Local limit theorem for large deviations and statistical box-tests

Local limit theorem for large deviations

and statistical box-tests

Igor Semaev

Department of Informatics, University of Bergen Hoyteknologisenteret, N-5020 Bergen, Norway

e-mail:[email protected]

Abstract: Letnparticles be independently allocated intoNboxes, where thel-th box appears with the probabilityal. Letµrbe the number of boxes

with exactlyrparticles andµ= [µr1, . . . , µrm]. Asymptotical behavior of

such random variables asNtends to infinity was studied by many authors. It was previously known that ifN alare all upper bounded andn/Nis upper

and lower bounded by positive constants, then µtends in distribution to a multivariate normal low. A stronger statement, namely a large deviation local limit theorem forµunder the same condition, is here proved. Also all cumulants ofµare proved to beO(N).

Then we study the hypothesis testing that the box distribution is uni-form, denoted h, with a recently introduced box-test. Its statistic is a quadratic form in variables µ−Eµ(h). For a wide area of non-uniform al, an asymptotical relation for the power of the quadratic and linear

box-tests, the statistics of the latter are linear functions of µ, is proved. In particular, the quadratic test asymptotically is at least as powerful as any of the linear box-tests, including the well-known empty-box test ifµ0is in

µ.

AMS 2000 subject classifications: Primary 62F03,60F10; secondary 62H15,60B12.

Keywords and phrases: random allocations, large deviations, box-test power.

1. Introduction

1.1. Prior Work

Letn particles be independently allocated intoN boxes(cells), where thel-th box appears with the probabilityal. We denotea=(a1, . . . , aN). Letµr(a) be

the number of boxes with exactlyrparticles andµ(a) = [µr1(a), . . . , µrm(a)] for some r1, . . . , rm. Asymptotical behavior of such random variables was studied

by Mises, Okamoto, Weiss, Renyi, B´ek´essy, Kolchin, Sevast’ynov, Chistyakov and others; see [10] for the bibliography before 1975, also [9] and [13] for later developments and applications. Poisson, normal and multivariate normal distri-butions are the most common limit distridistri-butions forµunder different conditions asN tends to infinity. Okamoto [16] and Weiss [23] were first who proved that if the cell distribution is uniform andα=n/N is a positive constant, then µ0

has asymptotically normal distribution. The fact got a number of generaliza-tions since then. In particular, Renyi [18] extended the area ofα, where µ0 is

still asymptotically normal. The normality ofµ0in case of the multinomial cell

distributiona, whereN alare upper bounded andαis upper and lower bounded

by positive constants, was proved by Chistyakov [2]; see global and local limit Theorems 3.4.2 and 3.4.3 in [10]. In somewhat weaker condition that was also proved in [8]. A multidimensional normal theorem forµwas established in [22] under the same conditions as in [2]; see Theorems 3.5.2 and 3.5.3 in [10].

Those limit theorems were mostly produced with either the method of mo-ments or characteristic functions. Both imply the convergence in distribution [11]. The latter is quite restrictive when it comes to studying the power of sta-tistical box-tests, where it is often required to estimate the probability of an interval which may vary asN grows and that probability tends to zero rapidly. Large deviation limit theorems are then necessary. Such a theorem is one of the goals of the present work.

Any statistical test based on the distribution ofµ is naturally to call a box-test. For instance, a test based on µ0 is called empty box(cell)-test and was

introduced by David in [6] to verify whether a sample of n independent ob-servations is taken from a continuous distribution F(x). The real axis is split intoN subintervals (zl−1, zl] of equal probability:F(zl)−F(zl−1) = 1/N. Then

µ0 is the number of intervals without observations. The test may have some

advantage over Pearson’s χ2 _{goodness-of-fit test, which originally requires N}

fixed while n→ ∞to approach limit distribution; see [24]. As the number of the intervals is fixed, some information on F may be lost. Though in practice

χ2_{-test is applicable forN growing as n}2/5_{; see [12], one can take advantage of}

a box-test by choosing the number of intervalsN as big asn, for instance. More general is a linear box-test whose statistic is the dot-productµc, where

cis a constant non-zero vector of lengthm, we use the same character to denote a vector and its transpose. The power of the linear test was studied in detail. Letαbe upper and lower bounded by positive constants,N32 PN

l=1(al−1/N)2

tend to a constant and some additional restrictions be fulfilled. Then c was constructed such that the µc-test overcomes in power any test based on the sameµ; see Chapter 5 in [10].

1.2. New box-tests

LetN al ≤C for all l = 1, . . . , N, where C >1 is a constant, and let α= _Nn

be upper and lower bounded by positive constants: 0< α0≤α≤α1 asN and

ntend to infinity. We assume that throughout the article: all statements below are true within this assumption. The random variable

ν(a) = µ(a)√−Eµ(a)

N =

_µ

r1(a)_√−Eµr1(a)

N , . . . ,

µrm(a)_√−Eµrm(a)

N

asymptotically has then multivariate normal distribution; see Theorem 3.5.2 in [10]. This basic fact is employed to test whether a multinomial sample was produced with prescribed box probabilities for large enoughN. The hypothesis

Let η(a) = [µ(a)−Eµ(h)]N−1/2_{, where a is unknown box distribution. A}

quadratic box-test, whose statistic is the quadratic form ηB−1η, where B is the covariance matrix for ν = η(h) = ν(h), is here studied. Asymptotically,

νB−1ν hasχ2

m-distribution asN tends to infinity. The limit covariance matrix

defined by (8.1) in Appendix may be taken forB. The test was found by this author and Hassanzadeh during a study on cryptographic hash-functions [21]. A good hash-function should have values indistinguishable from those produced with multinomial uniform probabilities. Hash-function values are considered as allocations into boxes, labeled with its different values. According to NIST, the total number of a hash function different values may be up to 2512 [15]. Therefore, to test a hash-function its values are split into N regions of equal probability as with the continuous distribution earlier.

The goal of this work is to prove Theorems 1.1 and 1.2. They imply in case of

N32PN

l=1(al−1/N)2→ ∞that the quadratic box-test is typically more powerful

compared to any linear box-test, whose statistic depends on the sameµ. The latter includes empty-box test ifµdepends onµ0.

The main technical tool is a local limit Theorem 1.3 for large deviations ofµ

proved in Section 3. The Theorem is of independent interest as it is a significant improvement over the Okamoto-Weiss result and a number of limit Theorems stated in [10].

