On Asymptotic Expansion in the Random Allocation of Particles ... - arXiv

On Asymptotic Expansion in the Random Allocation of Particles by Sets Saidbek Sh. Mirakhmedov1 and Sherzod M. Mirakhmnedov2

Abstract. We consider a scheme of equiprobable allocation of particles into cells by sets. The Edgeworth type asymptotic expansion in the local central limit theorem for a number of empty cells left after allocation of all sets of particles is derived. Key words. Asymptotic expansion, allocation scheme, empty cells, Bernoulli distribution. MSC: 62G20, 60F05

1. Introduction. Many combinatorial problems in probability and statistics can be formulated and best understood by using appropriate random allocation schemes (alternatively known as urn models). Such models naturally arise in statistical mechanics (Feller (1968)), clinical trials (Gani (1993)), cryptography (Menezes et al. (1997)) etc. The properties of various random allocation schemes have been extensively studied in the probabilistic and statistical literature, see books by Kolchin et al. (1976), Johnson and Kotz (1977) and survey papers by Ivanov et al. (1984) and Kotz and Balakrishman (1997). The classical allocation scheme assumes equiprobable allocation of n particles to a finite number N of cells, i.e. the probability of a particle falling into any particular cell is equal to N −1 . There are several generalizations of the classical scheme, see, for example, Ivanov et al. (1984). In the present paper we consider the following generalization of the classical model. Let cells be indexed by 1, 2,..., N and sets of particles be indexed by 1, 2,..., s , a set of particles with index j containing n j particles. We assume that • Particles are allocated one set at a time • Sets of particles are allocated independently of each other. • No two particles from a given set can be allocated to a common cell.

1

Saidbek Sh. Mirakhmedov

Institute of “Algoritm and Engineering”. Fayzulla Hodjaev-45, Tashkent. Uzbekistan e-mail [email protected] Sherzod M. Mirakhmedov (former Sh.A. Mirakhmedov)( ) GIK Institute of Engineering Sciences and Technology. Topi-

2

23460. Swabi, N.W.F.P. Pakistan. e-mail [email protected] ;

2 • All C Nni possible variants of the allocation of particles from the i-th set have the same

( )

probability equal to C Nni

−1

, i = 1, 2,..., s .

This paper deals with the random variable (r.v.) μ 0 = μ 0 ( N , n1 ,..., n s ) giving the number of empty

cells left after all the s sets of particles have been allocated. Note that this allocation scheme corresponds to an urn model when we deal with s independent samples of sizes n1 , n2 ,..., ns respectively, where each sample is drawn by simple sample scheme without replacement from a population of size N; here the r.v. μ0 is a number of untouched elements of the population. The classical allocation scheme corresponds to the case n1 = n2 = ... = ns = 1 .The random allocation of particles by sets was also studied in the literature. For detailed surveys of theory and applications see Ivanov et al. (1984) and Kotz and Balakrishman (1997). The most relevant to the present paper references are Park (1981), Mikhhaylov (1980, 1981), Vatutin and Mikhaylov (1982) and Mirakhmedov (1985, 1994, 1996). In particular, the most general limit theorem for r.v. μ0 were obtained by Mikhaylov and Vatutin (1982): they proved the central limit theorem and Poisson limit theorem for μ0 under rather weak conditions, giving lower bounds for the remainder terms. But the topics of interest to us here are not readily available in the literature. In the present paper, we will derive an Edgeworth type asymptotic expansion in the local central limit theorem for μ0 using a combination of two approaches. The first approach is based on a Bartlett type formula for the characteristic function of μ0 and the second one is based on the Frobenius-Harper technique, exploiting the fact that the r.v. μ0 can be represented as a sum of independent Bernoulli r.v.’s. Thus in this paper our purpose is twofold. On the one hand, we wish to derive a better local approximation formula for the distribution of μ0 . On the other hand, we would also like to develop the above mentioned methods to obtain an asymptotical expansion type result. The organization of the paper is as follows. In Section 2, a Bartlett type formula is presented. An asymptotic expansion of the characteristic function of μ0 is given in Section 3. In Section 4, an asymptotic expansion is obtained in the central local limit theorem for μ0 in the case of a fixed s. The case when s can increase as min nl → ∞ is discussed in Section 5. 1≤l ≤ s

3 In what follows, all the asymptotic relations assume that min nl → ∞ and N → ∞ . By the symbol C 1≤l ≤ s

(with and without subscripts) we denote absolute positive constant. The relation ξ ∼ B (1, p ) means that the r.v. ξ has the Bernoulli distribution with parameter p, 0 < p < 1 .

