Large deviations by poisson approximations

0 for CY>p. By setting t = t* in (3.2), we obtain readily the following equalities: P(S,3na)=exp

>C k>na

e-(k-m)l*p(S;*

=

/lo,

(3.7)

80

P. Deheuvels / Large deviations by Poisson approximations

and P(S,
c

Lj(t*)-nCXt*

i j=l

,,(x-na)l*p(S;*=q*

(3.8)

k
From this point on, the classical theory of large deviations uses a normal approximant for F,‘*(y) =P((SL*-na)/a,(t*)
function

and

density,

respectively,

k~ue-(k~n(l)‘*p(s~*=k)= /

bv

this approach

leads

to approx-

rme~Y/*u,;““‘d~~*(y)

(3.9)

-0

-_I

I‘@=

e-Yt*%,(r*)d@(y)

1- @(t*a,(t*))

=

for P
1,

(3.10)

@GHt*o,(t*))

to and

e-Y%,(f*)d~;*(y)

k~~ae-‘k~“a”*P(s:,*=k)=

80

!

e-Yl*‘&,((*)

d@(y)

=

1- @(-t*o,(t*))

(3.12)

fziqqt*a,(t*))

. --m

(3.11)

for o
The fact that t*>O in (3.10) while t*
that

,Im II
e~y’*ullc’*)(dF’*(y)-d@(y)) n

<2d,(SA*,

UA*)

for p
(3.13)

and ‘0

e-“*“~l(“)(dF,‘*(y)-do)

<2dK(SA*, (/A’“) for O
(3.14)

IL where UA* denotes a random variable following a normal N(nM,(t*),a,(t*)) distribution. Combining (3.9)-(3.14) with (2.9) yields the following theorem. Theorem 3.1. For any j?<(r<

1,

1- @(t*a,(t*))

P(S,>na)=

i *@(t*%(t*))

+~,‘(a)

n C Lj(t*)-nat*

exp 1

(

j=l

>

(3.15)

,

and for any O<(r
1- @(-t*a,(t*))

,

(3.16)

fiW*%(t*)) where, for A as in (2.9), ly,‘(a)l$2Aa,(t*)-‘.

(3.17)

Remark 3.1. Since lim

Z-m

41 - @(z)) @(z)

=

ly

(3.18)

P. Deheuvels / Large deviations by Poisson approximations

it follows from (3.15)-(3.18) that if It*1 + co and on(f*)+ assumptions leading to (3.15) and (3.16), respectively, P(kS,

3 em)

=

1+0(l) exp

t*(T,(t*)1/ZfE

i

c

81

03, we have,

Lj(t*)-nCYt*

J=I

under

the

(3.19)

. >

In particular, in the i.i.d. case wherep, = ... =pn =a, we obtain t*=log[(cr(l @(l-o))], (~,(t*)=n~(l-CZ), and Lj(r*)=lOg((l-p)/(l-cr)) for j=l,...,n. follows that (3.19) holds when (Y+ 0, p+O, na+ ~0, together with p/(r+ p/u+co.

-p)/ It 0 or

Remark 3.2. Theorem 3.1 is far from the best result one could obtain here, and serves mainly to illustrate the method. By taking care of the lattice structure of S, (see e.g. Petrov (1965, 1975)), sharper evaluations of the coefficients of exp( Cy=, Lj(t *) - nat*) in (3.15) and (3.16) can be made. However the corresponding results fail to give any reasonable estimate for P(tS, 2 +na) when the asymptotic normality of Si* does not hold, i.e. when a,(t*) it co. If it is the case, then the most reasonable approximant for the distribution of Si* is given by a Poisson distribution. This motivated the sequel, where such an analysis is achieved. We will make use of the following t*= t,:(a) is as in (3.6). Let G=Ji,

(Ppi,

notation,

At=&

;Ik*=g

cr~(O, 1) is arbitrary

for k-1,2,...

( l_;+‘;.e,)k J

where

and

and

-m
J

for k= 1,2, . . . and A*=A~=na.

(3.20)

Denote by T,’ a random variable following a Poisson PO@‘) distribution, and set T,,*= T,‘* (note that T,,* follows a Poisson Po(ncx) distribution). Set further dk=d,(SA,

T,‘),

dL=d,(SA,

T,‘),

di=d$

and

d,“=dc.

(3.21)

Let also, for pu>O and A >O, (3.22) and P-(/4~)=

c O
where Z7(p) denotes The main theorem proximations. Theorem

E

emV =

P(I7(p)


(3.23)

k!

a random variable following a Poisson PO(U) distribution. of this section, stated below, gives the basic form of our ap-

3.2. For any p
1,

P(S, 3 na) = {p+ (na em’*,e’*)+d,+(a)} exp

jJ L,(t*)-na(l-em’*) j=l

,

>

(3.24

82

P. Deheuvels

/ Large deviations

by Poisson

approximations

exp

Lj,(t*)-nU(l-eP’*)

and for any 0 < cz
i (

where IS+(a)/ <2d; <-

exp(-na(e-‘*-

lt

(1- ePa) exp(-ncr(ee’*-

By (3.2) taken

with t=t*,

(3.26)

1+ t*)).

we have for p
L,(f*))

+exp(i]

1,

k~ueCk’*(P(S~*=k)-P(T,;‘=k))

=:D,+D,. By (3.22), we obtain

readily

(3.27) that

D, =P+(ncx eC’*,e-‘*) exp

i (

while,

(3.25)

t*))

ncx Proof.