Also, as a Corollary to Theorem 3.1, we show that within the assumptions all cumulants ofµareO(N). This was not proved before.

1.3. Results

Let

δ=

_Eµ

r1(a)_√−Eµr1(h)

N , . . . ,

Eµrm(a)_√−Eµrm(h)

N

. (1.1)

and βc(a), β(a) denote second error probabilities of the linear ηc-test and the

quadraticηB−1η-test with the same significance level; see Section 2.2. Theorem 1.1. Let |δ| → ∞ such that|δ|=o(N1/2). If |δc| → ∞, then

lnβc(a)

lnβ(a) =

|δc|2

(cAc)(δA−1_δ)(1 +o(1)), (1.2)

where A is a matrix defined by Theorem 8.2 in Appendix. If |δc| is bounded, thenlnβc(a) =o(lnβ(a)).

The matrix A is positive definite so, by the Cauchy-Schwarz inequality,

|δc|2_≤(cAc)(δA−1_{δ). That inequality is often strict; see examples in Sections}

5.1 and 5.2. Anyway,βc(a)≥β(a)1+o(1) and the quadratic test is typically of

greater power than any linear in the Theorem assumptions. The next statement definesa, where the latter are fulfilled. We sayN al→1 if for anyε >0 there

existsNεsuch that|N al−1|< εfor allN > Nεandl= 1, . . . , N.

Theorem 1.2. LetN al→1, then|δ|=o(N1/2). Let in addition(α−ri)26=ri

for somei. Then|δ| → ∞ if and only ifN32PN

Theorem 1.1 is true foral =N−1+γlN−5/4, where |γl| all tend to infinity

such that γl =o(N1/4). The power of the quadratic box-test then tends to 1

by Theorem 4.1. In other words, the quadratic test can distinguish such non-uniform distributions with the probability tending to 1 and provides with an asymptotically smaller error probability than linear box-tests.

Letk= (k1, . . . , km) be an integer vector andx= (k−N p)N−1/2. We here

denoted p(a) = [pr1(a), . . . , prm(a)], where pr(a) are defined by Theorem 3.1.5 in [10](Theorem 8.1 in Appendix), so that Eµ = N p+O(1). We study the probability Pr(µ= k), where |x| may grow asN grows. By|x| the euclidean length ofxis denoted.

Numerous local and global limit theorems for µ were proved; see [10] for a good account. The value of|x| is there always assumed bounded. Also Berry-Esseen type estimates like those in [5] and [17] are too rough to estimatePr(µ=

k) for large enough|x|. However a large deviation limit theorem, that is when

|x| is allowed to grow withN, is necessary for Theorem 1.1. In Section 3 we prove the following statement, where a modification of the saddle-point method due to Richter, see [19, 20], was used.

Theorem 1.3. Assume |x|=o(√N). Then uniformly in α, al

Pr(µ=k) = e

−xA−1x

2 +N λ(√xN)

p

(2πN)m_detA

1 +O

_{|x|+ 1} √

N

,

whereλ(τ) =O(|τ|3_{)for all small realm-vectorsτ. The latter bound is uniform}

inα, N, al. The matrix A= (σrt)is defined by (8.3).

This is a significant improvement over local limit Theorem 3.4.3 forµ0 and

multivariate Theorem 3.5.3 in [10]. Stronger versions of the global limit the-orems, stated in [10] in case N al ≤ C and α0 ≤ α ≤ α1, are now easy to

deduce.

By small latin and greek characters we denote both complex and real numbers andm-dimensional complex and real points. We hope it is clear from the context what this or that character means. Anyway,x, s, τ, t, v, δ, c, palways(or almost always) denotem-dimensional real points,kis anm-dimensional integer point, andu=s+itis anm-dimensional complex point, whilezis usually a complex number andθ, σare reals.

2. Quadratic Test

Letm-variate non-singular normal distribution be given. It is well known ([11], Section 15.10) that the quadratic form in the exponent of its density function hasχ2_{-distribution with m degrees of freedom when the quadratic form}

vari-ables are substituted by the multivariate normal vector entries. Therefore the random variableνB−1_{ν asymptotically has χ}2_{-distribution with m degrees of}

freedom. Let 0< ε <1 be a required significance level(criterion first kind error probability). The quantileCεsuch that

is taken from theχ2

mdistribution. Let an allocation ofnparticles intoNboxes be

observed. The statisticηB−1_{η is computed, whereη(a) = [µ(a)−N p(h)]N}−1/2

andais unknown box distribution. The former has the same asymptotical dis-tribution as [µ(a)−Eµ(h)]N−1/2 _{and is denoted byη(a) again.}

If ηB−1_{η ≤ C}

ε, then a =h is accepted, otherwise rejected. The first error

probability Pr(νB−1_{ν ≥C}

ε) is close toε for large enoughN. For low N that

may differ fromεand is computed directly; see Section 7 and [21].

2.1. Example

Letµ= (µ0, µ1) andα= 1. Then by (8.1),

B= 1

e2×

e−2 −1

−1 e−1

, B−1= e

2

e2_{−3e+ 1} ×

e−1 1

1 e−2

and

η(a) =

_µ

0(a)−N e−1

√

N ,

µ1(a)−N e−1

√ N

.

So

ηB−1η= e

2

(e2_{−3e+ 1)N} ×

µ0−N e−1

µ1−N e−1

t

e−1 1

1 e−2

µ0−N e−1

µ1−N e−1

.

2.2. Second kind error probabilities

The criterion second error probability is β(a) = Pr(ηB−1_{η ≤ C}

) for

non-uniform a. In other words, that is the probability to accept a= h whereasa

is not uniform. Remark that β(h) = 1−ε. When it comes to compare two criterions, one is said more powerful if its second error probability is lower while the first error probability is the same for both.

The dot-productηcis the linear statistic undera. The second error probabil-ity isβc(a) =Pr(|ηc| ≤D), whereDεis theN(0,

√

cBc)-quantile of significance levelε. We will base linear and quadratic criteria on the sameµ and compare their power in Section 4 by proving Theorem 1.1. Some limit theorems for large deviations are then required.

