Statement of the theorems The theorems of this chapter do not have a collective character, and are related to Theorem 6 .1 .1.. We shall call such variables those of class C, and disting
Trang 1Chapter 7
RICHTER'S LOCAL THEOREMS AND BERNSTEIN'S INEQUALITY
„ 1 Statement of the theorems
The theorems of this chapter do not have a collective character, and are related to Theorem 6 1 1 We shall consider a sequence of independent, identically distributed random variables XX with
E(X;)=0,
V(X,)=c2>0
(7 1 1) satisfying Cramer's condition
(C) E {exp(a l X;I) < oo ,
(7 1 2) where a is a positive constant
We shall call such variables those of class (C), and distinguish the subclass (C, d) of variables with a bounded continuous probability density g (x), and the subclass (C, e) of lattice variables, i e those taking only the values
b + kh (k = 0, ± 1, ), h being maximal Assuming, as before, that
Z©= (XI +X2+ .+X.)/6nZ ,
we remark that, for (C, d) variables, Z© has a probability density &W,
while for (C, e) variables, Z© takes only the values
x = Xnk = (kh+bn)/cn 2
The local theorems of Richter [147], [148] treat the asymptotic behaviour
ofp© (x) and P (Z© = x©k) respectively We shall consider only the simplest formulation of these theorems, in order to make the proofs reasonably simple (cf „„ 4 2, 4 3)
Theorem 7 1 1 If the variables X3 belong to (C, d) then, for x
x = o (n 2 )
as n -+ oo, we have
Trang 27 2
A LOCAL LIMIT THEOREM FOR PROBABILITY DENSITIES
161 3
P" (z)
Po ( )
=exP
n2 2 n-,)] 11 +0
(7 1 3)
n
Pn(X) = exp
-n2
3 - n [1+0
x
(7.1 4)
Po ( )
n
Here p0(x)=(27r) 1e-'x' and 2(z)=2o+2lz+22 2 2 +
is Cramer's power series, convergent for Izl< E (a),where e(a)depends only
on a (cf (6.1 11)). The construction of this power series will be detailed later
Theorem 7 1 2 If the variables X, belong to (C, e), and x =xnk = (kh + bn)/ ant, then for x >, 1, x = o (n2) as n-+ oo, we have
The symbols po(x), A (z) have the same meanings as before Theorems (7 1 1) and (7 1 2) will be proved by the method of steepest descents
„ 2 A local limit theorem for probability densities
Let the X j belong to (C, d) and denote their characteristic function by
00
~(t) = M(it) =
~-00
e"xg(x)dx
We remark that 4 (t) e L 2 (-oo, oo), i e that
10 (t) 1 2dt < oo
(7 2 1)
Indeed, 1 0(t)1 2 is the characteristic function of X 1 - X2 , which has a bounded continuous density g (x) Then (7 2 1) follows from the following
cn 2 P
(Zn = xnk) = Po (x) exp x3 2 ( x 1 + 0
x
(7 1 5)
For x -1, x = o (0), we have
cn 2 P(Zn
=xnk) = Po (x) exp x
n 2) (7 1 6)
n2
Trang 3
RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY
Chap 7
lemma from the theory of Fourier transforms (for the proof of which see [11], page 20)
Lemma 7 2 1 If a bounded continuous function g (x) E L 1 (- oo, oo) has
a non-negative Fourier transform h(t), then h(t)EL 1 (-co, oc) The relation (7 2 1) permits us to express p" (x) (n>,2) by the inversion formula
Pn (x) = 2rri ~_jxt
~ M (z)" exp (-cn 2xz) dz ,
( 7 2 2)
the integral being taken along the imaginary axis Since g is bounded and continuous, M(z) *0 as z-,~+ioo Moreover,
j M (z) l < 1 for z00 Hence for any s>0, n>2,
00
~M(it)1 2 dt
~r~ >E
I
M(it)I"dt < {1ri(c)}"2
-o0