,

>

j=l

in view of t *>O, a summation

Lj(t*)-na(1

-em’*)

(3.28)

, >

j=l

by parts yields

ID,I<2d,*exp(-ncut*)<2dcexp(-ncrt*).

(3.29)

Combining (3.27), (3.28) and (3.29) with (2.5) taken with A =I*= n and A2=,l,* yields (3.24) and (3.26) in the ‘ +’ case. A similar argument used for P(S,
i=log(E)

and

(3.30)

i=log(z),

where p=n-tji,p,

and

j=l-[n-ljt,&jm’. J

(3.31)

83

P. Deheuvels / Large deviations by Poisson approximations

Since M,( .) is nondecreasing, analysis

it follows

from

(3.30)-(3.31)

and by elementary

that t^< T< t

d
and

M,,(f)
(3.32)


It may be verified that ?(respectively ior t^, is the value of t* obtained when S, follows a binomial B(n,b) (respectively B(n,P) or B(n,$)) distribution. In general, t * is related to < T and t^ via the following inequalities. Theorem

For any O
3.3.

1, we have

0<7
(3.33)

while for any 0 < a

(3.34) pi, . . . , pn, there does not exist a simple expressince M,(t) and ML(t) are easily computed for to evaluate t* by a numerical algorithm (for solving M,,(t*) = a in either of the intervals bounds are sharp in the sense that, in the i.i.d. ?=i=t^=t*.

Remark 3.3. For unequal values of sion of t* in closed form. However, each value of t, it is straightforward instance of Newton-Raphson type) given by (3.33) and (3.34). The latter case where p, = ... =p,,=p, we have Proof. similar

We limit ourselves to the proof of (3.33), since (3.34) may be obtained arguments. Set xJ=pj/p and y,=fi/p, for j= 1, . . . ,n, and let G(u,, . . . . u,)=a(l-Pi)

i

j&l

uj

by

(3.35)

l-a+Uj(a-Y)’

and H(U,, ee.9 U,)=a(l-f?)

i j=l

1

(3.36)

(l-a)Uj+a-fi3

By the Lagrange multiplier method, we see that the extrema of G under the constraint C,“=, uj = n are obtained by solving in u,, . . . , u,, and A the equations

a

UJ

-(

-A=0

l-a+Uj(a-p)

f3Uj

forj=l,...,

n, and

i

Uj=n.

j=l

(3.37)

Since the first n equations of (3.37) imply that ul = ... =u,, it follows that G(l, . . . . 1) = na is an extremum. To see whether it is a maximum or a minimum, it suffices to compute the corresponding Hessian matrix, which is diagonal since (a2/dui auj)G = 0 for i # j, the diagonal terms being given by a*G(l, au:

. . . . 1)

2a(l -a) = -

(1-p)2

(a-p)

We see that G(l, . . . . 1) = na is a maximum follows that

for j= 1, . . ..n. for a>p

G(l, . . . . l)=na=nM,,(t*)>G(x,,...,x,,)=nM,T)

and a minimum

for a
for a>p.

It

84

P. Deheuvels / Large deviations by Poisson approximations

A similar

argument

based on (3.36) shows likewise

that

H(l, . . . . l)=na=nM,(t*)6H(y,,...,y,)=nM,(~). This readily

proves

(3.33) as sought.

0

In the remainder of this section, we investigate the asymptotic sharpness of the bounds given in Theorem (3.2) for P(aS, > *no). We will make use at times of the following additional assumptions on p, =pI, ,,, . . . , p,, =p,,,, , n 2 1, p =p, and (Y= an, as n-w. (Al) (A2) (A3) (A4) Theorem

max(pr ,...,p,)/min(p,,...,p,)=O(l); JY-+OO; ncxa 1 is integer; ncr 3 + 0. 3.4.

Under (Al)-(A4),

we have for p<(~< 1,

P(S,~ncr)=(l+o(l))p+(ncre~‘*,e’*)exp

i

Lj(t*)-na(l-e-‘*)

(3.38)

Lj(t*)-nCX(l-ep’*)

(3.39)

j=l

and for O
as n-03,

whereP,(.,

.)

i

is as in (3.22)-(3.2)1/j=’

Proof. Using Stirling’s formula (i.e. m! = (m/e)“‘@ exp(BJ(12m)) with O< 8,,, < 1 for m 2 l), we obtain that for all p > 0 and y > 0 such that y~(Z 1 is integer,

-P

_yp)L& (YP)! By (3.40) and (A3), it follows

exp(-fi(l

- Y+ y log v)).

