Chapter 14INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE „ 1.. The role of the linear functionals aj, b j was played by moments of the random variables XX.. In the case 0 n = n" p n, the co
Trang 1Chapter 14
INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
„ 1 Formulation
In the preceding chapters we have studied theorems of a collective type concerning large deviations in zones of the form [0, 0 (n)] and [ - (n), 0], where 0(n) = o (n 2) The role of the linear functionals aj, b j was played by moments of the random variables XX In the case 0 (n) = n" p (n), the con-dition
E{exp (A IXXI4a/(2a+ 1)) ) < 00 (14 1 1) appears as a condition for normal attraction ; this implies that all the moments ofXi exist and the probability of a large deviation in XX itself falls off very sharply
In this chapter we study theorems in which x is not restricted to any zone, but allowed to range over the whole real line Thus let X1 , X2 , be independent and identically distributed with
E(Xj) = 0,
V (Xj) = U2>0
(14 1 2)
We shall seek classes of such variables for which collective limit theorems hold which assert that, uniformly in x > 1 as n->00,
P(Z§>x)/0(x, a 1 , a,, n)-* 1
(14 1 3) and
P(Z§<x)/i(-x, b 1 , , b 1 , n)-*1
(14 1 4) Here the limiting tails c depend on linear functionals aj , b ; of F (x) =
P (Xl < x) We remark that the restriction x > 1 is harmless, since in Ix I < 1 the classical theorems hold
For simplicity we shall restrict attention to the case in which F is symme-tric, having a bounded continuous density g (x) such that, for x >, 1,
Trang 214 2 PROBABILITY OF VERY LARGE DEVIATIONS : ELEMENTARY RESULT 2 5 5
P(X1 >x) _
{ 00
g(u)du = r Ar + O( -6 a - E) ,
(14.1 5)
Jx
r-a
and thus
P(X 1< - x) =
-x
g(u)du = Z xr + O(x-6a-E)
(14.1 6)
-Co
r-a
Herea >,3 (since the variance exists), the A,are constants, with Aa> 0,and
s>0 The class of such probability densities we call (A) Such variables have only a finite number of moments, and the role of the linear functionals
aj , biis layed by pseudomoments defined in „ 5 below Theorem 14 1 1 For x >, 1 we have, uniformly in x as n -* co,
P(Z§>x)I (27c)-'
J e
- iu2du+r(x, n+) -+1
(14 1 7)
x
where r(x, n+) is a rational function in both arguments For x>,nZ+a-1+E, n>n o (8),
(2rc)-Z
e-Za2 du+r(x,ni)-nP(X j >axn+) ; x
r(x, n+) is determined by a finite number of linear functionals of the distri-bution of X1 , called pseudomoments
f
This theorem has a collective character since the asymptotic form is determined by a finite number of pseudomoments For x < -1, of course, another analogous relation holds, and for IxI< 1 the classical theorems hold
„ 2 An elementary result on the probability of very large deviations
We shall be concerned with the deduction of the asymptotic forms of P(X1+ +Xn > unZx)
for very large x We begin with the first of these expressions, setting
(14 1 8)
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INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
Chap 14
y = axn 2 and supposing that y > n Thus we consider the probability of the event
X, +X2+ +Xn>y
( 14 2 1)
This can only occur if at least one of the events
X, > y/n
(i 1, 2, , n)
(14 2.2)
occurs These events overlap, but their intersections have small probability
if Y's large More precisely, we shall find values of y for which the probab-ility that two or more of the events (14 2 2) occur is of order
Bi7nny -a ,
(14 2 3)
where i , -*0 as n-> oo For each k >, 2 the probability that exactly k of the events (14 2 2) occur is
P(Sn>YIH1),
(14 2 10)
n ) {
P (X >
} k = B n
k (2Aa) k n ka y/n)
-k
1
My ka
(2A )kn ka+k
The sum over k>,2 is bounded by (14 2 3) if
n ka+k
n
(14 2 5) yka 1< 'In Y
a
since (2A a ) k/k ! = Be -k.