2. Bartlett’s type formula.

The allocation of the l-th set of particles is determined by the random allocation indicators ηl ,m , m = 1, 2,..., N , where ηl ,m = 1 if a particle of the l-th set falls into the cell with index m, ηl ,m = 0

otherwise. It is clear that ηl ,m ∼ Bi (1, pl ) with p l = nl / N ; also ηl ,1 + ηl ,2 + ... + ηl , N = nl , l = 1,2,..., s , and N

μ0 = ∑ ⇑ {η1,m + η2,m + ... + η s ,m = 0} ,

(2.1)

m =1

where ⇑ { A} is the indicator of the event A. Let ξl ,m , l = 1, 2,..., s ; m = 1, 2,..., N be a collection of the mutually independent r.v.’s ,

ξlm ∼ Bi (1, pl ) , ζ l , N = ξl ,1 + ξl ,2 + ... + ξl , N . It is easy to check that for any k1 , k 2 ,..., k N with k l = 0 or 1 such that k1 + k2 + ... + k N = nl P {ηl ,1 = k1 ,ηl ,2 = k2 ...,ηl , N = k N } = P {ξl ,1 = k1 , ξl ,2 = k2 ..., ξl , N = k N / ζ l , N = nl } .

(2.2)

Let ηl = (ηl ,1 ,ηl ,2 ,...,ηl , N ) , ξl = (ξl ,1 , ξl ,2 ,..., ξl , N ) . For any measurable function f such that E f (ξ1 , ξ 2 ,..., ξ s ) < ∞ one has the following Bartlett’s type formula (see Bartlett (1938)): Ef (η1 ,η2 ,...,ηs ) =

π

1 s

∏ (2π P{ζ l =1

l ,N

π

⎧ s ⎫ Ef ξ ξ ξ τ l (ζ l , N − nl ) ⎬ dτ 1...dτ s . ... ( , ,..., ) exp ∫−π −∫π 1 2 s ⎨⎩i ∑ l =1 ⎭ = n })

(2.3)

l

Indeed, by the total expectation formula ⎧ s ⎫ Ef (ξ1 , ξ 2 ,..., ξ s ) exp ⎨i ∑τ l (ζ l , N − nl ) ⎬ = ⎩ l =1 ⎭ ⎡ ⎤ ⎧ s ⎫ Eζ 1, N ,ζ 2 , N ,...,ζ s , N ⎢ exp ⎨i ∑τ l (ζ l , N − nl ) ⎬ E ( f (ξ1 , ξ2 ,..., ξ ) / ζ 1, N , ζ 2, N ,..., ζ s , N )⎥ ⎩ l =1 ⎭ ⎣ ⎦

Integrating both sides of this equality with respect to τ 1 , τ 2 ,..., τ s over the s dimensional cube [−π , π ] s we get

4 π

π

⎧ s ⎫ ... Ef ( ξ , ξ ,..., ξ ) exp ⎨i ∑τ l (ζ l , N − nl ) ⎬ dτ 1dτ 2 ...dτ s ∫−π −∫π 1 2 ⎩ l =1 ⎭

= (2π ) s P {ζ 1, N = n1 , ζ 2, N = n2 ,..., ζ s , N = ns } E ( f (ξ1 , ξ 2 ,..., ξ ) / ζ 1, N = n1 , ζ 2, N = n2 ,..., ζ s , N = ns ) . Formula (2.3) follows due to (2.2) and the fact that ζ 1, N , ζ 2, N ,..., ζ s , N are independent r.v.’s . Bartlett’s type formula seems to be very effective in applications to problems of random allocation schemes see, for instance, Holst (1979) and Mirakhmedov (1985, 1996, 2007). In particular, if f (η1 ,η2 ,...,ηs ) = exp{it μ0 } formula (2.3) generates an integral representation of the characteristic function of μ 0 . It is important that in this case the integrand in (2.3) is the characteristic function of a sum of independent ( s + 1) -dimensional random vectors (r.vec’s) (see relation (3.10) below). This fact allows us to use the method of characteristic functions which is well developed for a sum of independent r.vec’s, see e.g. the book by Bhattacharya and Rao (1976) (from now on it is referred to as BR). 3. Asymptotic expansion of the characteristic function of μ 0 .

Put p l = nl / N ,

q l = 1 − p l , Q s = q1 q 2 ...q s , s

g (u1 , u 2 ,..., u s ) =⇑ {u1 + u 2 + ... + u s = 0} − Q s + Q s ∑ q l−1 (u l − p l ) ,

(3.1)

l =1

and define an r.v. ν 0 by N

ν 0 = ∑ g (η1m ,η 2 m ,...,η sm ) .