(3.40)

that 1

p*(ncx ee’*, e’*) >-

1 2fi

exp(-na(e+-

1 + t*)).

(3.41)

2fi Combining (3.24), (3.25), ((3.39) holds for O
(3.26) and (3.41), we see that (3.38) holds for ~
1

(AS) ,I;=o(fi). We will show that the conditions sake, recall that n;=

i

of the theorem

Pj exp(t*)

j= 1 ( 1 -pj+Pj

eXp(t*)

which by (3.30) and (3.33) is for p
ensure

that (A5) holds.

For this

2

> ’ 1 less than

or equal to (3.42)

P. Deheuvels / Large deviations by Poisson approximations

85

we use (A4) to show that (x+0, and (Al) to obtain that Next, (l/p’) Cq,, p]!=O(n). This, when combined with (3.42), shows that L:=O(ncr2). By (A4), it follows that (A5) holds. A similar argument with the forma1 replacements of p by p in (3.42), and of (3.33) by (3.34) yields, for O
n;< i J=]

Pjew@)

(

l-p;+pjeXp(F)

=(g)2j,

2

1

[PI/(’ -Pj(z))]‘-

(3.43)

By (Al), we have pj = O(p) uniformly over j= 1, . . . , n. Moreover, by (A2), p- a
pi(na em’*,e’*) in

It is interesting to seek a simple approximant of the coefficients (3.38)-(3.39). For this sake, introduce the function exp(x(e’< - 1 - it))

‘iI

u(x, Y) = & This function

satisfies

(see e.g. Hoglund

lim u(x, y) * x-to In view of (3.44)-(3.45), Theorem

3.5. Assume

(3.44)

d<.

l-yei

I_,

(1979)) for O
1,

1

~ l-y’

(3.45)

we have the following

that (Al)-(A4)

approximation

of P(+S,>

km).

hold, together with

(A6) np+m. Then, for any p
1, we have

P(S,>na)=(l+o(l))u(na,e-‘*)exp

i

Lj(t*)-nat*

j=l

,

(3.46)

>

while for O
i

c

j=l

Lj(t*)-nat*

. >

(3.47)

86

Proof.

P. Deheuvels

By Theorem

/ Large deviations

3.3 and (3.32),

by Poisson

approximations

we have (3.48)

By (3.30)-(3.38),

(3.48) and (Al)-(A3),

it follows

that (3.49)

Thus, by (A5) and (3.49), n(xe-‘* + m. This last condition enables to apply Theorem A of Hoglund (1979) to evaluate P&(ncue-‘*,e’*) as deviations of the Poisson distribution. Note that the original assumptions in Hoglund’s theorem require e’* to be bounded. However, in the particular case of the Poisson distribution, a direct calculus based on Stirling’s formula shows that this restriction is not necessary. In view of Theorem 3.3, we so obtain the conclusion of Theorem 3.5. 0

4. Simple approximations

of P(*S,<

km)

In order to avoid the technicalities introduced by the numerical solution of the equation (3.6), i.e. M(t*) =a, or alternatively by the replacement of t* by an upper or lower board selected among 6 i or t, we will assume here that p, = ... =p,, =p. It follows from Section 2 that our arguments may be extended without difficulty through simple inequalities in the general case where pi, . . ..p., are arbitrary. Following the notation introduced in Section 1, we shall denote by s, a random variable following a binomial B(n,p) distribution. A direct application of the results of Section 3 shows that

t*&a(l-P) JY(l-a)’

Lj(t*)=lOg

1-p ~ I-C!

( >

for j=

1, . . . . n,

(4.1)

An application of Theorem 3.3 in combination with (4.1) and routine tions yields the following corollary of this theorem. Corollary

4.1.

Under (A2)-(A4),

while for 0 < ct
we have, as n+ 03, for p
A:=na2.

1,

computa-

P. Deheuvels

/ Large deviations

by Poisson

approximations

87

Remark 4.1. A comparison of (4.2)-(4.3) with the exact binomial probabilities shows that these expressions are obtained by replacing the coefficient n!/(n - k)! by nx ePa/(l - a)n-k in the expression of tl!

_

P(Sn=k)= k!

An alternative

$(I

_p)“-k.

(n-k)!

proof of (4.2)-(4.3)

could be obtained

by using Stirling’s

formula

for

n!and(n-k)!,whichyields,whenn~co,n-k~~,k/n-tOandk=ncu(l+o(l)), n” e-“u

n! P=(l+o(l))(l_Cl),-k’ (n-k)!

Remark 4.2. By combining Corollary 4.1 with Theorem 2.1, one obtains simple upper bounds for P(+-S,> +na) under very general conditions. We do not offer here a detailed numerical investigation of such bounds, for the sake of brevity.

Acknowledgement We thank paper.

the referees

for helpful

comments

concerning

the contents

of this

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