3. Local Limit Theorem for Large Deviations

By the characteristic function argument, see Theorem 3.5.3 in [10], the distri-bution of (µ−N p)N−1/2 _{tends to the multivariate normal distribution with 0}

expectations and covariance matrixAdefined in Theorem 8.2. So uniformly for any measurable areaG

Pr

_{µ−N p} √

N ∈G

= _p 1

(2π)s_det(A) Z

G

e−xA −1_x

IfGvaries withN, the main term may tend to 0 faster than the uniform esti-mate foro(1). That makes the probability estimate useless. Stronger statements, where

Pr

_{µ−N p} √

N ∈G

= _p 1 +o(1) (2π)s_det(A)

Z

G

e−xA −1_x

2 dx, (3.1)

are then necessary. They are called limit theorems for large deviations and well-known for the sums of independent(especially identically distributed) ran-dom variables; see for instance [4, 7, 19, 20]. In [20] the sums of independent identically distributed multi-dimensional random variables were treated. Such a representation forµis unknown, so the result isn’t directly applicable.

Local limit Theorem 1.3 for large deviations of µ is here proved. Global estimates like (3.1) are then not difficult corollaries. The proof will follow that in [20] with some modifications. First, we have to prove Theorem 3.1, where the moment generating function for µ is represented in a form suitable for a saddle-point estimation, as in [20], of the integral in (3.2).

Lets= (s1, . . . , sm) be a vector of real variables and

MN(s) =Eesµ= X

k

Pr(µ=k)esk

the moment generating function for µ, where sk = s1k1+. . .+smkm. There

are finite number of terms in the sum as µ may take only finite number of valuesk.MN(s) is a function inswhich also depends onN, n=αN, and the

probabilitiesal. One can write

Pr(µ=k) = 1 (2π)m

Z π¯ −π¯

MN(s+it)e−(s+it)kdt, (3.2)

wheret = (t1, . . . , tm), and the integral is over|tj| ≤π, so ¯π= (π, . . . , π). By

Theorem 3.1.1 in [10]( Theorem 8.3 in Appendix) and Cauchy Theorem,

Eeuµ = n! 2πiNn

I N Y

l=1

Fl(z, u)

dz zn+1,

Fl(z, u) = eN alz+ m X

j=1

(N alz)rj

rj!

(euj₋₁₎

for any complexu= s+it, where u= (u1, . . . , um), and complex variable z.

The integral is over any closed contour encircling the origin. So one can write

MN(u) =

n! 2πiNn

I

eN f(z,u)dz

z , (3.3)

where

f(z, u) = 1

N ln

N Y

l=1

Fl(z, u)−nlnz !

= 1

N

N X

l=1

We take the main branch of ln for this definition.

Lemma 1. Letsandz0take values from a closed bounded set inRm and from

a closed bounded interval of positive reals respectively. Let forε >0 either

1.ε≤ |tj| ≤π for at least onej and|θ| ≤π, or

2.ε≤ |θ| ≤π andt= 0

be true. Then there existsq <1 such that

N Y l=1

Fl(z0eiθ, s+it)

Fl(z0, s)

< qN. (3.4)

Proof. Denote z=z0eiθ, then

Fl(z, s+it)

Fl(z0, s)

= e

N alz₋Pm

j=1

(N alz)rj

rj! +

Pm j=1

(N alz)rj

rj! e

sj+itj

eN alz0−Pm

j=1

(N alz0)rj

rj! +

Pm j=1

(N alz0)rj

rj! e

sj

.

Assumeal6= 0. Asz0 is positive,

eN alz₋

m X

j=1

(N alz)rj

rj!

≤eN alz0₋

m X

j=1

(N alz0)rj

rj!

.

If θ 6= 0, then the last inequality is strict and so |Fl(z, s+it)| < Fl(z0, s). If

θ = 0 , then the former inequality becomes equality, while the latter remains strict if at least onetj6= 0.

LetC1<1 be a positive constant andC1≤N al≤C. Then for either of two

conditions we have|Fl(z, s+it)|< Fl(z0, s). So

Fl(z, s+it)

Fl(zα, s)

≤q1

for a positive q1 <1. This is true because the function is continuous and its

variablest, θ, z0, sandN alare in a closed bounded set in both cases. More than

N(1−C1)(C−C1)−1 probabilitiesal satisfyC1 ≤N al≤C. For every suchl

the product term in (3.4) is bounded byq1. The rest are≤1 in absolute value.

Therefore (3.4) is true, where

q=q(1−C1)(C−C1)−1

1 <1.

Lemma 2. There exist two closed bounded regions: multidimensionalU ⊆_Cm

around u = 0 and one-dimensional V ⊆ _C that does not include z = 0. The regions do not depend onα,N andal. There exists a unique functionzα(u)on

U with values in V such thatf_z0(zα(u), u) = 0. The function zα(u)is analytic

Proof.

f_z0(z, u) = 1

N

N X

l=1

N al

eN alz₊Pm

j=1

(N alz)rj−1

(rj−1)! (e

uj ₋₁₎

eN alz+Pm

j=1

(N alz)rj

rj! (e

uj −₁₎

−α

z.

is an analytic function in complex variablesz∈_Candu∈_Cm_{in a small region}

around its zero (α,0) and f_z002(α,0) = α−1_{. It follows from the Weierstrass}

Preparation Theorem [1] there are two regions: multidimensional U and one-dimensional V, such that for any u∈U there exists only one zα(u) ∈V and

f_z0(zα(u), u) = 0. The regions U and V may be taken the same for any α, N

andalwithin the assumptions.

We will prove that. LetV ⊆Cbe the closed region bounded by the circlez=

α0+ρeiθof radius (α1−α0)/2< ρ <(α0+α1)/2 centered atα0 = (α0+α1)/2.

There existsγ >0 such that for anyα0≤α≤α1the modulus offz0(z,0) = z−αz

on the border ofV has its minimal value≥γ. We now write

|fz0(z, u)−fz0(z,0)|=

1

N

N X

l=1

N alPm_{j=1 (N a}

lz)rj−1

(rj−1)! −

(N alz)rj

rj!

(euj ₋₁₎

eN alz+Pm

j=1

(N alz)rj

rj! (e

uj−1)

.

Letz=α0+ρeiθ_{. AsN a}

l≤C, there existsr >0 such that

|f_z0(z, u)−f_z0(z,0)|< γ

for any|u| ≤r,α,N, aland any pointzon the border ofV. LetU denote the

region|u| ≤r. By Rouch´e’s Theorem, the functions fz0(z, u) and fz0(z,0) have

the same number of zeroszinsideV, that is just one, for anyu∈U. This defines a function zα =zα(u) on U, which takes values in V. The Implicit Function

Theorem [1] then states thatzα(u) is analytic on U.