HereB is bounded and r1 (c) > 0 The right-hand side of (7 2 3) can be writ-ten as B exp (-nri 1 (c)), whereri l (c) > 0 Substituting into (7 2 2) and using the fact that, on the imaginary axis, exp (- cn+ z) j = 1, we have
p"(x) =
2~
M(it)"exp(-cn itx)dt+O{exp(-n1~1(c))} (7 2 4)
It,sE
Because of condition (C) in (7 1 2) M(z) has an analytic continuation to the strip IRe z( < a, which has a power series expansion about z = 0 convergent in tzI <2a-a 1 The integrand in (7.2.4) has the form
We shall suppose that c is chosen so small that c < a 1 and that (M(z)I >
in IzI < c (this being possible since M is continuous and M (0) =1) In IzI < c define K (z) as the branch of log M (z) with K (0) = 0 Then (7 2 5) may
be written as exp {n(K (z) - czz) } ,
(7 2 6) where i = x/n 2 ; we assume that x j 1 Because (M (z) I z in z < e, K (z)
is a regular function of z in this circle, and has a Taylor expansion
cC
K (z) _
y k z k / k ! ,
(7 2 7)
k=2
Trang 47 2
A LOCAL LIMIT THEOREM FOR PROBABILITY DENSITIES
163
where
72=62 , 73=P3, Y4=°4-354 , Y5 = °5 - 1O°352 , etc are the cumulants of Xi and ° j are the moments of Xj Turning now to (7 2 6), we assume that x = o (n4), so that T >0 as n > oo The saddle point equation (see for instance [24]) is
K' (z) -5T = 0
(7 2 8) or
or
2
Z2
Z 3
5T = G Z+Y3
2 + 74 6 + , Z2
Z 3 T=CZ+ Y3 +Y44G +
2c
6
(7 2 9)
(7 2 10)
IfT is sufficiently small, and this will be true for large n, (7 2 10) may be inverted as a power series in T, converging for sufficiently small T This gives the position of the saddle point as
Z = ZO (T) G 2G4 + 3 y36c
2 4G
T 3+
(7 2 11)
(by the rules for manipulating power series) For sufficiently small T, z0 will lie inside the circle jz) <iE = E1, and from (7 2 11) will lie on the positive half of the real axis
We consider the rectangular contour
L1 + L2 + L3 + L4,
(7.2 12) where
L 1 = ( ie1, -iE 1 ) ,
L 2 = (- iE1, ZO - '-'J
L 3 = (z 0 -i9 1 , z 0+ie 1 ) ,
L 4 = (ZO+is1, is 1 )
By Cauchy's theorem the function (7.2 6) has zero integral around this contour, so that, replacing e by E 1 in (7 2 4),
pn
(x) = 27zi
+
+
M (z)" exp (- an') dz +
Lz
L3
La
+0{exp(- nil,(E 1 ))}
( 7 2 13)
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RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY
Chap 7
Because M(it) = exp K(it) = exp(-2 t 2 +0(t 3 )) ,
we have, for E1 sufficiently small, IM(±iE 1)I exp( -ac2E2) Because M is continuous, when z is sufficiently small, IM(z)l < exp(-sc2E1)
(7 2 14)
on L 2 and L4 Moreover, lexp (-cn2z) I = exp (-cn 2Re z) < 1 ,
(7 2 15) and therefore
+
J
= O(exp(-nr72(E1)))
(7 2 16)
L 4
on L 2 and L 4 Moreover, lexp (-un-1z) I = exp (-cn-1Re z) < 1 ,
(7 2 15) and therefore
= O(exp(-ni 2(E1)))
(7 2 16) for '72(E1) > 0 We therefore have, from (7 2 13),
1 Zo+LE1 p©(x) = cn
2ni
exp {n (K (z) -czz) } dz +
Zo-iEi
+ O {exp (-nq (E 1 )) } ,
(7.2 17) where i (E 1 ) =min [171 (E 1), q2(81)1' If z = z o +it, and t is small, then
K (z) -czz = K (z o) - czz o +
K(j)(zo)(ity
(7 2 18)
=2
J• Moreover,
K (z o ) czz o = K (zo) zo Ko (zo) _
-
m 1 TMzo
m=2
Using (7 2 11), we have
K (z o ) czzo =
-
+T3 2 (z) ,
(7 2 19)
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A LOCAL LIMIT THEOREM FOR PROBABILITY DENSITIES
165
where
2
2 (t) - 6G
73
3 +
74
24G6
.