(3.2)

m =1

We have ν 0 = μ 0 − NQ s and N

N

s

E μ0 = ∑ P {η1m + η2 m + ... + η sm = 0} = ∑∏ P {ηlm = 0} = NQs . m =1

(3.3)

m =1 l =1

Note that g (ξ1m , ξ 2 m ,..., ξ sm ) as well as (g (ξ 1m , ξ 2 m ,..., ξ sm ), ξ 1m , ξ 2 m ,..., ξ sm ) , m = 1,2,..., N , are i.i.d. r.vec’s. Moreover, it is not hard to check that N

∑Var g (ξ

1m

, ξ2 m ,..., ξ sm ) = N σ 2

(3.4)

m =1

with ⎛

s

⎞

⎝

l =1

⎠

σ 2 = Qs ⎜ 1 − Qs (1 + ∑ pl ql−1 ) ⎟ ,

(3.5)

Eg (ξ 1m , ξ 2 m ,..., ξ sm ) = 0 , Eg (ξ 1m , ξ 2 m ,..., ξ sm )(ξ lm − p l ) = 0 , l = 1,2,..., s ,

(3.6)

and also that

5 Eg (ξ 1m , ξ 2 m ,..., ξ sm )(ξ lm − p l ) 2 = 0 , l = 1,2,..., s .

(3.7)

For simplicity of notation we put ~

g~ m = g (ξ 1m , ξ 2 m ,..., ξ sm ) / σ ,

ξ lm = (ξ lm − p l ) / p l q l .

(3.8)

~ ~ Let g~ and ξ l be r.v.’s having common distributions with g~m and ξ lm respectively. Set

Θ N (t ) = ∫ Ψ N (t , τ 1 ,..., τ s )dτ 1 ...dτ 1

(3.9)

A0

with

{

}

A0 = τ 1 , τ 2 ,..., τ s : τ l ≤ π Np l q l , l = 1,2,..., s , ⎧ it ~ s iτ l ~ ⎫ Ψ (t , τ 1 ,..., τ s ) = E exp⎨ ξl ⎬ . g +∑ N ⎭ l =1 ⎩ N

{

Put f (η1 ,η2 ,...,ηs ) = exp itν 0 / σ N

} in the formula (2.3) and use the inversion formula for the local

probability P{ζ lN = nl } to get ⎧ ν 0 ⎫ Θ N (t ) . ⎬= ⎩ σ N ⎭ Θ N (0)

def

ϕ N (t ) = E exp ⎨it

(3.10)

Define polynomials G1 (t ) and G 2 (t ) as Gk (t ) =

1 (2π ) s / 2

⎧ 1 s 2⎫ P t τ τ τ l ⎬ dτ 1...dτ s , k =1,2 ( , ,..., ) exp ∫ k 1 s ⎨⎩− 2 ∑ ⎭ l =1 Rs

(3.11)

where P1 (t , τ 1 ,..., τ s ) = P2 (t ,τ 1 ,...,τ s ) =

(

)

i3 ~ ~ 3 E tg~ m + τ 1ξ 1m + ... + τ s ξ sm , 6

i4 ⎡ E tg m + τ 1ξ1m + ... + τ sξ sm 24 ⎢⎣

(

)

4

(

(

− 3 E tg m + τ 1ξ1m + ... + τ sξ sm

) ) ⎤⎥⎦ 2 2

1 + P12 (t ,τ 1 ,...,τ s ) . 2 Taking into account (3.6) and (3.7) we obtain G1 (t ) =

(it ) 3 ~ 3 Eg , 6

G 2 (t ) =

(it ) 6 ~ 3 2 (it ) 4 ( Eg ) + 72 24 +

(

2 s ⎡ ~4 ~ 2 ξ~ ) 2 − 3⎤ + (it ) E g 3 ( E g − ∑ l ⎢ ⎥ 4 l =1 ⎣ ⎦

)

1 s ∑ 3Eξl4 + 5( Eξl3 )2 + 3(s + 2) . 24 l =1

∑ (Eg~ s

l =1

2

~ ~

~

)

ξ l Eξ l 3 − Eg~ 2 ξ l 2 + 1

6 Put ⎧ t 2 ⎫⎛ ⎞ 1 1 G1 (t ) + (G 2 (t ) − G 2 (0) ⎟⎟ , W N (t ) = exp⎨− ⎬⎜⎜1 + N N ⎠ ⎩ 2 ⎭⎝

(

TN = N min ⎛⎜ E g ⎝

)

3 −1

, p1q1 ,..., ps qs ⎞⎟ . ⎠

(3.12)

Theorem 1. There exist constants C0 and C1 such that if t ≤ C0TN

(3.13)

then ⎧ t2 ⎫ ⎬ ⎩ 12 ⎭

ϕ N (t ) − WN (t ) ≤ C1LN (1 + t ) exp ⎨ − 5

with s ⎛ ⎞ −3/ 2 5 LN = N −3/ 2 ⎜ E g + ∑ ( pl ql ) ( pl4 + ql4 ) ⎟ . l =1 ⎝ ⎠

(3.14)

Proof. Let s ⎧ 1⎛ 1 1 ⎞⎫ ⎛ ⎞ PN (t ,τ 1 ,...,τ s ) = exp ⎨ − ⎜ t 2 + ∑τ l2 ⎟ ⎬ ⎜ 1 + Ρ1 (t ,τ 1 ,...,τ s ) + Ρ 2 (t ,τ 1 ,...,τ s ) ⎟ , N N ⎠ l =1 ⎠⎭ ⎝ ⎩ 2⎝

