Chapter 10MONOMIAL ZONES OF LOCAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS „ 1.. Formulation The theorems to be discussed in this chapter are important generalisations of those of
Trang 1Chapter 10
MONOMIAL ZONES OF LOCAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS
„ 1 Formulation The theorems to be discussed in this chapter are important generalisations
of those of the last chapter There only elementary theorems (Taylor's theorem and elementary results in complex analysis) were used ; here we shall make use of the method of steepest descents
Theorem 7 1 1 shows that, for variables of class (C, d) (i e (d) variables satisfying the very stringent condition of Cramer), the limiting relations (7 1 3) and (7 1 4) are satisfied in the ranges [0, ~(n)] and [ - ( n), 0] so long as f(n) =o (nz) These relations involve the Cramer series ).(z) defined at (7 2 20)
Now let n(Z) =rc o +7r l z+n 2 z 2 + (10 1 1)
be any given power series with real coefficients, with non-zero radius of convergence Let X; be variables of class (d), and let Sn, Zn, r and pn(x)
be as in the last chapter We shall be interested in the possibility of limiting relations of the form
x3 x
~ 1 (10.1 2)
P (x) (2n) 2 exp -2x2 +
n22
n nz and
Pn(-x)/(2rc) 2 exp 2x2
n2 71
n2
-' 1 ,
in 0 < x< n" We shall see that, as before, it is sufficient to consider a< 2
We remark that, if a<6,
x 3 7 x
?r
l
2 -r1 = Bn3a-z =0(1),
(10 1 3)
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so that relations like (10 1.2) and (10.1 3) imply local normal convergence
In the last chapter the zones of local normal convergence were character-ised, so that we may take this case as having been dealt with
Suppose therefore that 6<a<2
For 0<x<n",
x3 7 x
7E =
nz
n
ao
xs+3
n -# E 7Zs
n-i s s=0
FORMULATION
Let s be the unique non-negative integer with
1 s+1 <
1 s+2
2 s+3 ~ a < s+4
It is easily seen that
w
xt+3 n_+
it
= Bn_E
t=s+ 1 n
'
1 9 1
(10 1 4)
(10 1 5)
(10 1 6)
(10 1 7)
where e= 2(s + 2) - a (s +4) > 0, and thus n (z) may be replaced in (10 1 2) and (10 1 3) by the truncated series
s 7Z Es1 (z) _
7Z v Z 2 ~
=o
s being determined by (10 1 6)
(10 1 8)
Theorem 10 1 1 Let 6< a < 2, and define s by (10.1 6) Let p (n) + co as
n tends to infinity, and suppose that
E {expIX,I4"/(2"+ 1)l
< co
(10.1 9) Then uniformly in 0 < x < n"/p (n) as n-+ oo,
and
3
Pn (x) / (27Z) Z exp (_x2
Pn(-x)/(2n)Z exp -2x2 -where A (z) is Cramer's series
AN x +
(0)) 1 , (10 1 10)
3
x
~~SI
\ x
1 ,
(10 1 11) n+
n2
Trang 3192 CRAMER'S SYSTEM OF LIMITING TAILS Chap 10 That (10 1 9) is to some extent necessary is shown by the following result Theorem 10 1 2 If,for all IxI < n' p (n) and all n >, n o , we have
p (x) < exp(-a o x2) , ( 10 1 12) where ao is a positive constant, then (10 1 9) is satisfied (and then by the previous theorem (10 1 10) and (10 1 11) follow)
These theorems are, of course, of collective type in the sense of Chapter 7 ; the moments ofX~(up to order s+ 3) play the role of the linear functionals
aj , b
i. We see that, in the monomial zones, the only possible limiting tails are those determined by the segments of Cramer's series ~(z) Since s-+00
as a >2,the only possible series7r(z) is the Cramer series, whose coefficients are fixed polynomials of the underlying moments Thus not every sequence
of numbers 2 can be the sequence of coefficients of a Cramer series
„ 2 On the condition (10 1 9)