Lemma 3. Any order partial derivatives ofzα(u)andf(zα(u), u)are uniformly

bounded onU whileα,N andal may change.

Proof. By Lemma 2, the function zα(u), thereforef(zα(u), u), are analytic on

U and their values are uniformly bounded. By the Cauchy estimates [14], their any order partial derivatives are uniformly bounded onU too.

Theorem 3.1. 1.There exists a small regionU ⊆Cmaroundu= 0, the same

for allα,N andal, where

MN(u) =eNK(u)Y1(u)Y2(u), (3.5)

andK(u),Y1(u), Y2(u)are analytic functions onU.

2.K(u) =up+uA₂u +O(|u|3_{), Y}

1(u) = 1 +O(|u|) uniformly in α,N and

al.

3.Y2(u) =O(N1/2) for complexu∈U, and Y2(s) = 1 +O(N−1/2) for real

Proof. First we get the presentation (3.5) forMN(s), wheresbelongs to a small

closed bounded region in_Rm_{around 0. The expansion of}QN

l=1Fl(z, s) inz has

only positive coefficients. So by Lemma 2.2.2 in [10](Theorem 8.4 in Appendix), for any real pointsand positiveαthe equationf_z0(z, s) = 0 has a unique positive rootzα=zα(s) andα=zα(0). Also, by Theorem 8.5 in Appendix,f_z002(zα(s), s)

is then positive and, in particular,f_z002(α,0) =α−1.

By Lemma 2, for all small enough s the value zα(s) belongs to a closed

bounded interval of positive real numbers. One can take the interval that does not depend onα, N and al. Also it is not difficult to show the values of

par-tial derivatives fz(rr)(zα(s), s) are there uniformly bounded and f_z002(zα(s), s) is

uniformly upper and lower bounded by positive constants.

We estimate the integral in (3.3) for realswith the saddle-point method. The integration path is z =zα(s) exp(iθ), where |θ| ≤π. Then MN(s) = I1+I2,

where

I1 =

n! 2πiNn

Z

|θ|≤ε

eN f(z,s)dz z ,

I2 =

n! 2πiNn

Z

ε≤|θ|≤π

eN f(z,s)dz z

= n!e

N f(zα,s) 2πiNn

Z

ε≤|θ|≤π z_α

z

nYN

l=1

Fl(z, s)

Fl(zα, s)

dz z .

The valueεwill be chosen when estimatingI1. We estimateI2first. By Lemma

1,

QN

l=1Fl(z, s)Fl(zα, s)−1 < q

N_{, whereq <1. Therefore,}

|I2| ≤

n!eN f(zα,s)

Nn q

N_.

To estimate the first integral we expand

f(zαeiθ, s) =f(zα, s)−

f_z002(zα, s)

2 z

2

αθ

2_+O(|θ|3₎

around θ = 0. The remainder estimate is uniform in α, N and al, for all

small enough s. By a standard argument, see, for instance, Lemma 2.2.3 in [10](Theorem 8.6 in Appendix),

MN(s) =

n!eN f(zα(s),s)_{(1 +O(N}−1/2₎₎

Nn_{[2π N z}2

αfz002(zα(s), s)]1/2

, (3.6)

where (1 +O(N−1/2_{)) is uniform in α, a}

l and all small enough s. We now

considerzα(u) for complexu. With a tedious calculation one expands

zα(u) =α+ m X

j=1

1

N

N X

l=1

(N alα−rj)

(N alα)rj

rj!

e−N alα

!

and

f(zα(u), u) =−αlnα+α+up+

uAu

2 +O(|u|

3

)

atu= 0 uniformly by Lemma 3. We writeMN(u) =eNK(u)Y1(u)Y2(u), where

K(u) = αlnα−α+f(zα(u), u),

Y1(u) =

α z2

α(u)fz002(zα(u), u) 1/2

,

Y2(u) =

MN(u)

eNK(u)_Y 1(u)

.

From (3.6), uniformlyY2(s) = 1 +O(N−1/2) for reals∈U. That implies (3.5)

for reals.

By direct calculation,Y1(u) = 1 +O(|u|) uniformly inα,N andal. All three

above functions are analytic onU and the values ofK(u) are there uniformly bounded. That proves the first two statements of the Theorem.

We prove|Y2(u)|=O(N1/2) for complex u∈ U. It is enough to show that

uniformly|MN(u) exp(−NK(u))|=O(N1/2). Let’s define the integration path

in (3.3) byz=zα(u) exp(iθ) and|θ| ≤π. We get

MN(u)

eNK(u)

≤ n!

Nn_eN(αlnα−α) max_θ

eN f(z,u)

eN f(zα(u),u)

. (3.8)

By Stirling approximation ton!,

n!

Nn_eN(αlnα−α) =O(N

1/2_{). (3.9)}

Letsbe a small real point, sozα(s) belongs to a closed interval of positive reals

andu=s+it. Any order partial derivatives off(zα(u)eiθ, u)−f(zα(u), u) are

bounded and it is represented by a power series for all small complex u∈ U

and real|θ| ≤π. That series is a superposition of the convergent power series forzα(u),f(z, u) andeiθ. Therefore for every smalls, we deduce a power series

expansion in all smallt and|θ| ≤π. By direct calculation,

ef(z,u) ef(zα(u),u)

=e

− "

f00

z2 (zα(s),s)z2α(s) 2 +O(|t|

2_+|t|θ)

# θ2

,

where the reminder estimate is uniform in α, N, al. Also we remark that

f_z002(zα(s), s)zα2(s) > 0 and it is uniformly bounded from below for small s.

Therefore for all small enought

eN f(z,u)

eN f(zα(u),u)

≤1.

It now follows from (3.8) and (3.9) that|MN(u) exp(−NK(u))|=O(N1/2) in

Corollary 1. All cumulants of µareO(N).

Proof. According to [11], the cumulants are coefficients in the power series ex-pansion for lnMN(u). The statement follows from Theorem 3.1.

Lemma 4. Let |tj| ≤π, where 1≤j ≤m and ε≤ |tj| for at least onej. Let

sbelong to a small closed region in Rm around 0. Then there exists a positive

q <1, such that

|MN(s+it)| ≤eNK(s)qN,

for all large enoughN.

