Discrete Math. Appl., Vol. 15, No. 2, pp. 125–143 (2005)© VSP 2005.

Approximation of moments of arbitrary integerorders by generalised factorial powers

A. P. BARANOV and YU. A. BARANOV

Abstract — For non-negative integer random variables ξ, we consider approximations of the momentsEξm , where m are integers, including negative integers. We find estimates of the difference

Eξm −s∑

k=0

{m

m − k

}Eξm−k ,

where{ mm−k

}are extensions to all integers m of Stirling numbers of the second kind, the functions xm

are the generalised factorial powers, and s is a positive integer.

Considering the distributions of the power-divergence statistics introduced in [1], in partic-ular, for solving the problems of belonging observations to a given distribution, we meetwith the need for study of the moments of integer random variables ξ both for positive andnegative powers m. For positive m this problem is resolved by the use of the well-knownrepresentation of the powers x m in the form of decomposition in factorial moments withStirling numbers of the second kind as coefficients. Following [2], we denote these num-bers by

{mk

}, that is,

xm =m∑

k=0

{m

k

}xk, (1)

where

xk = x !(x − k)! = x(x − 1) . . . (x − k + 1). (2)

Using equality (1), we obtain the decomposition

Eξm =m∑

k=0

{m

k

}Eξk =

m∑

k=0

{m

k

}dk

dtk�(t)

∣∣∣∣t=1

,

where �(t) = Etξ and m is a non-negative integer.

Originally published in Diskretnaya Matematika (2005) 17, No. 1, 50–67 (in Russian).Received July 20, 2004.

126 A. P. Baranov and Yu. A. Baranov

We would like to apply a similar approach for calculating Eξ m for negative m. In [1], arepresentation of the function x m for arbitrary m is obtained in the form of the series

xm =∞∑

k=0

{m

m − k

}xm−k , x �= 0, (3)

where{m

k

}is an extension of the Stirling numbers of the second kind to the domain including

negative m, k, and x k is defined as before by (2) with the factorials defined in terms of thestandard gamma function �(x) in the form

x ! = �(x + 1), x ≥ 0.

Here we do not discuss the domain of definition of the numbers{m

k

}and the series (3);

a detailed discussion of this subject can be found in [2] and [3]. For us it is important thatthe numbers

{mk

}satisfy the recurrence relation

{m

k

}={

m − 1

k − 1

}+ k

{m − 1

k

}(4)

for any integers k, m, k < m, and{

m

0

}= 0, m �= 0,

{m

m

}= 1, m = 0,±1,±2, . . . ,

{−1

k

}= (−k − 1)!, k = −1,−2, . . .

We define the m-fold operator δmt �(t) acting on a function �(t) of real argument t for

m < 0 as the m-fold integral

δmt �(t) =

∫ t

0

∫ t1

0. . .

∫ t−m−1

0�(t−m) dt−m . . . dt1.

Below we consider only integer m, and then it is clear that

Eξm = δmt �(t)

∣∣t=1 (5)

for both negative m and positive m if we assume that

δmt �(t) = dm

dtm�(t)

if m ≥ 0.Property (5) shows that application of equality (3) can be efficient. However there are

some difficulties in such approach.First, Eξm is not defined for negative m if P(ξ = 0) > 0. Everywhere below we assume

that 0m = 0 for any integers m including the point m = 0. This definition differs from the

Approximation of moments of arbitrary orders 127

ordinary properties of power function at the point 0 (see [2, 3]), where it is assumed thatx0 = 1 for any x . For our purpose, in approximation of the functions x m for integer valuesof the argument x , this difference in not essential, but is necessary for the definiteness ofEξm for m < 0.

Secondly, the approximation by the series (3) is complicated since there are no explicitexpressions of the numbers

{mk

}. Only several representations are known, for example,

{m

m

}= 1,

{m

m − 1

}=

m∑

k=1

(k − 1) =(

m

2

),

{m

m − 2

}=

m∑

k=1

(k − 2)

(k − 1

2

)=(

m

3

)3m − 5

4,

{m

m − 2

}=

m∑

k=1

(k − 3)

(k − 1

2

)3k − 2

4

=(

m

4

)(m − 2)(m − 3)

2, m = 0, 1, 2 . . . (6)

These expressions can be extended with the use of recurrence relation (4), but there is nopossibility to obtain acceptable expression for the numbers

{ mm−k

}as functions of m and k.

However, it is easily seen that equalities (6) for m = −1,−2, . . . yield numbers satisfyingrecurrence relation (4) and, consequently, can be considered as extensions of the Stirlingnumbers of the second kind introduced in [3]. Thus, there is only a possibility to calculate,that is, to obtain explicit values of the sums

Ss(m) =s∑

k=0

{m

m − k

}Eξm−k =

s∑

k=0

{m

m − k

}δm−k

t �(t)∣∣∣t=1

. (7)

This suggests the problem of estimating the difference

Rs(m) = Eξm − Ss(m) +s∑

k=0

{m

m − k

}P(ξ = 0)

(k − m)!

for m < 0 and fixed s = 1, 2, . . .

In order to solve this problem, we first obtain an explicit expression for the difference

Rs(m, x) = xm − Ss(m, x), (8)

where

Ss(m, x) =s∑

k=0

{m

m − k

}xm−k .

Beforehand we consider a useful example, setting m = −1 and ξ ∼ �(α), that is,assuming that the random variable ξ has the Poisson distribution with parameter α.