We first show that (10 1 9) follows from (10 1 12) ; the proof follows that
of Theorem 9 2 2 almost verbatim. If (10 1 9) does not hold, there exists a sequence x m- ' oo such that either (9.2.6) or (9 2.7) holds Taking x,,,= unz + "p(n)+0 (101 < 1) we see from (10 1 12) that, for sufficiently large m,
P(in"p(n) < Zn < in"p(n)) < exp [-8ao n2 ap(n)2 ] (10.2 1) The event whose probability is so bounded certainly occurs if both of the independent events
IX2 + +XJ/Gn 2<1 , X m<Xl <2xm , occur, and this has probability greater than
co exp [ -co n2a p (n )4a/(2 a+1)] , ( 10.2.2)
if (9 2 6) holds Thus (9 2 6) (and likewise (9 2 7)) contradicts (10 2 1), since 4a/(2a + 1) < 1 This proves Theorem 10 1 2
„ 3 Derivation of the fundamental integral
We now proceed to the proof of Theorem 10 1 1, assuming as we may that
o =1 and replacing (10 1 9) by the weaker condition
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DERIVATION OF THE FUNDAMENTAL INTEGRAL
1 9 3
E {exp (A IX;I4a/(2a+ 1))}
<00
(10 3 1) for fixed A < 1 We shall begin along the lines of „„ 9 3, 9 4 Note the basic equation (9 3 6), and set
° = 2- a Following the arguments of the last chapter we arrive at an analogue of (9 4 4)
nz f
n-°
Pn(x)
= 27L _n-°
where K(t) has its usual meaning According to (9 4.12), if so is sufficiently small, and Ito l < 2E0, K(q) (t o) = B exp(Bq + kq log q),
(10 3 3) where
k = (1 +2a)/4a
(10.3 4) Moreover, (9 4.14) and (9 4.17) show that, if
m = [n 2"/K,] ,
( 10 3 5) and K1 is chosen sufficiently large, then
K (t) = 1 r K(r)
( 0 ) +
) m+ 1 ! Rm(t) ,
( 10.3 6)
r-0
(
where, for It) < n - " ,
tm+ 1 Rm
(m+1)!
Thus (as in (9 4 16) for Itk < n -'u,
+1Rm(t)
= B exp (-E l n2a) ,
(m+1)!
if K 1 is chosen sufficiently large From (10 3 6),
= B exp [m (B + (k -1) log m - ° log n)]
m
tr
nK(t) _ -2nt 2 +n I Wr - + B exp(-E1n 2a) ,
( 10 3 9)
r=3 r!
and Re(nK (t)) < 0 for Iti < n - " Write
exp(nK(t)-n 4 itx)dt+Bexp(-c 2 n2a),
(10 3 2)
(10 3 8)
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Km(t) _ E Or t - ;
r r=3 then from (10 3 9) we have (cf (9 5 2))
n2 n-u
2
pn (x) _ - exp(-4nt2 +nKm(t)-n+itx)dt+B exp(-E2 n 2 a)
7r f -n-P
„ 4 Application of the method of steepest descents
The integrand of (10 3 10) is an entire function, since Km(t) is a polynomial Set
z=x/n+, 1<x<n«, z<n"-z=n-'`, (10 4 1)
and z=it ; then the integrand of (10 3 10) becomes exp[n(2z2 + Km ( -iz) - zz)] (10 4 2) Here K m (iz) = Km§ )(z) is a polynomial in z with real coefficients The saddle point equation is
z + d K,(n§) (z) = r , (10 4.3) which for sufficiently large n will have a unique positive solution
Z= Z"
='L-2Y3,C2+6(3Y3-Y4)-r3+ ,
(10 3 10)
(10 4 4)
in which the first (m - 2) coefficients will coincide with those of the Cramer series (7 2 20) Write p1(n) = p (n)- 1 and suppose that
1 <x<n"/pl(n) , (10 4 5)
so that
n 1<i <n
-u
The contour of integration z =it, Itj < n -, u may now be given a parallel translation to the contour z = z§ +it, I tj < n - u, so long as we can estimate the integral over the horizontal segments z = ~ ± in -,", 0 < ~ < z § For
3< r < m we have (cf (9 5 5)) on these horizontal segments,
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THE METHOD OF STEEPEST DESCENTS : ITS APPLICATION
1 95
r
K(r)
_
)z) 1 (0)
= B exp r (B +
1 4a
a log r-° log n
(10 4.7) This may be estimated as in „ 9 5 ; when r < C1 it is
Bn"',
(10.4.8) when C1 <rl<E2 log n (10 4 7) it is
B n -84r ,
(10.4 9) and when E2 log n< r < m it is
B e-ESr .