Proof. From (3.3) we get|MN(s+it)|=

n!eN f(zα(s),s) 2πNn

I _z α

z

n_YN

l=1

Fl(z, s+it)

Fl(zα(s), s)

dz z

, (3.10)

where the integration path isz=zα(s) exp(iθ) and|θ| ≤π. By Lemma 1,

N Y

l=1

Fl(zα(s)eiθ, s+it)

Fl(zα(s), s)

< qN,

whereq <1. For smallswe havef(zα(s), s) =−αlnα+α+K(s). We use Stirling

approximation ton! and take a slightly largerq. The Lemma follows.

Lemma 5.

Pr(µ=k)≤ n!e

N f(zα(s),s)−sk

Nn

for any reals∈RmandPr(µ=k) =O(N1/2exp N[K(s)−skN]

)for all small enoughs.

Proof. One sees

Fl(zα(s)eiθ, s+it)

Fl(zα(s), s)

≤1,

for any real s, |t| ≤ π and |θ| ≤ π. From (3.10) we get |MN(s+it)| ≤

n!N−nexp(N f(zα, s)). The Lemma follows from (3.2).

One now proves Theorem 1.3.

Proof. Let K0(u) be the gradient of K(u) = αlnα−α+f(zα(u), u). From

Lemma 3, the entries ofK0(s) are functions in reals∈U whose any order partial derivatives are uniformly bounded inα, N andal. Therefore, by Theorem 3.1

uniformlyK0(s) =p+sA+O(|s|2_{). We can write (3.2) as}

Pr(µ=k) = 1 (2π)m

Z

eN[K(s+it)−(s+it)kN ]_{Y(s+it)dt, (3.11)}

where Y(u) = Y1(u)Y2(u) and estimate the integral with the saddle-point

method. The saddle-point equationK0(s) =kN−1_{is equivalent tosA+O(|s|}2_{) =}

By Theorem 8.2 in Appendix, the determinant ofA is bounded from below by a positive constant. Transformation Inversion Theorem [3] states that for any small enoughτthere exists unique solutions=s(τ) toK0(s) =kN−1_{in a small}

regionU1 ⊆U around s= 0. That region is taken the same for all α, N and

al. Any order partial derivatives of s(τ) are uniformly bounded. So uniformly

s=τA−1+O(|τ|2_).

For small complexu =s+itwe write MN(u) = exp(NK(u))Y(u), where

K(u) and Y(u) = Y1(u)Y2(u) are analytic functions defined in Theorem 3.1.

One takesεsuch that the following two Taylor expansions att= 0 are true for all|tj| ≤εands∈U1.

K(s+it)−(s+it)k

N =K(s)−

sk

N −

H2(t)

2! −

iH3(t)

3! +. . . ,

whereHr(t) are forms of degreer. For instance,H2(t) =Pi,jtitjK00si,sj(s).

Y(s+it)

Y(s) = 1 +iR1(t)−

R2(t)

2! +. . . ,

where Rr(t) are forms of degree r. Remark Y(s)6= 0 for small enough s and

all largeN. By Theorem 3.1, the values ofK(u) are uniformly bounded onU

whileα, N, al changes. Therefore, by the Cauchy estimates, the coefficients in

Hr(t) are uniformly bounded too. As |Y(u)|=O(N1/2) uniformly on U, then

the coefficients inRr(t) are uniformlyO(N1/2). We split the integral in (3.11):

Pr(µ=k) = 1 (2π)m

Z ε¯ −ε¯

eN[K(s+it)−(s+it)kN ]Y(s+it)dt

+ 1

(2π)m Z

some|tj|≥ε

MN(s+it)e−(s+it)kdt.

LetIdenote the first integral. By Lemma 4, the second integral absolute value is bounded by exp(N[K(s)−sk

N])q

N_{, whereq <1. We change variablestN}1/2_=v,

use the above expansions and represent

I= e

N[K(s)−sk N]Y(s) (2π√N)m

Z ε¯ √

N

−ε¯√N

(1 +iR√1(v)

N −

R2(v)

2!N +. . .)e

−H2 (v) 2 _e−

iH3 (v) 3!√N . . . dv.

So

I=e

N[K(s)−sk N]Y(s) (2π√N)m

Z ¯ε √

N

−ε¯√N

e−H2 (2v) T(v)dv,

where

T(v) = 1 +√1 N

_R

1(v)H3(v)

3!√N − R2(v)

2√N

+ 1

N(. . .).

zero. We integrate the series term-wise with infinite integration limits. That introduces an error of ordere−C1N _{for a constantC}

1>0. One now writes

I= e

N[K(s)−sk N]Y(s) (2πN)m/2√_detH 2

(1 +O(N−1/2)).

Therefore, asY(s) = 1 +O(N−1/2_{) and detH}

2= detA+O(|τ|), we have

Pr(µ=k) = e

N[K(s)−sk N]

[(2πN)m_detA]1/2(1 +O(|τ|+N

−1/2_{)) +O(e}N[K(s)−sk N]qN).

Asτ= k−N p_N , by Theorem 3.1,

K(s) − sk

N =−sτ+

sAs

2 +O(|s|

3_),

Taking into accounts=τA−1+O(|τ|2_{), we get for all small enoughτ}

K(s)−sk

N =−

τA−1τ

2 +λ(τ),

λ(τ) =O(|τ|3₎

uniformly inα, N, al. That implies the Theorem.

4. Quadratic and Linear Test Power

In this section we estimate the second error probability of the quadratic and lin-ear tests by using Theorem 1.3. That will imply Theorem 1.1. Asδ=N1/2_[p(a)−

p(h)] +O(N−1/2_{), we useδto denoteN}1/2_{[p(a)−p(h)] in the rest of the article.}

Theorem 4.1. Let |δ| → ∞ such that|δ|=o(N1/2_{). Then}

β(a) =e−δA −1_δ

2 (1+o(1)).