128 A. P. Baranov and Yu. A. Baranov

As noted in [3], as early as 1730, Stirling proved that

1

x=

s+1∑

k=1

(k − 1)!(x + 1) . . . (x + k)

+ (s + 1)!x(x + 1) . . . (x + s + 1)

=s∑

k=0

{ −1

−1 − k

}x−k−1 + (s + 1)!

xx−s−1, x �= 0. (9)

Hence it follows that

Rs(−1, x) = (s + 1)!x

x−s−1. (10)

Using (5), it is easy to show that

Eξ−k = α−kP(ξ ≥ k), k = 1, 2, . . . , (11)

therefore,

Ss(−1) = ESs(−1, ξ) =s∑

k=0

k!α−k−1P(ξ ≥ k + 1),

Eξ−1 =s∑

k=0

k!(α−k−1P(ξ ≥ k + 1) − e−α/(k + 1)!) + (s + 1)! E(ξ−1ξ−s−1). (12)

Note that (see [4])

Ss(−1) − e−αs∑

k=0

1

k + 1=

s∑

k=0

k!α−k−1P(ξ ≥ k + 2)

= e−α

(s∑

k=0

α

(k + 1)(k + 2)+ α2

(k + 1)(k + 2)(k + 3). . .

)

= e−α

∞∑

j=1

α j

j

(1

j ! − 1

(s + 2) . . . (s + j + 1)

)≤ 1 (13)

and the last estimate is valid for any s.Note also that for fixed s estimate (13) implies that, as α → ∞,

Ss(−1) − e−αs∑

k=0

1

k + 1=

s∑

k=0

k!α−k−1 + O(e−α).

Now it is clear that in order to obtain an asymptotic expansion of Eξ −1 as α → ∞, itremains to estimate Eξ−1ξ−s−1 in (12). A general formula below, which gives us an ap-proximation of Eξ −m , yields also an estimate of the remainder term in (12).

Approximation of moments of arbitrary orders 129

Lemma 1. For negative integers m and integers s ≥ 0, the difference Rs(m, x) forx �= 0,−1, . . . , m − s is of the form

Rs(m, x) =−m−1∑

r=0

s − r − m

xr+1xr+m−s

{r + m

r + m − s

}. (14)

Proof. Let

δs(m, x) = Rs(m, x) − 1

xRs(m + 1, x).

Then the obvious equality

Rs(m, x) = δs(m, x) + 1

xδs(m + 1, x) + . . . + xm+1δs(−1, x) (15)

is true, where the right-hand side consists of −m summands as well as the right-hand sideof the required equality (14). It follows directly from (8) that

δs(m, x) =s∑

k=0

x−1xm−k(

x(m − k + 1)

{m

m − k + 1

}− (m − k)

{m + 1

m + 1 − k

})(16)

and if (14) is true, then

δs(m, x) = x−1xm−s(s − m)

{m

m − s

}. (17)

In order to prove (17) we use induction on s, assuming that m is an arbitrary negativeinteger and x �= 0,−1, . . . , m − s, . . . It follows directly from (16) that

δ0(m, x) = R0(m, x) − 1

xR0(m + 1, x) = (−m)x−1xm,

and this expression corresponds to (17) for s = 0 if we take into account that for anyinteger m {

m

m + 1

}= 0.

Note that equality for δ0(−1, x) coincides with (10), since R0(0, x) = 0.

Using (16), we represent δs+1(m, x) in the form

δs+1(m, x) = δs(m, x) + x−1xm−s−1(

x(m − s)

{m

m − s

}− (m − s − 1)

{m + 1

m − s

}).

We substitute the expression for δs(m, x) corresponding to the assumption (17) into the lastequality and use recurrence relation (4) in the form

{m + 1

m − s

}={

m

m − s − 1

}+ (m − s)

{m

m − s

}.

130 A. P. Baranov and Yu. A. Baranov

Reducing the similar terms, we obtain

δs+1(m, x) = x−1(

xm−s(s − m)

{m

m − s

}+ x(m − s)xm−s−1

{m

m − s

})

+ x−1xm−s−1{

m + 1

m − s

}(s + 1 − m)

= x−1xm−s−1{

m

m − s

}(x − x + m − s − 1)

+ x−1xm−s−1{

m + 1

m − s

}(s + 1 − m)

= x−1xm+s−1(s + 1 − m)

({m + 1

m − 1

}− (m − s)

{m

m − s

})

= x−1xm−s−1(s + 1 − m)

{m

m − s − 1

}.

The last expression corresponds to δs+1(m, x) computed by formula (17), so that theinductive proof of the lemma is completed.

Lemma 1 extends the Stirling formula to the functions x m , m = −1,−2, . . . , since (10)is a particular case of (14) for m = −1.

The purpose which is served by the theorem below is to find approximations of themoments of integer-valued random variables of negative and positive orders.

Theorem 1. For any integers m, s, s ≥ 0, and positive integers x there exists a constantC(m, s) depending only on m and s such that

∣∣∣∣∣xm −

s∑

k=0

{m

m − k

}xm−k

∣∣∣∣∣ < xm−s−1C(m, s). (18)

Proof. Indeed, for m ≥ 0 and integers x > 0

xm = x(x − 1) . . . (x − m + 1) < xm .

On the other hand, for m < 0 and x > 0

xm = 1

(x + 1) . . . (x − m)< xm .