{

Α1 = τ 1 ,...,τ s : τ l ≤ C2 Npl ql ( pl4 + ql4 )

(

Lemma 1. If t ≤ 2 N E g

)

3 −1

−1/ 3

}

, l = 1, 2,..., s .

and (τ 1 ,...,τ s ) ∈ A1 then

⎧ 1 ⎫ Ψ N (t ,τ 1 ,...,τ s ) ≤ exp ⎨− (t 2 + τ 12 + ... + τ s2 ) ⎬ . ⎩ 6 ⎭

(

)

Proof. In view of (3.6), the r.vec. Vm = g m , ξ1m ,..., ξ sm has a unit correlation matrix. Due to the

well-known inequalities between Lyapunov’s ratios (see Lemma 6.2 of BR) we have 3

(

E ξl ≤ E ξl

)

5 1/ 3

and E ξl = ( pl4 + ql4 ) ( pl ql ) 5

−3/ 2

. Now Lemma 1 follows from Theorem 8.7 of BR.

Lemma 2. There exist constants C2 and C3 such that if

(

t ≤ C2 N E g

)

5 −1/ 3

(3.15)

and (τ 1 ,...,τ s ) ∈ A1 then s s ⎧ 1⎛ ⎛ ⎞⎫ 9 9⎞ Ψ N (t ,τ 1 ,...,τ s ) − PN (t ,τ 1 ,...,τ s ) ≤ C3 LN ⎜ 1 + t + ∑ τ l ⎟ exp ⎨− ⎜ t 2 + ∑τ l2 ⎟ ⎬ . l =1 l =1 ⎝ ⎠ ⎠⎭ ⎩ 4⎝

Proof. Lemma 2 follows from Theorem 9.10 of BR because (3.6) and Vm has unit correlation matrix.

7 Recalling (3.9) and (3.11) we have ⎧ t2 ⎫ 1 1 ⎛ ⎞ G1 (t ) + G2 (t ) ⎟ exp ⎨− ⎬ = ∫ Ψ N (t ,τ 1 ,...,τ s ) − PN (t ,τ 1 ,...,τ s ) dτ 1...dτ s Θ N (t ) − ⎜ 1 + N N ⎝ ⎠ ⎩ 2 ⎭ A1

(

∫

−

s

PN (t ,τ 1 ,...,τ s )dτ 1...dτ s +

)

def

∫

Ψ N (t ,τ 1 ,...,τ s )dτ 1...dτ s = ∇1 + ∇ 2 + ∇ 3 .

(3.16)

A0 − A1

− A1

Let t satisfy (3.15). Then, using Lemma 2, we have ⎧ t2 ⎫ ∇1 ≤ C4 LN (1 + t ) exp ⎨ − ⎬ . ⎩ 4⎭ 9

(

)

5 −1/ 3

Now assume that C2 N E g ∇1 ≤

∫Ψ

N

(3.17)

(

≤ t ≤C0 N E g

(t ,τ 1 ,...,τ s ) dτ 1...dτ s + ∫

A1

A1

)

3 −1

. Applying Lemma 1 we obtain

⎧ t2 ⎫ PN (t ,τ 1 ,...,τ s ) dτ 1...dτ s ≤ C5 LN exp ⎨ − ⎬ . ⎩ 12 ⎭

(3.18)

Due to the definition of PN (t ,τ 1 ,...,τ s ) , it is clear that ⎧ t2 ⎫ ∇2 ≤ C6 LN (1 + t 6 ) exp ⎨ − ⎬ . ⎩ 2⎭

(3.19)

We have ⎧ i Ψ (t ,τ 1 ,...,τ s ) ≤ E exp ⎨ ⎩ N ⎧ i ≤ E exp ⎨ ⎩ N

s

⎫

⎛⎛

⎧ i ⎧ it ⎫ ⎞ g ⎬ − 1⎟ exp ⎨ ⎩ N ⎭ ⎠ ⎩ N

∑τ lξl ⎬ + E ⎜ ⎜ exp ⎨ l =1

⎭

⎝⎝

s

⎫

t

l =1

⎭

∑τ l ξ l ⎬ +

⎫⎞

s

∑τ ξ ⎬⎭ ⎟ l =1

l l

⎠

s t ⎧ iτ ⎫ E g = ∏ E exp ⎨ l ξl ⎬ + Eg . N N ⎩ N ⎭ l =1

Use here the inequalities x ≤ exp{( x 2 − 1) / 2} and 1 + x ≤ e x to get 2 ⎧⎪ 1 s ⎛ t ⎧ iτ l ⎫ ⎞ ⎫⎪ ξl ⎬ ⎟ ⎬ + Ψ (t ,τ 1 ,...,τ s ) ≤ exp ⎨ − ∑ ⎜ 1 − E exp ⎨ Eg ⎜ ⎟ 2 N N ⎩ ⎭ 1 = l ⎪⎩ ⎝ ⎠ ⎪⎭ 2 s ⎧ iτ l ⎫ ⎞ e t ⎪⎧ 1 s ⎛ ⎪⎫ ξl ⎬ ⎟ + E g ⎬. ≤ exp ⎨ − ∑ ⎜ 1 − E exp ⎨ N ⎩ N ⎭ ⎠⎟ ⎪⎩ 2 l =1 ⎝⎜ ⎪⎭