Proof. By definition, β(a) =Pr ηB−1η≤C

=P

kPr(µ=k),wherek runs

over (xk +δ)B−1(xk+δ) ≤ C and xk = (k−N p)N−1/2 for p= p(a). The

matrixB−1 is positive definite. That implies|xk+δ| ≤ C1 for a constant C1

and so|xk|=o(N1/2). By Theorem 1.3,

Pr(µ=k) = e

θ(xk)_{(1 +o(1))}

p

(2πN)m_detA, (4.1)

where

θ(x) =−xA

−1_x

2 +N λ(

x √

Let k run over the set of integer m-vectors with non-negative entries, which satisfy (xk+δ)B−1(xk+δ)≤C. We write Pke

θ(xk) _{with repeated integrals} by repeatedly using the identity

b X

n=a

f(n) =

Z b

a

f(y)dy+

Z b

a

(y− byc)f0(y)dy+f(a), (4.2)

where nruns over integers from ato b andf(y) is one-variable function with continuous derivative. So we representβ(a) by an integral in case ofm= 1. For

m >1 the repeated integrals are written as multiple; see [3]. Anyway, we get

X

k

eθ(xk)₌

Z

eθ(xk)_{dk+H =}

Z

eθ(xk)_{dk(1 +o(1)),}

whereH is a finite sum of multiple integrals whose multiplicity is lower thanm

or exactlym and the integrand is bounded by the absolute value of a partial derivative ofeθ(xk)_{. All these integrals areo(}R_eθ(xk)_{dk). We now takex=x}

k+δ

as the integration variables. SoN1/2dx=dkand

X

k

eθ(xk)_=Nm/2

Z

xB−1_x≤C

eθ(x−δ)dx(1 +o(1)) =

= C1Nm/2eθ(y−δ)(1 +o(1)) =Nm/2e−

δA−1δ

2 (1+o(1)),

by the Mean Value Theorem, where y is a point within the integration limits andC1=

R

xB−1_x≤C

dx. Therefore,y=O(1) andθ(y−δ) =−

δA−1_δ

2 (1 +o(1)).

The Theorem now follows from (4.1) and the definition ofβ(a).

Theorem 4.2. Let c be any fixed real m-vector and |δ| → ∞ such that |δ| =

o(N1/2). If |δc| → ∞, then

βc(a) =e−

|δc|2

2cAc(1+o(1)).

If|δc| is bounded, thenβc(a)is bounded from below by a positive constant.

Proof. By definition, βc(a) = Pr(|ηc| ≤ D). Asymptotically ηc has a normal

distribution with the expectationδcand the variance cAc. From Theorem 8.3 and the Cauchy-Schwarz inequality,cAcis upper and lower bounded by positive constants. If|δc| is bounded, thenβc(a) tends to the probability of a bounded

interval of length 2D. Therefore,βc(a) is lower bounded by a positive constant

in this case.

We represent βc(a) = PkPr(µ = k), where the sum is over k such that

|xkc+δc| ≤D. Denotexk = (k−N p)N−1/2 = (xk1, . . . , xkm). Let|δc| → ∞,

then

βc(a) =

X

all|xkj|≤f N1/2

+ X

some|xkj|>f N1/2

wheref is a function such that|δc|2_=f4_{N. Obviously,f tends to 0 andf N}1/2

tends to infinity. The first sum is represented by a sum of integrals as in the above proof and equal toI+o(I), where

I=_p 1

(2π)m_detA Z

eθ(x)dx.

The integral is over|xc+δc| ≤D, and|xj| ≤f N1/2, wherex= (x1, . . . , xm).

With a standard transform of the integral and by the Mean Value Theorem,

I=e−|δc|

2

2cAc(1+o(1))_. The second sum in (4.3) is at most

m X

j=1

Pr(|µrj−N prj| ≥f N) =O(

√

N eN(−sf+O(s2))),

by Lemma 5, where we used the expansion ofK(s) at s = 0 for m = 1. Put

s=f2_{, so the second sum iso(I) too. That proves the statement.}

Theorem 1.1 now follows.

5. Examples

Two different non-uniform distributions a tending to uniform as N tends to infinity are considered in this Section. The power of the statisticsηcandηB−1η

on them are compared. The statistics only depend onµ0 andµ1.

5.1.

LetN1≤N, andal=N1−1forl= 1, . . . , N1, andal= 0 forl=N1+ 1, . . . , N.

Assumeθ=N N₁−1tends to 1 asN tends to infinity. By Theorem 8.2 one then computes the matrixAand the vectorδ:

A= 1

θe2θ ×

eθ_{−1−θ −θ}2

−θ2 _θeθ_−θ3_+θ2_−θ

.

and

δ=N1/2

_e−θ_{+θ−1−θe}−1

θ ,

θe−θ−θe−1 θ

.

Remark thatδsatisfies the condition of Theorem 1.1, that is|δ|=o(N1/2). For differentc, that is for different linear criterions, we study the expansion of the main term in the right-hand side of (1.2). Forc= (c1,1) the maximum of the

expansion main term is achieved atc1= (e−2)−1(e−1)−1= 0.8102..and

|δc|2

That impliesβ(c1,1)≥β

0.4272+o(1)_{for allc}

1. Forc= (1,0) we get 1+O((θ−1)2).

Soβ(1,0)=β1+o(1) and the empty-box test here is asymptotically as powerful

as the quadratic test. The reason for that is the vectors

cA = 1

e2(e−2,−1) +O(θ−1),

δ = (θ−1)N

1/2

e (e−2,−1) +O((θ−1)

2_N1/2₎

are tending to be collinear. However that is not true in the next example.

5.2.

LetN = 2N1 and al =N−1+θN−1 for l= 1, . . . , N1 and al =N−1−θN−1

forl =N1+ 1, . . . , N. Assume θ tends to 0 asN tends to infinity, thereforea

tends to be uniform. The covariance matrix

A=

σ00 σ01

σ01 σ11

and the expectation vectorδforη are computed by Theorem 8.2:

σ00 =

e−1−θ_−e−2−2θ_+e−1+θ_−e−2+2θ

2 +

(1 +θ)e−1−θ_{+ (1−θ)e}−1+θ2

4

σ01 = −

(1 +θ)e−2−2θ_{+ (1−θ)e}−2+2θ

2

−

(1 +θ)e−1−θ_{+ (1−θ)e}−1+θ _{(1 +θ)θe}−1−θ_{−(1−θ)θe}−1+θ

4 ,

σ11 =

(1 +θ)e−1−θ_{+ (1−θ)e}−1+θ

2 −

(1 +θ)2_e−2−2θ_{+ (1−θ)}2_e−2+2θ

2 ,

−

(1 +θ)θe−1−θ−(1−θ)θe−1+θ2

4 ,

and

δ=N1/2

_e−1−θ_+e−1+θ

2 −e

−1_,(1 +θ)e−1−θ+ (1−θ)e−1+θ

2 −e

−1

.