Therefore the validity of (18) and the existence of the constant C(m, s) follow from (1) form ≥ 0 and from (14) for m < 0.

It is also not difficult to see that both for the case m ≥ 0 and the case m < 0 the constantC(m, s) can be chosen explicitly in a finite form with the use of (1) or Lemma 1.

In our opinion, inequality (18) has a weakness in application to the problem of approx-imation of the moment Eξ m . The remainder term of the approximation

Eξm ≈s∑

k=0

{m

m − k

}Eξm−k

Approximation of moments of arbitrary orders 131

is of the form O(Eξm−s−1), so that in order to estimate the accuracy of approximation,estimates of the non-factorial moments are needed, whereas the factorial moments can becalculated with the use of the operator δm

t �(t), as we have seen from the example of thePoisson distribution and formula (11). Theorem 2 eliminates this weakness.

Theorem 2. For any integers m and s, s ≥ 0, and positive integers x there exists aconstant C1(m, s) depending only on m and s such that

∣∣∣∣∣xm −

s∑

k=0

{m

m − k

}xm−k

∣∣∣∣∣ < C1(m, s)(xm−s−1 + I (x < m − s − 1)) (19)

(here I (A) is the indicator of an event A).

Proof. In Theorem 1, we used the inequality x m < xm for integers m and x , x > 0. Nowwe use another inequality for the factorial power function of the integer positive argument

xm ≤ C(m)(xm + I (x < m)), (20)

where C(m) is a constant depending only on m, m and x are integers, x > 0.For m = 0, (20) is obviously true for C(0) = 1. Let m > 0. Then it is sufficient to

set C(m) = mm . Indeed, for 0 < x < m the inequality x m < mm and for x ≥ m theinequalities x < xm − km for k = 0, 1, . . . , m − 1 are true.

Therefore,

xm < xm(xm − m) . . . (xm − m2 + m) + mI (x < m)

= mm(x(x − 1) . . . (x + m − 1) + I (x < m)).

Let m < 0. Then it is sufficient to take C(m) = (1 − m)−m , since for x ≥ 1

x−1 ≤ (1 − m)(x − k)−1, k = −1, . . . , m.

Thus, inequality (20) under the conditions of Theorem 2 is proved and bound (19) followsfrom Theorem 1. In this case, it is sufficient to set C1(m, s) = C(m, s)C(m − s − 1). Notethat the constant C1(m, s) can be found in an explicit form.

Theorem 2 shows that it is possible to apply approximation (19) for calculating the mo-ments Eξm of positive and negative orders if we take into account the accepted assumptionthat 0m = 0 for all integers m:

∣∣∣∣∣Eξm −s∑

k=0

{m

m − k

}(Eξm−k − I (m < 0)

(k − m)! P(ξ = 0)

)∣∣∣∣∣

≤ C1(m, s)(Eξm−s−1 + P(ξ < m − s − 1)). (21)

Returning to the example where ξ ∼ �(α), we obtain in (12) the estimate of the re-mainder term

∣∣∣∣∣Eξ−1 −s∑

k=0

k!α−k−1P(ξ ≥ k + 2)

∣∣∣∣∣ ≤ C1(−1, s)α−s−2P(ξ ≥ s + 2),

132 A. P. Baranov and Yu. A. Baranov

which, as α → ∞, can be reduced to the form

Eξ−1 =s∑

k=0

k!α−k−1 + O(α−s−2).

As another illustration of (21), note that under the condition that α → ∞, ξ ∼ �(α)

for any fixed integer m

Eξm =s∑

k=0

{m

m − k

}αm−k + O(αm−s−1). (22)

Now consider the application of the described above approach to approximation of themoments Eξm by factorial moments Eξ k to the approximation of mixed moments of theform

µm,k̄ = Eξmr∏

i=1

ξkii ,

where ξ1, . . . , ξr are independent non-negative integer-valued random variables,ξ = ξ1 + . . . + ξr , and the parameters m, k1, . . . , kr are integers, k = k1 + . . . + kr ,k̄ = (k1, . . . , kr ). As before, for definiteness, we assume that 0m = 0 for any m. We usebound (19) given in Theorem 2 to construct an approximation of the function

f (x̄) = xmr∏

i=1

xkii

of r non-negative integer arguments x 1, . . . , xk , x̄ = (x1, . . . , xr ), x = x1 + . . . + xr . Let

M(x̄) =

s∑

j=0

{m

m − j

}xm− j

r∏

i=1

s∑

j=0

{ki

ki − j

}x

ki − ji ,

R(x̄) = xmr∏

i=1

xkii − M(x̄).

Then

M(x̄) =∑

0≤ j0<...< jr≤s

xm− j0

xk1− j11 . . . x

kr − jrr

{m

m − j0

} r∏

i=1

{ki

ki − ji

},

and for R(x̄) the representation

R(x̄) =r∏

j=0

(aj + bj + Ij ) −r∏

j=0

aj = I0 I1 . . . Ir

+r−1∑

n=0

∑

0≤ j0<...< jn≤r

Ij0 . . . Ijn

∏

i /∈{ j0,... , jn}(ai + bi) + aj0 . . . ajn

∏

i /∈{ j0,... , jn}bi

Approximation of moments of arbitrary orders 133

is valid, where

a0 =s∑

j=0

xm− j , b0 = xm−s−1, I0 = I (x < m − s − 1),

ai =s∑

j=0

xki − j , bi = xki −s−1i , Ii = I (xi , k − s − 1), i = 1, . . . , r.