(7.2 20) W
is completely determined by K (z), converges for sufficiently small -r, and is called Cramer's series
The series in (7 2 18) is the Taylor expansion of K (z) about zo, and its radius of convergence is at least 3E=3E1 From (7 2 11),
K" (zo) = G2+ O (zo) > 2 > 2 G2
for sufficiently large n For Jt) < el we have n
Y
K(')
(zo)(its' _ - i nK" (zo)t2 +nO (t 3 )
(7.2 21)
j=2
1
Consider t in the range
n - ' (log n)2 < I t(
8 1
(7.2.22) Because of (7 2 21) we have in this range,
Re n ~
00
K(')(zo)(itY
< - 4nK"(zo)t2 ,
(7 2.23)
j = 2
J'
if 8 1 is sufficiently small Further
~n'/z (tog n) 2 t _< s t
where cl is a positive constant Inserting (7 2 18) and (7 2 24) into (7 2 11),
we obtain
z
272 exp {n (K (zo) -Giz o ) } x
§§
( at
dt +
)'
x
exp ~n E K (j )(zo) ~ l
S jtj-<n - '/z(logn)z
j=2
cn 2
+ O
27r exp n(K (z,,) -a rzo) exp (- c1 logo n) +
+ O (exp { - nt (c 1 ) })
(7 2 25) exp { -4nK" (zo)t2} dt = 0 (exp (- c1 log n)), (7 2 24)
Trang 7166
RICHTER'S LOCAL THEOREMS
; BERNSTEIN'S INEQUALITY
Chap 7
„ 3 Calculation of the integral near a saddle point
From now on B will denote a bounded quantity, not necessarily the same from one occurrence to another If Itf <n - I(log n) 2 ,
00
E K(j)
(zo) ( ) _ -2nK" (zo) t 2 + 6nK,,, (zo)(it)
3 +Bn - 1
(log n 8 ,
I
) and
6nK,,
(zo)(it) 3 = Bn- (log n)6
so that
,
exp n E
00
K(j ) (z)o (it)' ldt =
/
J(tjEn
-f '/z (logn)2
j=2
jl
_
exp(-iK"(z o ) tn 2 ) x
fj t j Win- 1/z (log n)2
f1t1n
x (1+6nK"' (z o )(it)3+ Bn-1 log 8n) dt =
-'
exp (-ZnK"(z o )t2) x
/z(logn)2
x (1 +Bn-1
log 8 n)dt =
5
00
exp(- 2nK"(zo)t 2)dt+Bn-2 log 8n (7 3.1)
00
The integral is equal to (2n/nK"(zo))+ ,
(7 3 2)
so that (7 3.1) is equal to (27r/nK"(z o ))2(1 +Bn-1 log 8 n)
(7 3 3) Thus the first term on the right-hand side of (7.2 25) is equal to
a(2itK"(zo))-I- exp In (K(zo) -aTzo)}(1+Bn-1
log 8n) ,
(7 3 4)
or, because of (7.2.19), a(2nK"(zo)) 1 exp { -ini2+nT32(T)}(1+Bn-1 log 8n)
(7 3 5) Furthermore, (7.2.7) gives
K" (zo) = U 2+ Bz o = 0- 2 + BT
( 7.3 6)
Trang 87.4
A LOCAL LIMIT THEOREM FOR LATTICE VARIABLES
1 6 7
Substituting into (7 3 5) and noting that (1+Br)(1+Bn -1 log8n)= 1+O(x/n1'), (7 3 5) becomes
(27r)- -1 exp { in r 2 +ni t + ni 3A(z) } (1 + O (x/n*))
( 7.3 7)
We now remark that, for z = o (1), -2ni 2 +n' 3 a.(.r) < -4n T 2 < - 2n q (s1) , n+ exp (- c1 logo n) = O (x/n+) ,
so that (7 2 25)and (7 3 7) combine to give (7 1 3) To obtain (7 1 4) replace
Xj by -Xj ; Theorem 7.1 1 is proved
We shall make a few remarks about Cramer's series (7 2 20) It is easy to verify that the first k coefficients of this series determine the first (k+3) moments of Xj (assuming that EXj=O and that 62= VXj is known)
In fact if these coefficients are known, we have the first (k + 3) terms of the expansion of
K (z o ) - Qrz o = K (zo) - zo K' (zo)
in powers of 'L Hence from (7 2 9) we can determine the cumulants y,n (m < k + 3)and hence the moments /1m (m < k + 3) The argument reverses ;
if°3, , °k+3 are known, then 2o, A1, , 2k-1 are determined
„ 4 A local limit theorem for lattice variables
We now proceed to the proof of Theorem 7 1 2 We introduce
00
M (z) = E(ez Xi) =
Pkexp[(kh + b) z] ,
(7.4 1)
k=-co
defined in IRe zi < a because of (7 1 2), and periodic with period 27ri/h Write
n
Sn = E Xj ,
j=1
P (k) = P (S n = kh +bn) ,