We study the expansion of the main term in (1.2). For instance, forc= (1,1) that equalsO(θ4_).Soβ

(1,1)=βo(1)and (1,1)-test behaves very poorly on such

a. Generally, forc = (c1,1) the maximum of the expansion main term in (1.2)

is only achieved at c1 = (e−2)/(3−e) = 2.549.. and equals 1 +O(θ2). So

β(c1,1)=β

1+o(1) _{in this case. Forc= (1,0)}

|δc|2

(cAc)(δA−1_δ) = 0.7469 +O(θ

2_).

The empty-box test is asymptotically less powerful than the quadratic test and

6. Where Theorem 1.2 is Proved

Letal=N−1+bl. Denoteδr=N1/2[pr(a)−pr(h)] andRN =N

3 2PN

l=1b 2

l.

Theorem 6.1. Let N al→1, then

1.|δr|=o(N1/2),

2.|δr|=O(RN),

3.assume in addition(α−r)2_{6=r, thenR}

N =O(|δr|).

Proof. By (8.2),

pr(a) =

1

N

N X

l=1

(αN al)r

r! e

−αN al_.

Let xl = N bl. We put f(x) = (1 +x)re−αx. By Taylor expansion, f(x) =

f(0) + (r−α)x+f00(θx)x2_{/2, where 0 ≤ θ ≤ 1 around x= 0. For all large}

enoughN

δr=

√

N[pr(a)−pr(h)] =

αre−α

r!√N

N X

l=1

[f(xl)−f(0) ]

= α

r_e−α

r!√N

N X

l=1

(r−α)xl+f00(θlxl)x2l/2

= α

r_e−α

r!√N

N X

l=1

f00(θlxl)x2l/2,

where 0≤θl≤1 andP N

l=1xl= 0. That implies the first statement. There exist

two constantsc1, c2 such thatc1≤f00(x)≤c2 for all smallx. Therefore,

αr_e−α_c

1

2r! N 3 2

N X

l=1

b2_l

!

≤√N[pr(a)−pr(h)]≤

αr_e−α_c

2

2r! N 3 2

N X

l=1

b2_l

!

.

That implies the second statement. One computes f00(0) = (α−r)2 _{−r. If}

(α−r)2 _{6= r, then c}

1, c2 may be taken both positive or both negative. That

implies the last statement.

Theorem 1.2 now follows from Theorem 6.1.

7. Tables

LetCεbe defined by (2.1). AsN → ∞, the quadratic test first error probability

Pr(νB−1ν≥Cε)→ε.In this Section we experimentally study the convergence

rate. Exact values of Pr(νB−1ν ≥ Cε) for some µ and moderate N are

pre-sented, whereε= 0.01 and 0.05. Table 1 contains data for µ= (µ0, µ1, µ2, µ3)

and Table 2 forµ= (µ2, µ4, µ5), wheren=N. The probabilities are computed

with the method explained in [21], which contains more computations. Numer-ical results demonstrate that the convergence rate depends onm,ri and may

Table 1

(µ0, µ1, µ2, µ3)-quadratic test

ε, N 24 ₂5 ₂6 ₂7 ₂8 ₂9 ₂10 ₂11 ₂12

0.05 0.0564 0.0403 0.0621 0.0579 0.0527 0.0515 0.0507 0.0503 0.0500 0.01 0.0200 0.0229 0.0178 0.0171 0.0153 0.0134 0.0120 0.0111 0.0105

.

Table 2

(µ2, µ4, µ5)-quadratic test

ε, N 24 ₂5 ₂6 ₂7 ₂8 ₂9

0.05 0.0449 0.0907 0.0376 0.0561 0.0510 0.0522 0.01 0.0412 0.0163 0.0181 0.0134 0.0142 0.0120

.

Acknowledgements

I am grateful to V.F. Kolchin, B.A. Sevast’ynov and V.P. Chistyakov, the au-thors of ”Random Allocations” [10], the reading of which was very stimulating.

References

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[3] _{Courant, R. and John, F.}(1974).Introduction to calculus and analysis

2. J. Wiley & Sons, New York.

[4] Cram´er, H.(1938).Sur un nouveau th´eoreme-limit de la th´` eorie des prob-abilit´es.Actual. Sci. et Ind. No. 736, Paris.

[5] Englund, G.(1981).A remainder term estimate for the normal approxi-mation in classical occupancy.Ann. Probab.9684–692.

[6] David, F.N. (1950).Two combinatorial tests whether a sample has come from a given population.Biometrika.3797–110.

[7] _{Feller W.}(1970).An Introduction to Probability Theory and Its Applica-tions 2. 2nd ed. J. Wiley & Sons, New York.

[8] _{Hwang, H.-K. and Janson, S.}(2008).Local limit theorems for finite and infinite urn models.Ann. Probab.36992–1022.

[9] _{Johnson, N.L. and Kotz, S.}(1977).Urn models and their applications.

J. Wiley & Sons, New York.

[10] _{Kolchin, V.F., Sevast’ynov, B.A. and Chistyakov, V.P.} (1978).

Random Allocations.V.H. Winston & Sons, Washington, D.C.

[11] _{Kendall, M.G. and Stuart, A.}(1969).The Advanced Theory of Statis-tics 1. 3rd ed. Ch. Griffin & Company Limited, London.

[12] Kendall, M.G. and Stuart, A.(1973).The Advanced Theory of Statis-tics 2. Ch. Griffin & Company Limited, London.

[14] _{Markushevich, A.I.}(1985).Theory of functions of a complex variables.

2nd ed. Chelsea Pub Co.

[15] http://csrc.nist.gov/groups/ST/hash/documents/FR Notice Nov07.pdf [16] _{Okamoto, M.}(1952).On a non-parametric test.Osaka J. Math.477–85. [17] _{Quine, M.P. and Robinson, J.} (1982).A Berry-Esseen bound for an

occupancy problem.Ann. Probab.10663–671.

[18] _{Renyi, A.} (1962). Three new proofs and generalization of a theorem of Irving Weiss.Magy. tud. akad. Mat. kutat´o int.k˝ozl. 7203–214.

[19] Richter, W.(1957).Local Limit Theorems for Large Deviations.Theory Probab. Appl.2206–220.

[20] Richter, W.(1958).Multi-Dimensional Local Limit Theorems for Large Deviations,Theory Probab. Appl.3100–106.

[21] Semaev, I. and Hassanzadeh, M.(2011).New statistical box-test and its power. available at http://www.ii.uib.no/publikasjoner/textap/pdf/2011-397.pdf

[22] _{Viktorova, I.I. and Chistyakov, V.P.} (1965). Asymptotic normality in a problem of balls when the probabilities of falling into different boxes are different.Theory Probab. Appl.10149–154.