It is not difficult to see that the sums of factorial powers of all arguments x, x 1, . . . , xr inthe products of the form a j1 . . . ajn

∏i /∈{ j1,... , jn} bi after multiplying the sums aj1, . . . , ajn

do not exceed m − s − 1 + ∑rj=1 kj for 0 ≤ n ≤ r − 1.

In all other summands in the representation of R(x̄), an indicator function presentsas a factor. These summands for all integer non-negative x, x 1, . . . , xr admit estimatesx

yx

y1

1 . . . xyrr I (xi < a), i = 1, . . . , r , or x

yx

y1

1 . . . xyrr I (x < a), where a, y, y1, . . . , yr are

some integers.Note that it follows from (20) and the inequality (x y)2 ≤ x2y , which is valid for any

integer y and x > 0, that

(x y)2 < C(2m)(x2y + I (x < 2y),

where C(2m) is defined in (20).The last inequality can be used for estimating the summands with indicator factor. For

example,

Eξ y I (ξ1 < a) <

(C(2y)(Eξ2y + I (ξ < 2y) + I (y < 0)

(−y)!√

P{ξ = 0})

P{ξ1 < a}.

It is easily seen from the definition of the function M(x̄) and the estimate of R(x̄) thatthe important role in the approximation of the mixed moment µ m,k̄ is played by the mixedfactorial moments of the form

µ(m, k̄) = Eξmr∏

i=1

ξki

i .

The general method of calculating µ(m, k̄) is based on the application of the operators δmt

to the generating function

�(z0, z1, . . . , zr ) = Ezξ

0

r∏

i=1

zξii =

r∏

i=1

E(z0zi )ξi .

It is obvious that

µ(m, k̄) = δmz0

δk1z1

. . . δkrzr

�(z0, z1, . . . , zr )

∣∣∣z0=z1=...=zr =1

for the generating functions � i (z) = Ezξi , i = 1, . . . , r , which converge in the circle|z| ≤ 1. Therefore,

µ(m, k̄) = δmz0

zk0

(r∏

i=1

δkizi

�i (zi )

∣∣∣zi =z0

)∣∣∣∣∣z0=1

. (23)

134 A. P. Baranov and Yu. A. Baranov

The calculation of (23) for m ≥ 0 and any k i , i = 1, . . . , r , creates no difficulty, since form-fold differentiating one can use the well-known Leibniz formula

dm

dzmzk f (z)

∣∣∣∣z=1

=m∑

j=0

(m

j

)k j dm− j

dzm− jf (z)

∣∣∣∣z=1

(24)

provided f (z) has a needed number of derivatives.However, for m < 0 there is no formula for m-fold integration of a product of two

functions. Therefore, further we will calculate the mixed moments µ(m, k̄) with regard tothe form of the function f (z) for the random variables ξ i such that ξi ∼ �(αi ), i = 1, . . . , r .

We begin with the simplest but demonstrative case where r = 2, k 2 = 0. In this case,

k̄ = k, µ(m, k) = Eξk1 (ξ1 + ξ2)

m, �1(z) = eα1(z−1), �2(z) = eα2(z−1).

For k ≥ 0, the result of action of the operator δ kz �1(z) is obvious:

δkz �1(z) = αk

1eα1(z−1).

For k < 0, fulfilling the (−k)-fold integration, we obtain

δkz �1(z) =

∫ z

0

∫ t1

0. . .

∫ t−k−1

0�(t−k) dt−kdt−k−1 . . . dt1

= αk1

eα1(z−1) −−k−1∑

j=0

(α1z) j

j ! e−α1

.

Combining the two last representations of δ kz �1(z), assuming that

∑kj=0 = 0 for k < 0, we

obtain

µ(m, k) = αk1e−αδm

z zk

eα1z −−k−1∑

j=0

(α1z) j

j !

eα2z

∣∣∣∣∣∣z=1

, (25)

where α = α1 + α2.The investigation of expression (25) for arbitrary m, k and α 1, α2 is very complicated

and, as it will be shown later, this analysis can be realised only in the case where m ≥ 0,k ≥ 0. Therefore, we restrict ourselves to obtaining the asymptotic approximation of thefactorial moment µ(m, k) only as α = α1 + α2 → ∞.

Consider the action of the operator δmz on the function

f (z) =

eα1z −−k−1∑

j=0

(α1z) j

j !

zkeα2z . (26)

Theorem 3. Let α1, α2 → ∞ in such a way that there exist constants C1, C2 such that

0 < C1 < α1α−1 < C2 < 1.

Approximation of moments of arbitrary orders 135

Then for arbitrary integers m, k, s, s ≥ 0, and the function f (z) defined in (26),

δmz f (z)

∣∣z=1 =

s∑

i=0

(m

i

)αm−i eαki + O(αm−s−1eα). (27)

Proof. For arbitrary k, we consider the simplest case where m ≥ 0. Using equality (24),we obtain

dm

dmzf (z)

∣∣∣∣z=1

=m∑

i=0

(m

i

)ki dm−i

dm−i zeαz

∣∣∣∣∣z=1

−−k−1∑

j=0

m∑

i=0

(m

i

)(k + j)i α

j1

j !dm−i

dm−i zeα2z

∣∣∣∣∣∣z=1

= eα

s∑

i=0

(m

i

)ki (α1 + α2)

m−i + O((α1 + α2)m−s−1eα)

