16-18 being sufficient for the purpose

It is important to note that, since

A (n)2 = M (n) log n > po(n) log n,

we havenx+1 exp(-c 11 A(n)2)=Bexp(-c 12 A(n)2),

(11 18 13)and we can argue as in „ 15

Theorem 11 2 5 is derived by classical methods, the asymptotic expansions