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RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY
Chap 7
(these being the only values taken by S") Then, if IRe zj <a,
x
m(z)n = E P" (k) exp [z (kh + bn)]
(7 4 2)
k=-crj
For any c in -2a < c < 2a, multiply (7.4.2) by exp [ -z (k o h + bn)] to obtain (after writing k for k o ),
c + in/h
P" (k) =
27[1 f
-
M (z)" exp [ -z ( kh + bn)] dz
(7 4 3)
c u t/h
Writing x =xnk=( kh + bn) / an', we have
h
c+in/h
P" (k) _ 2
-iri j
M (z)" exp (-zan z x) dz
( 7 4 4)
c-in/h
We now remark, that, for It{<7r/h, t =A 0, the strict inequality
IM (c +it)I < M(C)
(7 4 5)
obtains The weak inequality is obvious, and if there is equality we must have
ekhit = 1
(7 4 6) whenever pk0 0, which contradicts the maximality of h
Assuming that x > 1, x = o (nZ) and keeping the notation of the previous sections, we find that the saddle point is at z o , determined by (7 2.10) For sufficiently small e l , we take c=z o and study the integral
h f`O+1`1
M (z)" exp (-zu rn) dz
( 7 4 7)
27ri z -it,
This differs from (7 2 17) only by a factor ant/h, and consequently, according to (7 3 7), is equal to
for E I t< 7r/h, where J(E I ) is a positive constant not depending on z o
Further, according to (7 2 19),
M (z o )" exp (-z o a rn) = exp { -Zn r 2+ni 3 )? (i)} (7 4 9) Because of (7 4 5) and the continuity of M we have
Trang 107.5
BERNSTEIN'S INEQUALITY
1 69
Hence
SEiSItISrz/h
IM (z o+lt)I'Iexp(-z6in)I ldzI =
=BIM(zo)In exp( - zorin)(1 - il(E1))n
(7.4.11) Since x>1, x/n>(1-rl(E1))' for sufficiently large n, and this, together with (7 4 9) and (7 4 8) gives (7 1 5) ; (7 1 6) follows on replacing Xj by -XX
„ 5 Bernstein's inequality
We have remarked before that useful results of the theory of large devia-tions are not always asymptotic expansions, but are sometimes inequali-ties These are particularly useful if they admit effective computation, and
if the constants in them are best possible, or nearly so Important among these is Bernstein's inequality ([7], pages 161-165)
We assume that the independent random variables X1 , X2 , satisfy
E (Xl ) = a 1 ,
V (Xl) _ #j ,
( 7.5 1) and write
ZI =XI -a1 , Bn=/31+fl2+ +/3n,
Sn = Z I +Z2+ +Z n
Proof It is sufficient to prove the first of the inequalities (7 5 3) ; the other two follow from it in an obvious way From (7 5 2) it is clear that
Theorem 7 5 1 Suppose that, for some H > 0 and all k >2,
E(Zk) <
Then,for 0 < t < 2BnH-1 , P(S,, > 2tBn) < e_t2 , P(Sn < - 2tBn) < e - '2 , (7 5 3) P(ISnj > 2tBn) < 2e - t2
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RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY
Chap 7
E exp (yZi) < oo
(7 5 4)
if IyI < H -1 Take 0 < y < ( 2H) - ' Then
I (T)= E exp [y (Z 1 +Z2+ -+Z,)] = E exp (ySn) Consider the inequality
exp (TS ) > e`2I (y) ,
(7 5 5)
or
exp (TS,,)/I (y) > e`2
( 7 5 6)
Since the left-hand side of (7 5 6) has expectation 1, Chebyshev's inequality shows that this inequality has probability at most e- ` 2 , i.e
PITS,, > t 2 +log I (y) } < e_,2 .
(7 5 7)
But E(eyz') < 1 +
kl 2~,H k-2kI <
k= -2
< 1 + y 2f3i < exp (y2
so that
n
I (y) _ F1 E (e yz ') <
j=1
(7 5 8)
< exp (y2 Bn)
( 7 5 9)
Thus P(S n > t2 +y 2Bn)< P(yS,, > t2 +log I(y))e - ` 2
( 7 5 10)
Now take y = tB,~ - 1 , so that yH <2 by the condition assumed of t Then
from (7 5 10) we deduce that P(S,,>,2tB©-1)<e-`2,
(7 5 11)
and (7 5 3) is proved