+ O(αm2 α−k−1

1 I (k < 0)eα2).

The last equality corresponds to (27), since(m

s

) = 0 for s > m and α2 < (1 − C1)α. Theproof for the case where m < 0, is more complicated and carried out by induction on m forany fixed k. We give an inductive proof for the slightly more general formula

δmz f (z) = eαz

s∑

i=0

(m

i

)αm−i ki zk−i + O(αm−s−1eαz) + O(α|k|eεz), (28)

where 0 < ε ≤ z ≤ 1, ε is a constant, 1 − C1 < ε < 1. The usual symbol O(·) for thefunction �(α, z) in the expression y(α, z) = O(�(α, z)) means that there exists a constantC , which does not depend on α and z, such that

y(α, z) < C�(α, z)

in the domain where ε ≤ z ≤ 1 and α → ∞. The constant C may depend on the form ofthe function �(α, z).

Consider the case m = −1 and divide the domain of integration into two parts:

δ−1z f (x) −

∫ z

ε

f (t) dt =∫ ε

0f (t) dt =

∞∑

j=max(−k,0)

αj1

j !∫ ε

0t j+keα2t dt ≤ εkeεα. (29)

Then

∫ z

ε

f (t) dt −∫ z

ε

tkeαt dt = O

−k−1∑

j=0

αj1

j !∫ z

ε

t j+keα2t dt

+ O(α|k|eεz). (30)

Taking the integral by parts s + 1 times, we obtain∫ z

ε

tkeαt dt = tk

αeαt∣∣∣∣z

ε

− k

α

∫ z

ε

tk−1eαt dt = . . .

= eαz

α

s∑

i=0

(−1

i

)α−i ki zk−i + O(α−s−2eαz). (31)

136 A. P. Baranov and Yu. A. Baranov

Combining (29), (30), and (31), we arrive at (28) with m = −1.Assume that (28) holds true for m ≤ −1 and prove it for m − 1. Let

S = Sα(k, z) =s∑

i=0

(m

i

)αm−i ki

∫ z

ε

tk−i eαt dt . (32)

Then

δm−1z f (z) =

∫ z

0δm

t f (t) dt = S + O

(∫ ε

0δm

t f (t) dt

)+ O

(αm−s−1

∫ z

ε

eαzdt

),

and it suffices to consider S and estimate the integral∫ ε

0δm

t f (t) dt =∫ ε

0

∫ t1

0. . .

∫ t−m−1

0f (t−m) dt−m . . . dt1, (33)

where 0 < tj ≤ ε, j = 1, . . . , t−m−1.

Expanding in a Taylor series and comparing terms in the left-hand and the right-handsides of the inequality

ex −n∑

i=0

xi

i ! ≤ xn+1ex , (34)

we conclude that it is valid for n ≥ 0 and x ≥ 0. For negative k inequality (34) implies thatfor 0 ≤ t−m−1 ≤ ε

∫ t−m−1

0f (t−m) dt−m =

∫ t−m−1

0

(eα1t−m −

−k−1∑

i=0

(α1t−m)i

i !

)tk−meα2t−m dt−m

≤∫ t−m−1

0(α, t−m)−k tk−meαt−mdt−m = O(α−k

1 eαε).

Since the integration in (33) is taken (−m) times over domains of size at most ε, fornegative k and ε ≤ z ≤ 1

∫ ε

0δm

t f (t) dt = O(α−k1 eαε). (35)

For non-negative k and 0 < 1 − C1 < ε ≤ z,∫ ε

0δm

t f (t) dt ≤∫ ε

0. . .

∫ t−m−1

0tk−meαt−m edt−m . . . dt1 ≤ eαε − 1 = O(eαε).

Thus, estimate (35) of integral (33) is valid for any integer k.Using the equalities

(−1

j

)(m

i

)= (−1) j (−1)i

(−m + i − 1

i

),

ki (k − i) j = k!(k − i − j)! = ki+ j ,

Approximation of moments of arbitrary orders 137

we apply (31) to the sum

S =s∑

i=0

(m

i

)αm−i ki eαz

α

s∑

j=0

(−1

j

)α− j (k − i) j zk−i− j + O(αm−s−2eαz)

= eαzs∑

i=0

s∑

j=0

(−m + i − 1

i

)(−1)i+ j αm−i− j−1zk−i− j ki+ j + O(αm−s−2eαz).

We change the summation indices, setting i + j = γ, and taking into account thatm < 0, obtain

S = eαz2s∑

γ=0

(−1)γ αm−γ−1zγ kγ

γ∑

i=0

(−m + i − 1

i

)+ O(αm−s−2eαz)

= eαzs∑

γ=0

(−1)γ αm−γ−1zγ kγ

(−m + γ

γ

)+ O(αm−s−2eαz)

= eαzs∑

γ=0

αm−γ−1zγ kγ

(m − 1

γ

)+ O(αm−s−2eαz).

The last expression for S together with (35) leads to (28) for δ m−1z f (z). Thus, the

inductive on m proof is completed.

Note that the equality for indeterminate m-fold integrals of real argument z

∫. . .

∫zkeαzdz = eαz

αm

k∑

i=0

(−1)i

αi

(i + m − 1

i

)ki zk−i + cm−1(z) (36)

for m ≥ 0, k ≥ 0, and an arbitrary polynomial cm−1(z) of degree m − 1 was obtainedby I. V. Kharlamov (private communication). This result is of its own interest; it, in largemeasure, inspired the expansion given in Theorem 3.