(3.20)

Because ξl ∼ B(1, pl ) and sin 2 we have

z z2 ≥ , z ≤π , 2 π2

(3.21)

8 2

τl τ2 ⎧ iτ ⎫ E exp ⎨ l ξl ⎬ = 1 − pl ql sin 2 ≤ 1 − 2l . π N 2 Npl ql ⎩ N ⎭ Hence from (3.20) we obtain

⎧ 1 Ψ (t ,τ 1 ,...,τ s ) ≤ exp ⎨ − 2 ⎩ 2π N

s

∑τ l2 + l =1

es t E g ⎫ ⎬. N ⎭

(3.22)

(

If (τ 1 ,...,τ s ) ∈ A0 − A1 then there exists an index l0 such that C2 Npl0 ql0 pl40 + ql40

)

−1/ 3

≤ τ l0 ≤ π Npl0 ql0 .

Therefore, using (3.22), we get ⎧ ⎧ C ⎫ ⎡C ⎤⎫ ∇3 ≤ 2( s +1) 2 π s exp ⎨ − Npl0 ql0 ⎢ 22 − C0e s ⎥ ⎬ ≤ 2( s +1) 2 π s exp ⎨ − 22 Npl0 ql0 ⎬ ⎩ 8π ⎭ ⎣ 4π ⎦⎭ ⎩

(3.23)

because of (3.13), (3.12), the choice of C0 ≤ C2 / e s 8π 2 and E g ≤ 1 due to the well-known inequalities

for moments. Thus by (3.17), (3.18), (3.19) and (3.23) from (3.16) we have

(

)

⎧ t2 ⎫ ⎧ t2 ⎫ 1 1 9 ⎛ ⎞ Θ N (t ) − ⎜ 1 + G1 (t ) + G2 (t ) ⎟ exp ⎨ − ⎬ ≤ C7 LN 1 + t exp ⎨ − ⎬ . N N ⎝ ⎠ ⎩ 2⎭ ⎩ 12 ⎭ In particular, 1 ⎛ ⎞ Θ N (0) − ⎜ 1 + G2 (0) ⎟ ≤ C7 LN . N ⎝ ⎠

(3.24)

Theorem 1 follows from these relations and formula (3.10). Remark 1. Let, for some C,

(

Eg Eg

)

3 −1

≤ C min( pl gl ) . 1≤ l ≤ s

(3.25)

Then condition (3.13) of Theorem 1 can be replaced by

(

t ≤ C0 N E g

)

3 −1

.

Indeed, in this case we should only give an alternative bound for ∇3 as follows: ∇3 ≤ 2

( s +1) 2

⎡C ⎤ ⎪⎫ Eg ⎪⎧ s 2 π exp ⎨ − Npl0 ql0 ⎢ 2 − C0e 3 ⎥⎬ pl0 ql0 E g ⎥⎦ ⎪⎭ ⎢⎣ 4π ⎪⎩ s

⎧ ⎧ C ⎫ ⎡C ⎤⎫ ≤ 2( s +1) 2 π s exp ⎨ − Npl0 ql0 ⎢ 22 − CC0e s ⎥ ⎬ ≤ 2( s +1) 2 π s exp ⎨ − 22 Npl0 ql0 ⎬ , ⎩ 8π ⎭ ⎣ 4π ⎦⎭ ⎩ because of (3.25), (3.26) and due to the choice of C0 ≤ C2 / Ce s 8π 2 .

(3.26)

9 4. Asymptotic Expansion in the Local Limit Theorem for μ0 . Case of fixed s. We set s

(

)

s

((

M 2 = ∑ Eg 2ξl Eξl3 − Eg 2ξl2 + 1 , M 3 = Eg 3 , M 4 = Eg 4 − 3∑ Eg 2ξl l =1

l =1

)

2

)

+1

and define Wˆ N ( x) as the inverse Fourier transform of above defined function WN (t ) . The function 2 2 dν Wˆ N ( x) can be obtained by formally substituting (−1)ν ν e− x / 2 instead of (it )ν e − t / 2 for each ν in the dx

expression for WN (t ) . Let Hν ( x) stand for the ν-th order Hermit-Chebishev polynomial. Recall that H 2 ( x ) = x 2 − 1 , H 3 ( x ) = x 3 − 3x , H 4 ( x ) = x 4 − 6 x 2 + 3 , H 6 ( x ) = x 6 − 15 x 4 + 45 x 2 − 15 . We have 2

1 − x2 ˆ WN ( x ) = e 2π

H 3 ( x) 1 ⎛1 1 ⎛ ⎞⎞ 2 M3 + ⎜ H 6 ( x)M 3 + H 4 ( x)M 4 + H 2 ( x)M 2 ⎟ ⎟ . ⎜1 + 4 N ⎝ 18 8 6 N ⎠⎠ ⎝

Recall (3.1), (3.5), (3.8) and additionally put n = max nl , Ps = p1 ⋅ ... ⋅ ps , α = max min ( Qs2 pl ql−1 , Ps2 pl−1ql ) , 1≤l ≤ s

1≤l ≤ s

xk = ( k − NQs ) / N σ .