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[24] _{Wilks, S.S.}(1962).Mathematical Statistics.J. Wiley & Sons, New York.

8. Appendix

In this Section we collect known auxiliary results mostly from [10], Chapter 3 with the exception of Theorem 8.2, which was not proved there. In Theorem 2.1.1 [10] it was proved that in casea=h

Eµr=N pr+O(1),

Cov(µr, µt) =N σrt+O(1),

where

pr =

αr

r!e

−α_,

σrr = pr−p2r−p

2

r

(α−r)2

α , (8.1)

σrt = −prpt−prpt

(α−r)(α−t)

α .

We putB(α) = (σrirj). The matrix is positive definite and there exists a positive constant ρ such that detB(α)> ρ for allα0 ≤ α≤ α1 by Corollary 2.2.1 in

[10]. More general are formulae (8.3) and Theorem 8.1 for box probabilities

a= (a1, . . . , aN). Let

prl=

(αN al)r

r! e

−αN al_{, p}

r(a) =

1

N

N X

l=1

and

σrr =

1

N

N X

l=1

prl−

1

N

N X

l=1

p2_rl− 1 α " 1 N N X l=1

prl(αN al−r) #2

, (8.3)

σrt=−

1

N

N X

l=1

prlptl−

1 α " 1 N N X l=1

prl(αN al−r) # " 1 N N X l=1

ptl(αN al−t) #

.

Theorem 8.1. (Theorem 3.1.5 in [10]) Let N tend to infinity, N al≤C for a

constantC, andα0≤α≤α1. Then

Eµr(a) =N pr+O(1),

Cov(µr(a), µt(a)) =N σrt+O(1),

wherepr andσrt are defined by (8.2)and (8.3).

Theorem 8.2. LetA= (σrirj), where1≤i, j≤m, asN tends to infinity such

that N al ≤C for a constant C and α0 ≤α≤α1. Then A is positive definite

and there exists a positive constantρsuch that detA≥ρ.

Form = 1 the statement is true by Theorem 3.1.4 in [10]. We here give a general proof.

Lemma 6. xAx≥ 1

N PN

l=1xB(αN al)xfor any realm-vectorx, whereB(αN al)

is defined by (8.1).

Proof. From (8.3)

xAx = 1

N N X l=1 X r

prlx2r−

1 N N X l=1 X r

prlxr !2 − 1 α 1 N N X l=1 X r

prl(αN al−r)xr !2

,

whererruns overr1, . . . , rmandx= (xr1, . . . , xrm). Letγl=αN al, then

1 α 1 N N X l=1 X r

prl(αN al−r)xr !2 = α N X l=1 al X r

prl(γl−r)

γl xr !2 ≤α N X l=1 al N X l=1 al X r

prl(γl−r)

γl xr !2 = 1 N N X l=1 1 γl X r

prl(γl−r)xr !2

by the Cauchy-Schwarz inequality. Therefore,

xAx ≥ 1

N N X l=1   X r

prlx2r− X

r

prlxr !2

− 1

γl X

r

prl(γl−r)xr !2

 = 1 N N X l=1

asprl=γrle −γl_/r!.

SoAis positive definite. We will prove Theorem 8.2 now.

Proof. LetA1, . . . , Ambe the principal minor ofA. SoAm= detA. By Lemma

6,A1 ≥N−1PN_l=1B1(γl), whereB1(γl) =σr1r1 are principal minors of order 1 in matricesB(γl). LetC1 <1 be a positive constant. There are more than

N(1−C1)(C−C1)−1 probabilitiesal such that C1≤N al≤C and, therefore,

α0C1 ≤ γl ≤ α1C. By Corollary 2.2.1 in [10], there exists positive ρ1 such

that B1(γ) > ρ1 for any α0C1 ≤γ ≤ α1C. So A1 ≥ ρ1(1−C1)(C−C1)−1.

TriangulatingxAx, one constructsx= (x1, . . . , xm−1,

p

Am−1), where xAx=

Am. So

Am≥

1

N

N X

l=1

xB(γl)x.

By Cauchy-Schwarz inequality, (xB(γl)x)(cB(γl)−1c) ≥ |xc|2. We put c =

(0, . . . ,0,1) and get

Am≥

Am−1

N

N X

l=1

1

bl

,

whereblis the bottom diagonal entry ofB(γl)−1. ForC1≤N al≤Cthe entries

ofB(γl) are upper bounded and its determinant is lower bounded by a positive

constant. Therefore, all entries of B(γl)−1 are upper bounded. In particular,

bl< ρ2 for suchland someρ2>0. SoAm≥Am−1(1−C1)ρ−21(C−C1)−1. The

Theorem now follows by induction.

Theorem 8.3. (Theorem 3.1.1 in [10])For any variables z, x1, . . . , xm ∞

X

n=0

(N z)n

n! E



 m Y

j=1

xµ_jrj(n,N)



= N Y

k=1



eN alz+ m X

j=1

(N alz)rj

rj!

(xj−1) 

.

LetF(z) =P∞ k=0alz

k _{be analytic function, wherea}

l≥0 for allkandal>0

for all large enoughk. Denotef(z) = lnF(z)−αlnz.

Theorem 8.4. (Lemma 2.2.2 in [10])Ifa0=a1=. . .=ak0−1= 0andak0>0,

then the equationf0(z) = 0has a unique real positive root zα for anyα > k0.

Theorem 8.5. We havef00(zα)>0 for anyα > k0.

Proof.Forz=zα one represents

f00(z) =zF

00_F−zF02_+F0_F

zF2 .

Here zF2 >0 and zF00F−zF02+F0F =P∞

l=0blzl, wherebl =P l

i=0(l−i+

1)(l−2i+ 1)aial−i+1 = P

bl+1 2 c

i=0 (l−2i+ 1) 2_a

ial−i+1. Therefore bl >0 for all

Theorem 8.6. (Lemma 2.2.3 in [10]) Suppose there exists a γ > 0 such that f00(zα)> γ andzα> γ, whereα0≤α≤α1. Then

1 2πi

I

|z|=zα

_F(z)

zα N _dz

z =

eN f(zα)

zα(2πf00(zα)N)1/2 h

1 +O(N−1/2)i.

Theorem 8.7. (Cauchy-Schwarz inequality) Let A be a symmetric positive definite matrix of size m×m. Then |xa|2 _{≤ (xAx)(aA}−1_{a) for any real}