Corollary 1. Let α1, α2 → ∞ in such a way that there exist constants C1, C2 such that

0 < C1 < α1(α1 + α2)−1 < C2 < 0.

Then for arbitrary m, k, s, s ≥ 0, mixed factorial moments can be approximated in thefollowing way:

µ(m, k) = αk1αm

s∑

i=0

(m

i

)α−i ki + O(αm+k−s−1). (37)

Equality (37) follows from (25) and (27).

138 A. P. Baranov and Yu. A. Baranov

The availability of approximation of factorial moments µ(m, k) provides a way of con-structing the approximation of ordinary moments

µm,k = Eξk1 (ξ1 + ξ2)

m (38)

=s∑

e=0

∑

0≤i, j≤1,i+ j=l

{k

k − i

}{m

m − j

}µ(m − j, k − i) + O(αm+k−s−1),

which follows from the definition of the function M(x, y, z) and the estimate of the dif-ference R(x, y, z). At the same time, it should be noted that the derivation of an explicitexpression for (38) is very complicated even for the case r = 2. However application ofsymbolic transformations with the use of equalities (6) and a computer makes this problemsolvable.

Below we demonstrate the possibility to use (38) for computing concrete moments fors = 3. For s = 2, r = 2, k2 = 0 this work can be performed without computer:

µm,k = µ(m, k) +{

m

m − 1

}µ(m − 1, k) +

{k

k − 1

}µ(m, k − 1)

+{

m

m − 2

}µ(m − 2, k) +

{k

k − 2

}µ(m, k − 2)

+{

m

m − 1

}{k

k − 1

}µ(m − 1, k − 1) + O(αk+m−3)

= αk1αm + αk

1αm−1mk +{

m

m − 1

}αk

1αm − 1 +{

k

k − 1

}αk

1αm

+ αk1αm−2

(m

2

)k(k − 1) +

{m

m − 2

}αk

1αm−2 +{

k

k − 2

}αk−2

1 αm

+(

m

2

)(k

2

)αk−1

1 αm−1 +(

m

2

)(m − 1)kαk

1αm−2

+(

k

2

)m(k − 1)αk−1

1 αm−1 + O(αk+m−3)

= αk1αm + αk

1αm−1((

m

2

)+ mk

)+(

k

2

)αk−1

1 αm

+ αk1αm−2

(m

2

)(k(k − 1) + (m − 2)(3m − 5)

12

)+(

k

3

)3k − 5

4αk−2

1 αm

+(

k

2

)αk−1

1 ωm−1((

m

2

)+ m(k − 1)

)+ O(αk+m−3). (39)

It is not difficult to see that, as it must, (39) for m = 0 or k = 0 turns into (22) with s = 2.Consider the possibility to approximate the mixed factorial moments of the more com-

plicated form µ(m, k̄). Note that the ordinary moments µm,k̄ can be expressed in terms ofµ(m, k̄).

Let

ϕ(z) =r∏

i=1

fi (αi , z), (40)

Approximation of moments of arbitrary orders 139

where

fi (αi , z) = zki

eαi z −−ki −1∑

j=0

(αi z) j

j !

, i = 1, . . . , r.

Then equality (23) for the random variables ξ i ∼ ∏(αi ), i = 1, . . . , r , takes the form

µ(m, k̄) = αk11 . . . αkr

r e−αδmz ϕ(z)|z=1. (41)

It is easily seen that, as could be expected, expression (41) for r = 2, k 2 = 0 is reducedto (25).

Theorem 4. Let α1, . . . , αr → ∞ in such a way that there exist constants c1, c2 inde-pendent of α1, . . . , αr such that

0 < c1 < αiα−1 < c2 < 1.

Then for arbitrary integers s, m, ki , i = 1, . . . , r , s > 0, and functions ϕ(z), defined byequality (40),

δmz ϕ(z)|z=1 =

s∑

i=0

(m

i

)αm−i eαki + O(αm−s−1eα). (42)

Proof. Conceptually and in many technical details the proof of Theorem 4 repeats theproof of Theorem 3 with corresponding complications. We consider only the momentswhere some new features appear.

The case m ≥ 0 is considered similarly with the use of (24).Validity of (42) for m < 0 is proved by induction on m for arbitrary k 1, . . . , kr as the

corollary of the approximation

δmz ϕ(z) = eαz

s∑

i=0

(m

i

)αm−i ki zk−i + O(αm−s−1eαz) + O(αmax |ki |eεz) (43)

for ε ≤ z ≤ 1 and some fixed value ε, 1 − c1 < ε < 1.Consider the case m = −1. Represent the functions f i (αi , z) in the form

fi (αi , z) = zki

∞∑

j=max(−ki ,0)

(αi z) j

j !

Then, for any fixed k̄ and s,

δ−1z (ϕ(z)) −

∫ z

ε

ϕ(t) dt =∫ ε

0ϕ(t) dt =

∫ ε

0

r∏

i=1

∞∑

j=max(−ki ,0)

(αi t) j

j ! dt

=∞∑

j1=max(−k1,0)

. . .