(4.1)

Also keep in mind that all of pl , σ , g etc depend on N , n1 ,…, ns . Theorem 2. Let (3.25) hold and sup σ E g ≤ C . 3

(4.2)

n1 ,...,ns

Then there exist constants C8 and C9 such that max

0≤ k ≤ N − n

(

)

− C9 N α N σ P {μ0 = k } − Wˆ N ( xk ) ≤ C8 LN + N ( s +1) / 2σ PQ , s se

with LN and σ from (3.12) and (3.5) respectively. From Theorem 2 immediately follows Corollary 1. If pl are bounded away from zero and one for all l = 1, 2,..., s then

max

0≤ k ≤ N − n

⎛ 1 ⎞ N σ P {μ0 = k } − Wˆ N ( xk ) = O ⎜ 3/ 2 ⎟ . ⎝N ⎠

(

Proof of Theorem 2. Put TN = C0 N E g

N σ P {μ0 = k } − Wˆ N ( xk ) ≤

1 2π

∫

t ≤TN

)

3 −1

. Due to the inversion formula, we have

ϕ N (t ) − WN (t ) dt +

∫

TN ≤ t ≤π N σ

ϕ N (t ) dt

10 +

∫

def

WN (t ) dt = Δ1 + Δ 2 + Δ 3.

(4.3)

t ≥TN

Use Theorem 1 and Remark 1 to get Δ1 ≤ C10 LN .

(4.4)

Δ 3 ≤ C11LN .

(4.5)

It is obvious that

Let X ∗ be the symmetrization of the r.vec. X , i.e. X ∗ = X − X ′ , where X ′ and X are mutually independent r.vec.’s having a common distribution , a be the distance between the real a and the integers; DX (u ) = E

( X ,u) ∗

2

(

)

, where X ∗ , u is the scalar product of the vectors X ∗ and u.

Lemma 3. For any real u one has

⎛ u 4 DX ⎜ ⎝ 2π

2 ⎞ ⎛ u 2 ⎟ ≤ 1 − E exp {i ( X , u )} ≤ 2π DX ⎜ ⎠ ⎝ 2π

⎞ ⎟. ⎠

Proof of Lemma 3 was presented by Mukhin (1984).

For simplicity of notation, let g and ξl stand for r.v.’s having the same distributions as g (ξ1m ,..., ξ sm ) and ξlm , respectively. Let C12 ≤ t ≤ π . Putting X = ( g , ξ1 ,..., ξ s ) and u = ( t ,τ 1 ,...,τ s ) , we have 1 ⎛ u ⎞ DX ⎜ g ∗t + ξ1∗τ 1 + ... + ξ s∗τ s ⎟=E 2π ⎝ 2π ⎠

(

)

≥ P {ξ1 + ... + ξ s = 0, ξ1′ = 1, ξ 2′ + ... + ξ s′ = 0}

2

(

+ P {ξ1 = 0, ξ 2 = ... = ξ s = ξ1′ = ξ 2′ = ... = ξ s′ = 1} = Qs2 p1q1−1

1 ⎡(1 + Qs q1−1 ) t − τ 1 ⎤ ⎦ 2π ⎣

2

)

1 ⎡ 1 + Qs q1−1 t − τ 1 ⎤⎦ ⎣ 2π

2

+ Ps2 q1 p1−1

because t is bounded away from zero and hence

2

1 ⎡⎣Qs q1−1t − τ 1 ⎤⎦ 2π 1 ⎡⎣Qs q1−1t − τ 1 ⎤⎦ 2π

(

)

2

≥ C13α ,

1 ⎡ 1 + Qs q1−1 t − τ 1 ⎤⎦ ⎣ 2π

{

2

and

}

(4.6) 1 ⎡⎣Qs q1−1t − τ 1 ⎤⎦ 2π

2

can

not be equal to zero simultaneously. Now use inequalities x ≤ exp ( x 2 − 1) / 2 , (4.6) and Lemma 3 to get

(

Ψ tσ N ,τ 1 Np1q1 ,...,τ s Nps qs

)

11

(

⎧ 1 ≤ exp ⎨ − ⎛⎜ 1 − Ψ tσ N ,τ 1 Np1q1 ,...,τ s Nps qs ⎩ 2⎝

)

2

⎞ ⎫ ≤ e −C14α / 2 . ⎟⎬ ⎠⎭

(4.7)

Hence using (3.10) and taking into account (3.9), (3.10), (3.24) and condition (4.2), under which TN / σ N ≥ C0 / C = C12 , by simple manipulations we obtain

∇2 ≤

π s +1 Θ(0)

( s +1) / 2 PQ exp {−C14 N α / 2} ≤ exp {−C15 N α } . s sN

(4.8)

Theorem 2 follows from (4.3),(4.4), (4.5) and (4.8). 5. Asymptotic Expansion in the Local Limit Theorem for μ0 when s may increase.