(10 4 10) Thus on the horizontal segments we have
n )K ( ' ) ( z)) = Bn'
= Bn3 a -2
(10.4.11) Moreover, since z= ±in
-Re [n(?z2-•rz)] =2n(~2-n-2°-2•t~)
< in' 2 u= -4n2« ,
(10.4 12) since ~ < n - '`/ p 1 (n) and i <n - °/ p 1 (n)(10 4 6)
Moreover, 2a > 3a - 2, and comparing (10 4.11) With (10 4.12) we see that (10 4 2) is, on the horizontal segments, equal to
B exp (-8n 2 a)
(10.4 13) Thus (10 3 10) transforms into
Pn (x) = 2n
n
zo+in - °
exp[n(2z2+Km(-iz)-iz)]dz+B exp(-ES n 2 a)
zo-in_°
(10 4 14) From (10 4 3)
2z0+K m(-izo )-Tz o = K ;no) (zo) - zo and, on z = zo + it ,
2z2 +Km (iz)-'CZ =
d Km""(z o ) ,
( 10.4 15)
=izo+K (0)(z0) tz o + E it - Kmo)(zo)(it}~
(10 4 16)
Using the estimates (10.4 8), (10.4 9) and (10 4 10) in (10.4 7), we have
Trang 7196 CRAMER'S SYSTEM OF LIMITING TAILS Chap 10
1 d2 2 dz2 K,(n§)(zo) = z + Bz , (cf (9 3 6)) We separate the contour of integrationz= zo+it, I tI <
n-° into two parts,
ItI < n_ 2(1og n) 2 ,
n-"
(log n)2 < Itl < n - ,u
as in „ 7 3 According to (10 4 8), (10 4.9) and (10.4.10) we have, for I ti <n -, u,
11 dz K.n §) (zo) (its' _ J=3J'
= B ItI3
( E n-n(r-3)° + y n-E4(r-3) + L.r
n -ESr
3<r5C1 C1<r1<E21ogn E2logn<r
Thus
m
1 (d\)J exp [n -2t 2 + - Km§)(zo)( it}j dt =
n-Ihlog2n_<jtj n-° j=3 ) dZ
= B
Jn - 141og2n5Itl _< n -IA exp[n(-Zt2 +BItI 3)]dt=
Inserting this into (10.4 14), we have
Pn(x)
= 2n exp[n(izo + K(~§)(zo)- zzo)] x
x
f
(10 4 17)
'" d ; exp (-Znt2)exp n
1 i dz K(m) (zo)(it}' dt ItJ <,n 1/2(1ogn) 2 j=33
+ B0exp n (2z02+ Km§) (zo) -'czo) exp [ - (log n)3] +
From (10 4 18), for Itj <O(log n) 2 ,
n it d Kmo)(zo)(it}' = Bnt3 = Bn-o
.49
j=3 J' Inserting this into (10 4 20) and using the computations of „ 3 3, we have
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COMPLETION OF THE PROOF OF THEOREM 10 1 1
1 9 7
Pn (x) =
- exp [n(izo +K,1 © §1(zo)
-iz0)]
2n
# (1 + Bn-0 48) + (n
+ B exp (-e5 n 2a) _
_ (2n)- exp [n(2zo+Km§)(z0)- zzo)](1+Bn -0.48 )
+
+ B exp (-85n2 )
„ 5 Completion of the proof of Theorem 10 1.1
(10 4.22)
We now consider the expression whose exponential appears in (10 4.22) Using (10.4.8), (10 4 9) and (10.4.10), we have
n(izo+Km§) (zo)) = nK[C1 (-izo)+Bn -1 ,
( 10.1.1) whereKtcl denotes the sum of the first C terms of K According to(10.2 18), n(Ktcl(zo)-izo)=n(-2c2+z3 tCl3 (z)+Bn -2),
(10 5 2) where C 1 becomes arbitrarily large for large C If 1 <x < n§`/p1 (n) and s satisfies (10 1 6), then
nr 3 AEC 1I (i) = ni 3 2ts] (i)+Bn_E
( 10 5 3) Substituting (10 5.1), (10.5.2) and (10 5 3) into (10.4.19) completes the proof of the theorem for 1 < x <na/p( n) The case -n"/p(n)<x,<-1 follows on replacing Xf by -Xj , and the case jxi < 1 is a consequence of the classical theorem