∞∑

j2=max(−kr ,0)

αj11 . . . α

jrr

j1! . . . jr !∫ ε

0t j1+...+ jr+kdt = O(eεα). (44)

140 A. P. Baranov and Yu. A. Baranov

Now, for arbitrary fixed k̄, s, z, ε ≤ z ≤ 1, we obtain in the same way as relation (30)

∫ z

ε

ϕ(t) dt −∫ z

ε

tkeαt dt =∫ z

ε

tkeαt

r∏

i=1

1 −−ki −1∑

j=0

(αi t) j

j ! e−αi t

− 1

dt

= O

∫ z

ε

tkeαtr∑

i=1

−ki −1∑

j=0

(αi t) j

j ! e−αi t dt

= O(αmax |ki |eεz).

The integral∫ zε

tkeαt dt satisfies equality (31), therefore we obtain relation (42) for m = −1and any fixed k.

Using (32), we obtain

δm−1z ϕ(z) =

∫ z

0δm

t ϕ(t) dt = Sα(k, z) + O

(∫ ε

0δm

t ϕ(t) dt

)+ O

(αm−s−1

∫ z

ε

eαtdt

).

Recall that in contrast to (32) α = α1 + . . . + αr and k = k1 + . . . + kr . We applyestimate (44) obtained in the consideration of the case m = −1. Then for fixed ε, m < 0,ε ≤ z ≤ 1∫ ε

0δm

t ϕ(t) dt ≤∫ ε

0. . .

∫ ε

0ϕ(t−m) dt−m . . . dt ≤ ε−m+1

∫ ε

0ϕ(t) dt = O(αmax |ki |eεz),

and we obtain an estimate similar to (35), although we use (44) instead of inequality (34)used in the derivation of estimate (35).

It remains to transform the sum Sα(k, z) with the use of (31) as it was demonstrated atthe end of the proof of Theorem 3. These transformations give us expression (43) for m − 1and complete the inductive proof of Theorem 4.

Corollary 2. Let α1, . . . , αr → ∞ in such a way that there exist constants c1, c2 suchthat

0 < c1 < αiα−1 < c2 < ∞, i = 1, . . . , r. (45)

Then for arbitrary fixed sets of integers k̄, m, s, s ≥ 0, the mixed factorial moments areapproximated by the expression

µ(m, k̄) = αk11 . . . αkr

r αms∑

i=0

(m

i

)α−i ki + O(αm+k−s−1). (46)

Approximation (46) follows directly from (41) and (42).Note that under conditions (45) the estimate of the remainder term O(α m+k−s−1) is less

in order than the smallest term contributing to the main part of expansion (46).The availability of the approximation of factorial moments provides a way of construct-

ing the approximation similar to (38) of ordinary moments

µm,k̄ = E(ξ1 + . . . + ξr )m

r∏

i=1

ξkii = EM(ξ̄ ) + ER(ξ̄ ).

Approximation of moments of arbitrary orders 141

It follows from the definition of the functions M(x̄) given above that

EM(ξ̄ ) =∑

0≤ j0,... , jr≤s

µ(m − j0, k̄ − ̄ )

{m

m − j0

} r∏

i=1

{ki

ki − ji

}, (47)

where ̄ = ( j1, . . . , jr ).The estimate of ER(ξ̄ ) for ξ ∼ �(αi ), i = 1, . . . , r, under restrictions (45) is of the

formER(ξ̄ ) = O(αm+k−s−1),

which follows directly from the estimate of the function R(x) and (46).

Corollary 3. Let α → ∞ and α1, . . . , αr satisfy (45) for i = 1, . . . , r . Then forarbitrary fixed sets of integers k̄, m, s, s ≥ 0, the mixed moments are approximated by theexpression

µm,k̄ = EM(ξ̄ ) + O(αm+k−s−1), (48)

where α = α1 + . . . + αr and EM(ξ̄ ) has representation (47).

It is easy to see that expression (38) is a particular case of (48) for r = 1. Abovewe demonstrated technical difficulties arisen in application of representation (39) for thecalculation of µm,k without use of computers. Representation (47) is more cumbersome.However calculations with the use of this representation are not complicated in nature. Thedifficulties are connected with enormous number of simple algebraic transformations suchthat substitution into (47) of expressions (46) and of the explicit representations of Stirlingnumbers of the second kind calculated, for example, with the use of formulas (6) and theirextensions into (47). After a change of variables it is needed to perform the reduction ofsimilar terms and write the result in the order of increasing of powers of α. Nevertheless,the experience of derivation of (39) shows that this work can be fulfilled without use ofcomputers only for small values of the parameters s and r , namely, for s = 2, r = 1.

We entrusted this work to a computer and used the package MAPLE V6. However evenwith the use of computer it is impossible to obtain the representation for arbitrary s and r .

We present the result of the described routine work for s = 3, r = 3 setting

αi = xiα, 0 < xi < 1, i = 1, 2, 3, x1 + x2 + x3 = 1.