In this Section we apply another approach (known as Frobenius-Harper technique) first considered by Frobenius (1910) and rediscovered by Harper (1967); it was also used, for instance, by Park (1981), Vatutin and Mikhaylov (1984) and Gani (2004). We wish to use the distributional equality of μ0 to a sum of independent Bernoulli r.v.’s. Vatutin and Mikhaylov (1984) showed that the probability generating function F ( z ) = Ez μ0 satisfies the Frobenius-Harper property: F ( z ) is a polynomial of degree

N − n and has only negative real roots, which we denote by − d1 , −d 2 ,..., − d N − n . Hence z − d m N − n Ym = ∏ Ez . m =1 1 + d m m =1

N −n

F ( z) = ∏

(5.1)

where Y1 , Y2 ,..., YN − n is a sequence of independent r.v.’s and Yl ∼ B(1, al ) with al = (1 + d l ) , −1

l = 1, 2,..., N − n . Thus the r.v.’s μ0 and μ0′ = Y1 + Y2 + ... + YN − n have a common distribution. Using this fact we can obtain an asymptotic expansion result without any restrictions on s – the number of the sets of particles. Indeed, by Fourier inversion formula P {μ0 = k } =

1 2π Var μ0

∫

e −ituk γ (t )dt ,

t ≤π Var μ0

where uk = (k − E μ0 ) / Var μ0 , ⎧⎪ μ0 − E μ0 ⎫⎪ N − n ⎧⎪ Ym − am ⎬ = ∏ E exp ⎨it Var μ0 ⎭⎪ m =1 ⎩⎪ ⎩⎪ Var μ0′

γ (t ) = E exp ⎨it

(5.2) ⎫⎪ ⎬, ⎭⎪

(5.3)

Because of (5.1) and E μ0 = E μ0′ = a1 + a2 + ... + aN − n , Var μ0 = Var μ0′ . Due to the second equality in (5.3), for t ≤ π Var μ0′ we find, using (3.21), that

12 ⎛ ⎜ m =1 ⎝

N −n

γ (t ) = ∏ ⎜ 1 − am (1 − am ) sin 2 2

t 2 Var μ0′

⎞ ⎧ t2 ⎫ ≤ exp ⎟⎟ ⎨− 2 ⎬ . π ⎭ ⎩ ⎠

Therefore, applying the classical method of the proof of asymptotical expansion results in the central local limit theorem for sums of independent lattice r.v.’s (see, for instance, Petrov (1995), and also Section 4 of the present paper), we can easily derive an asymptotic expansion result for P {μ0 = k } without any restrictions on s . This result has the following form. Put L3 (n, N ) = (Var μ0′ )

−3/ 2

L4 (n, N ) = (Var μ0′ )

N −n

∑ E (Y

m

m =1

−2

∑ ( E (Y

N −n m =1

⎛ N −n ⎞ = ⎜ ∑ am (1 − am ) ⎟ ⎝ m =1 ⎠

m

− am )

3

⎛ N −n ⎞ = ⎜ ∑ am (1 − am ) ⎟ ⎝ m =1 ⎠

(

− am ) − 3 E (Ym − am ) 2 4

−2 N − n

∑a m =1

m

)

2

−3/ 2 N − n

∑a m =1

m

(1 − am )(1 − 2am ) ,

)

(1 − am ) (1 − 6am (1 − am ) ) .

Theorem 3. There exists a constant C such that 2

1 − u2k Var μ0 P {μ0 = k } − e 2π

max

0≤ k ≤ N − n

+

H 3 (uk ) ⎛ L3 ( n, N ) ⎜1 + 6 ⎝

H 6 (uk ) 2 H (u ) C ⎞ L3 ( n, N ) + 4 k L4 ( n, N ) ⎟ ≤ . 3/ 2 72 24 ⎠ (Var μ0 )

However, such a result is not practical, because the terms of the asymptotic expansion depend on d1 , d 2 ,..., d N −n the calculation of which is a difficult problem. Theorem 2 of Section 4 assumed that the number s of the sets of particles is fixed. This is a technical condition, as we used the method of asymptotic expansions for the characteristic functions of the sums of independent r. vec.’s, where s is the dimension of those vectors. Actually, the basic formula (3.10) is still correct without any restrictions on s . Hence it can be used for formal construction of the terms of an asymptotic expansion of the characteristic function of ( μ0 − E μ0 ) / Var μ0 . Recalling (2.1) and that η1m ,η2 m ,...,ηsm are mutually independent r.v.’s with ηlm ∼ Bi (1, pl ) for each m = 1, 2,..., N , we have N