This is equivalent to
y
qn1/(n-1 )ank/(k-1)+1/a
and is certainly satisfied if
y > f1n 1 n 2+a - 1
(14 2 7)
In particular, we may take
y
> Yn = n 2 + a - ' log n ,
' I n = (log n)-1 (14 2 8)
Let H1 be the event {X 1 > y/n} Then in view ofthe discussion,
P(S n > y) = nP(H1 ) P(Sn >yIH I ) +Bq n ny - a (14 2 9)
We now investigate the expression
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PROBABILITY OF VERY LARGE DEVIATIONS : ELEMENTARY RESULT
257
which if
L = y,,/n-' log n
(14 2 11) may be written
P(Sn>ylH1)=P(X1+ +Xn>ylH1) =
For the first of these two expressions we have the inequalities
P(
P(
P(
X2 + 1 + Xn
< L P ( s>n Y
an ©
X2+ +Xnl ant
X2+ - + X n
an`
= P (Sn
>y,
+P(sn >y,
X2 + +Xn an2
ant
< L PCS n >y
X2+ -+Xn
H1,
L H1 +
> LI H i
( 14.2 12)
X2+ -+Xn an2 (1 +o(1)) P(X1 >y+Lan 4 l H i ) , (14 2 13)
H1, X, + +-xn
ant
is independent of H 1 , and implies that for some i,
Xl > yna/log n Arguing as before, and using (14 2 8) and (14 2 7), we have
<L ) <1
< (1 +o(1)) P(X1 <y-Lan 2 j Hi ), (14 2 14)
by virtue of the central limit theorem Further, under (14 2.7), P(X1 >y„Lan 2 I H 1 ) = P(X 1>y„Lan2)/P(X1 >y/n)
=n - all +0(1)) ,
( 14 2 15) because of (14 1 5), (14.2 8) and (14 2 11)
We now examine the second term in (14 2 12) The event X2+ +Xn > L
(14 2 16) ant
> L) < nP Xi > log
)
+ Bq n n y =
g n
= Bn 1-2 alog n=Bn - © - 1 , as a<3
(14.2 18)
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INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
Chap 14
We* now use (14 2 14), (14 2 15) and (14 2 18) to rewrite (14 2 9) in the form
P(Sn>y) = nP(HI) P(X1 Y) (1 +o(1)) _
(HI)
= nP(X1 >y)(1 +0(1))
(14.2 19) Thus, for y >, y,,,
P(S,,> y) = A,,ny - ©(1 +o(1)) ,
(14.2 10) where o (1) is uniform in y as n -+ cc
This simple result has an immediate probabilistic significance ; it asserts that if S§ takes a very large value this is most likely to be because exactly one of the summands is very large ; the probability of S§ being large as a result of an accumulation of moderately large summands is comparatively small
Since the underlying distribution is symmetric, we also have P(S§<-y)=nP(Xl<-y)(1+o(1)) =Aany-a(l+0(1)), (14 2 22) where o (1) is uniform in y >, y,,
± 3 Radial extensions
We now set
whereE< 10 -4 is a small positive constant Because of the previous results
we have, for x >, x,,,
P (Z§ > x) - nP (Xl > xan 1 )
(14.3 2) Since the range x < 1 is dealt with by the central limit theorem, it is suffi-cient now to examine the range
1<x<x,,
(14 3 3)
We shall do this with the help of the analytic method of Chapter 9 Con-sider the characteristic function
00
0(t) _
f- e`txg(x)dx ,
(14 3 4)
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RADIAL EXTENSIONS
2 5 9
which by (14.1 5) and (14 1 6) is differentiable for all t at least (a-1) times
We now introduce the concept of a radial extension of 0 (t) A function ,,/ (t) will be called a radial extension in t >, 0 if is it defined in some neigh-bourhood [ -to , to] of t =0 and coincides with 0(t) on [0, to] A radial extension in t < 0 is similarly defined For example, the characteristic function 0 (t) = e- Itt (ItI + 1) corresponding to the probability density
g (x) = 2/n (1 +x2)2 has radial extensions y(t)=e-'(t+1) in t>,0 and
y (t) =e` ( -t + 1) in t < 0 Both are entire, neither is even
We now prove that, under the conditions here assumed, 0 (t) has a radial extension y + (t) in t > 0 which is everywhere differentiable at least (4a + 2) times, and a similar radial extension y - (t) in t < 0 From (14 1 5) and (14 1 6)
it is immediately clear that it is sufficient to prove that, for any r >, 3, the expression
eit
-1
eir
J1
r
d
+~-
r d
(14 3 5) ma
has radial extensions which are differentiable any number of times It is clearly sufficient instead to consider
0o eit
- 3
o
eit
J br d + ~- br
d
3
a
since J
3 eit
0 r
d
is an entire function For j 3 we can expand -r as a power series in
W = 1+~2 = 1+
~-2 Thus
-r_
Z
00
K
ekWk-I' ek l+ 2
k
+0(~-K-1 )
(14.3 7)
k=r
k=r
The question of the differentiable of the radial extensions of (14 3 5) there-fore reduces to that of the radial extensions of