Under these conditions we obtain the expression of µm,k1,k2,k3 as an expansion in powersof α. Below we give the expansion in the first three powers of α:

µm,k1,k2,k3 = αm+k1+k2+k3 xk11 xk2

2 xk33 + (1/2)αm+k1+k2+k3 x (k1−1)

1 x (k2−1)2 x (k3−1)

3

× (−mx1x2x3 + k23x1x2 + m2x1x2x3 − k1x2x3 + 2mk1x1x2x3

+ 2mk2x1x2x3 + 2mk3x1x2x3 + k22x1x3 − k2x1x3 + k2

1x2x3 − k3x1x2)/α

+ (1/24)αm+k1+k2+k3 x (k1−2)1 x (k2−2)

2 x (k3−2)3 (−14k3

1x23 x2

2 − 10mx23x2

1 x22

+ 3m4x23 x2

1 x22 + 24m2k1k2x2

3 x21 x2

2 + 24m2k1k3x23 x2

1 x22 + 21k2

1x23 x2

2

142 A. P. Baranov and Yu. A. Baranov

+ 24m2k2k3x23 x2

1 x22 − 24mk1k2x2

3 x21 x2

2 − 24mk1k3x23 x2

1 x22 − 14m3x2

3 x21 x2

2

+ 21m2x23 x2

1 x22 + 12m3k1x2

3 x21 x2

2 + 12m3k2x23 x2

1 x22 + 12m3k3x2

3 x21 x2

2

+ 12m2k21x2

3 x21 x2

2 + 12m2k22x2

3 x21 x2

2 + 12m2k23x2

3 x21 x2

2 − 36m2k1x23 x2

1 x22

− 36m2k3x23 x2

1 x22 − 36m2k2x2

3 x21 x2

2 − 12mk21x2

3 x21 x2

2 − 12mk22x2

3 x21 x2

2

− 12mk23x2

3 x21 x2

2 + 24mk3x23 x2

1 x22 + 24mk1x2

3 x21 x2

2 − 24mk2k3x23 x2

1 x22

+ 24mk2x23 x2

1 x22 + 12mk1k2

3x3x21 x2

2 − 6m2k3x3x21 x2

2 + 12mk21k3x2

3 x1x22

+ 18mk3x3x21 x2

2 + 12mk2k23x3x2

1 x22 − 6m2k2x2

3 x21 x2 − 12mk1k2x2

3 x21 x2

− 30mk22x2

3 x21 x2 − 12mk1k3x2

3 x1x22 + 18mk2x2

3 x21 x2 + 3k4

1x23 x2

2

+ 12mk1k22x2

3 x21 x2 − 10k1x2

3 x22 − 12mk2k3x2

3 x21 x2 + 12mk3

3x3x21 x2

2

+ 12mk31x2

3 x1x22 + 12mk2

2k3x23 x2

1 x2 − 12mk2k3x3x21 x2

2 + 6k1k3x3x1x22

+ 6m2k22x2

3 x21 x2 + 3k4

3x21 x2

2 − 10k3x21 x2

2 − 12mk1k3x3x21 x2

2

− 12mk1k2x23 x1x2

2 − 6k1k23x3x1x2

2 − 14k33x2

1 x22 + 21k2

3x21 x2

2 − 10k2x23 x2

1

+ 12mk21k2x2

3 x1x22 + 6m2k2

3x3x21 x2

2 − 30mk23x3x2

1 x22 + 12mk3

2x23 x2

1 x2

+ 21k22x2

3 x21 − 14k3

2x23 x2

1 + 6m2k21x2

3 x1x22 + 3k4

2x23 x2

1 − 6m2k1x23 x1x2

2

− 30mk21x2

3 x1x22 + 18mk1x2

3 x1x22 + 6k2

1k22x2

3 x1x2 − 6k22k3x3x2

1 x2

− 6k2k23x3x2

1 x2 + 6k2k3x3x21 x2 − 6k1k2

2x23 x1x2 + 6k2

1k23x3x1x2

2

+ 6k22k2

3x3x21 x2 − 6k2

1k3x3x1x22 + 6k1k2x2

3 x1x2 − 6k21k2x2

3 x1x2)/α2

+ O(αm+k1+k2+k3−3).

In order to check correctness of the obtained expression, we can compare it with (39) settingk3 = k2 = 0.

In conclusion we notice that in fact equality (47) does not depend on the distributionof the random variables ξ1, . . . , ξr and, consequently, can be used in calculations of mixedmoments µm,k̄ for arbitrarily distributed random variables taking with positive probabilitiesnon-negative integer values. For the distributions of random variables ξ 1, . . . , ξr , whichdiffer from the Poisson distribution, theorems similar to Theorems 3, 4 and estimates ofER(ξ̄ ) will be needed. We guess that a similar result can be obtained for random variablesξ1, . . . , ξr with binomial distribution, negative binomial distribution including geometricdistribution, hyper-geometric distribution, and a series of other distributions, for which it ispossible to obtain an explicit expansion of the main integral in (23) of the form

∫ z

0tm

r∏

i=1

ϕi (t) dt, (49)

where ϕi (t) is obtained as a result of application of the operator δkit to the generating func-

tion of the corresponding distribution. In its turn, the main difficulty in the calculation ofthe operator δk

t �(t) is the calculation of the k-fold integral∫ t

0

∫ t1

0. . .

∫ tk−1

0�(t−k) dt−k . . . dt1, (50)

Approximation of moments of arbitrary orders 143

where �(t) is the corresponding generation function, k < 0.The method of calculation of mixed factorial moments described in this paper is accept-

able for obtaining particular expansions if the generating functions of the correspondingdistributions admit explicit calculations of (50) and expansions for (49), similar, for ex-ample, to (42).

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1. T. R. G. Read and N. A. S. Cressie, Goodness-of-Fit Statistics for Discrete Multivariate Data.Springer, New York, 1988.

2. D. E. Knuth, Two notes on notation. Amer. Math. Monthly (1992) 99, 403–422.

3. R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics. Addison–Wesley, Reading,Mass., 1994.

4. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Gordon & Breach,New York, 1988.