E μ 02 = ∑ E ⇑ {η1m + η2 m + ... + ηsm = 0} m =1

N

+ ∑ E ⇑ {η1m + η2 m + ... + ηsm = 0} ⇑ {η1l + η2 l + ... + ηsl = 0} m ,l =1 m ≠l

13 N

= NQs + + ∑ P {η1m = 0,η1l = 0,η2 m = 0,η2 l = 0,...,ηsm = 0,ηsl = 0}. m ,l =1 m ≠l

The r.vec.’s (η1m ,η1l ), (η2 m ,η2l ),..., (η sm ,η sl ) are mutually independent, because the sets of particles are allocated independent of each other. Hence E μ 02 = NQs +

∑ ∏ P {η s

N

m ,l =1 j =1 m ≠l

jm

= NQs + N ( N − 1)Q − NQ 2 s

s nj ⎞ ⎛ nj ⎞⎛ = 0,η jl = 0} = NQs + N ( N − 1)∏ ⎜1 − ⎟ ⎜1 − ⎟ N ⎠ ⎝ N −1 ⎠ j =1 ⎝

2 s

s

∑pq i =1

−1 i i

( −1)ν p p ... piν ( qi1 qi2 ...qiν ) −1. + NQ ∑ ν −1 ∑ i1 i2 ν = 2 ( N − 1) s

2 s

where ∑ denotes summation over all ν -tuples (i1 ,..., iν ) with components not equal to each other and such that im = 1, 2,..., s ; m = 1, 2,...,ν . From this and (3.3), (3.5) we have s N ( −1)ν −2 ∗ Qs2 ∑ pi1 pi2 ... piν ( qi1 qi2 ...qiν ) −1 ν −2 ∑ N − 1 ν =2 ( N − 1)

Var μ0 = N σ 2 +

⎛ ⎞ ⎜ ⎟ s Qs ( −1)ν −2 −1 ⎟ 2⎜ = Nσ 1 + p p ... piν ( qi1 qi2 ...qiν ) ∑ ν − 2 ∑ i1 i2 s ⎜ ⎟ ⎛ ⎛ −1 ⎞ ⎞ ν = 2 ( N − 1) ⎜⎜ ( N − 1) ⎜ 1 − Qs ⎜ 1 + ∑ pl ql ⎟ ⎟ ⎟⎟ l =1 ⎝ ⎠⎠ ⎝ ⎝ ⎠ def

= Nσ 2 (1 + bN ) .

Note that γ (t ) = ϕ N

(( Nσ

(5.4) 2

/ Var μ0 )

1/ 2

)

(

t = ϕ N (1 + bN )

−1/ 2

)

t (see (3.10) and (5.3)) owing to (3.4) and

(

(5.4). So due to (3.10) we formally have γ ( t ) = WN (1 + bN )

(

−1/ 2

)

t + ε N (t ) = WN (t ) +

(it ) 2 bN + ε N (t ) , 2

)

where ε N (t ) and ε N (t ) are of the order O N −3/ 2 . Hence the first three terms of asymptotic expansion of Var μ0 P ( μ0 = k ) equal to Wˆ N (uk ) + N −1 H 2 (uk )bN , where bN = NbN . This, in combination with Theorem 3, implies the following result Theorem 4. There exists a constant C such that max

0≤ k ≤ N − n

1 C Var μ0 P {μ0 = k } − Wˆ N (uk ) − H 2 (uk )bN ≤ 2 N Nσ (1 + bN )

(

)

3/ 2

.

14 Remark 2. In Theorem 4 exact normalization by Var μ0 is used, see uk in (5.2), whereas in

Theorem 2 we deal with normalization by σ N ( see xk in (4.1)), which is an asymptotic of Var μ0 . The extra term N −1H 2 (uk )bN appeared in Theorem 4 for that reason. N

Remark 3. The r.v. μ0 is the special case of the statistics of the form ∑ h(η1,m ,η2,m ,...,ηs ,m ) , were m =1

h is a measurable real function. For this general statistics Bartlett type formula is also hold. Hence the

formal construction of the terms of asymptotic expansions can be established. Some applications of Bartlett type formula for so called “generalized allocation schemes” are presented in Mirakhmedov (1985, 1994, 1995, 1996, 2007) . Remark 4. We restricted ourselves to the first three terms of the asymptotic expansion. It is obvious

that the presented approach can be used to derive asymptotic expansions of any length. Acknowledgement.

We would like to express our gratitude to the referees for useful comments. We also thankful to Prof. Kostya Borovkov for careful reading of our manuscript and help us to improve the presentation of the paper.

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