00
eit4 ~k -00 (1 +Ok
(14 3 8)
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INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
Chap 14
for r <k < K, for if the continuous function p () = 0( -s-1 ), its Fourier transform is differentiable at least (K - 1) times
In the integral (14 3 8) the integrand is rational, non-zero on the real axis, has poles „ i and is of order O (~-"`) at infinity Moving the contour of integration upwards for t>,0 and downwards for t,<0 we obtain radial extensions in the form of entire functions (as in the example) Hence (14 3 5) has radial extensions which are infinitely differentiable, and
° (t) has radial extensions which are differentiable at least (4a + 2) times
± 4 Investigation of the fundamental integral Because 0 (t) is real, the probability density p, (x) of Zn is given by
n
00
Pn(x) = - ~
o(t)ne-an'/zitxdt =
-00
n2
f
00
_
Re
0( t)n e-an'/zitxdt 7r
o Moreover, since there is a bounded continuous probability density g (x),
we have (cf (9 3 7))
2
E0
n
n -an'zitx
E n
pn (x) = -ReRe
(t)n
dt+Be- '
(14 4.2)
f0
We note that fort >, 0, 0 (t) = y (t),and that y (t) is differentiable in the neigh-bourhood at least b >, 6a- 3 times From (14 4.2) we have
Pn(x) =
n2
Re J
Eo
y(t)ne - an'/2i txdt+Be -E'" ,
( 14 4 3) 71
o where y (t) is differentiable b times in [0, g o] In t > 0,
Y (t) _ (t) = 1- it2 +o (t3) ,
( 0 < t <go)
In view of this we find that, for n- 2 log n< t< g o ,
y (t) < 1- 2n- 1 (log n) 2
Y (t)" = B exp (- E2 (log n)2) , and from (14 4 3) that
(14 4 1)
Trang 814 4
INVESTIGATION OF THE FUNDAMENTAL INTEGRAL
n -2
-n - 11ogn
p n (x) =
n Re
y(t)ne-and/2itxdt+B exp [-•2(log n) 2 ]
0
If for t <n- z log n we write
K (t) = log y (t) , then
nz
-n- flogn _ -Re
exp [nK (t) - an' itx] dt +
n
o
+ B exp [ -E2(log n) 2 ] (14 4.5) Note that y (t) is not necessarily even Since it is b times differentiable,
b - 1 (q)
y(t)= 1-2t2 + Y Y 0) tq+Bt b
(14 4 6)
q =3 q
in JtI 5 e o If ltl ,<n - 2log n, nBt b = nB(n-ib log n)b = BEn- 26+ i +E ,
( 14.4.7) and since
exp(B n
-,b+
1+B n -26+1+E E
E
~
we have, writing
Yo (t)
2 + b- 1 y(q)(0)
tq
= 1-2t1 ,
q=3 q
that
K(t) =log yo(t)+BEn-26+E ,
( 14.4.9) since in our interval 2< y o (t) <2
For !tl <n 2 log n,
b - 1
tq
log y o(t) _ -2t2
g q - + Btb ,
( 14 4 10)
q=3
q~
where
gq = [log Yo(t)](q)It-o Moreover,
Bntb = B n-2b+E +E
7
and substituting into (14 4 5) we find that
2 6 1
(14.4 4)
(14 4 11)
(14 4 12)
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INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
Chap 14
We write
b- i
tq K3 (t) _
gq i , q=3
q«
and examine the entire function exp [nK3 (t)]
(14 4.15) For Itl < n-2 log n,
InK3 (01 = BEn 2+` ,
(14 4 16) and if we work to accuracy
BEn-2b+1+E
(14 4.17)
we can ignore [nK3 (t)]b, and write exp [nK3 (t)] = 1 + K4 (t, n) + Bn-Zb+ 1 +E ,
(14 4.18) where.
K4(t, n) = b [nK3 (t) ]q
(14 4 19) Y
q=1
q Substituting in (14 4.5) we obtain
0
n-1/zlogn p,, (x) = - Re J
e - -In` { 1 + K4 (t, n) } exp (- n2 itx) dt + 0
+BEn-Zb+1+E (14 4 20) Substituting ~ = tn2 ,
log n pn(x) = 1 Re I
e-22(1+K4(~n-2, n))e-'4xd~+BEn-Zb+1+E n
o
(14 4.21) and since
n 2
n-'/zlogn
f
b-1
t9 _ ~ Re
exp n - 2t2 + q gq
- un2 itx dt + 0
q=3
q +BEn-,b
(14 4 13)
(14.4 14)
for r<C1,
j logn J
e-Z4z~rd~ = B exp(-4 (log n)2)
(14 4 22) 0
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INVESTIGATION OF THE AUXILIARY INTEGRALS
26 3
1
©©
Pn(x W =
n Re
e -442 (1+K4(cn -Z, n))e-` 4xdx+BEn -+6+i+E,
fo
± 5 Investigation of the auxiliary integrals
In this section we investigate more thoroughly the expression
00
E(x, r) = Re
J0 e- _J42 ~re-i4x
If r is even, then
E (x, r) = 2
00
e --1-42Zr
i4x
-00
is expressed in terms of e- Zx2Hr ©~ (x), where H,(©) is the r th Hermite poly-nomial We also remark that the assumption that g (x) be even is not essen-tial, though it simplifies the calculations
Ifr is odd, then E (x, r)does not fall off so sharply asx-+oo (for a discussion
of this function see [150]) For even r we have, for r bounded, E(x, r) = Bxr e-Zx2
,
(14 5 1) while for r odd,
E(x, r) = (-1)Z(r+1)r_
x-r- +Bx-r-2 .
(14 5.2) Let us now turn to the integral (14 4 25) The terms involving even powers of cn -Z are bounded by B(xn -Z)re -Zx2 and for x5 log n are therefore negligible compared with e- -L2X2 and for x > log n smaller than the remainder term in (14 2 25) Then (14.2.25) will have the form
Pn(x)=(27r) Ze Zx2 +r1(x,n')+BEn -Z6+1+E , where rl is a rational function of x
(14 4 24) 1
©©
_ (27r)-Ze -Ix2 + _ Re [7t
o
e -Z42 K4(~n -4 , n)e- `4xd~+
+BEn-Zb+i+E (14 4.25)
We therefore have to investigate
00
Re e-ZS2 re -i~x d (14 4.26)
J o0
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INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE
Chap 14
Now let
2+a -1 +E
1<x<xn = n (14 3 1) Then, from the last equation,
X n2 pn(x)dx= 1-O(x)+r2(x, n 2 )+BE n -2b+3+E
(14 5 3)
where r 2 (x, n2 ) is a rational function, since r 1 (x, n 2) can be represented,
up to accuracyB E n-2a+ 1 +E,
in the form of a power series in x -1 ,beginning with x-k (k >, 2), which may be integrated term by term to give r2 (x, 172) For y > x n ,
f
00
Pn(x) dx - nP (X 1 > yun 2 ) - nAa/o-a y a n 2a
Y
(14 5 4)
In particular, taking y = xn , nAa/faxan2a = BE n -2' 3+E ,
( 14.5 5) and so, combining (14 5 3) and (14 5.4),
f"o p
n (x) dx = P (Z,, > x) _
X
= 1-~(x)+r2(x, n2 )+BE n -26+3+E,
(14 5 6) for x < xn This formula is also true moreover for x n < X<, n4 , and conse-quently, for such values of x,
r 2 (x, n2 ) nA a /Q©xan2a
(14.5 7)
It is not difficult to see that (14 5 7) is also true forx > n , so that for x > 1,
CIO
P(Z n >x) - (2 )
X
e-2"2du+r(x, n 2 ) ,
(14.5 8)
X
where r is a rational function
We notice that the coefficients of the rational function are expressed in terms of a finite number of the derivatives at zero of the radial extension
y (t) of 0(t) These derivatives are called the pseudomoments of X,
If 0 (t) is differentiable h times at 0 (i.e if h < a - 1) then the first (h - 1) pseudomoments differ from the corresponding moments only by powers
Trang 1214 5
INVESTIGATION OF THE AUXILIARY INTEGRALS
2 6 5
of i The pseudomoments play the role of the linear functionals a j , b;
described in Chapter 2
We remark that similar conclusions may be drawn when the densities have asymptotic expansions as x *co;
P (X, > x) = f g ( u)du =
6a dG (v)
+ 0 61 +
Jx
J a
X
x and similarly for x-* - oo, where G is of bounded variation (but not neces-sarily monotonic)
± 6 An example Suppose that
g (X) = 2
7r(1 +x2)2
so that a =1 and, for t > 0, q5(t) = e-'(t+ 1) Then
log 4(t) = -t+log (1+t),
so that, in 0 < t < l,
K(t) = - t -f- t - 2t 2 +3t 3- 4t4~
= -Zt 2 +K 3 (t) ,
where
K3(t)= -t+2t2+log (1+t), exp[nK 3 (t)] = e-nt+'Int 2(1 + t)n
Thus K4(t) will be a truncation of
e-5nlz+-z52(1 +fin ) _3 +
30 . Now
Re
f
00
e-z52 3e`5x
6
B
d ~ = - +
x5 ,
0
(14 6 1)
(14 6 2)
(14 6 3)
11