Some properties of weight functions in tauberian theorems
The rate of convergence for weight functions series in Tauberian theorems is obtained. Numerical results demonstrate required rate of convergence. Some asymptotic properties of hypergeometric functions are obtained as auxiliary results.
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| Цитувати: | Some properties of weight functions in tauberian theorems / A. Olenko, B. Klykavka // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 123–136. — Бібліогр.: 17 назв.— англ. |
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irk-123456789-44622009-11-12T12:00:32Z Some properties of weight functions in tauberian theorems Olenko, A. Klykavka, B. The rate of convergence for weight functions series in Tauberian theorems is obtained. Numerical results demonstrate required rate of convergence. Some asymptotic properties of hypergeometric functions are obtained as auxiliary results. 2006 Article Some properties of weight functions in tauberian theorems / A. Olenko, B. Klykavka // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 123–136. — Бібліогр.: 17 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4462 en Інститут математики НАН України |
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The rate of convergence for weight functions series in Tauberian theorems is obtained. Numerical results demonstrate required rate of convergence.
Some asymptotic properties of hypergeometric functions are
obtained as auxiliary results. |
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Article |
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Olenko, A. Klykavka, B. |
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Olenko, A. Klykavka, B. Some properties of weight functions in tauberian theorems |
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Olenko, A. Klykavka, B. |
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Olenko, A. |
| title |
Some properties of weight functions in tauberian theorems |
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Some properties of weight functions in tauberian theorems |
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Some properties of weight functions in tauberian theorems |
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Some properties of weight functions in tauberian theorems |
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Some properties of weight functions in tauberian theorems |
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some properties of weight functions in tauberian theorems |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/4462 |
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Some properties of weight functions in tauberian theorems / A. Olenko, B. Klykavka // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 123–136. — Бібліогр.: 17 назв.— англ. |
| work_keys_str_mv |
AT olenkoa somepropertiesofweightfunctionsintauberiantheorems AT klykavkab somepropertiesofweightfunctionsintauberiantheorems |
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2025-07-02T07:42:05Z |
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2025-07-02T07:42:05Z |
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1836520178924388352 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 123–136
ANDRIY OLENKO AND BORIS KLYKAVKA
SOME PROPERTIES OF WEIGHT FUNCTIONS IN
TAUBERIAN THEOREMS. I
The rate of convergence for weight functions series in Tauberian theo-
rems is obtained. Numerical results demonstrate required rate of con-
vergence. Some asymptotic properties of hypergeometric functions are
obtained as auxiliary results.
1. Introduction
In numerous problems in actuarial and financial mathematics asymptotic
behavior of random processes, fields, and limit theorems for some their
functionals are of great importance. Abelian and Tauberian theorems find
numerous applications in obtaining various asymptotic properties of random
processes and fields. The majority results of such type (see, for example,
[1]-[4]) describe relations between asymptotic behavior of spectral and cor-
relation characteristics at the infinity and in zero. Such relations do not
always exist for long memory random processes and fields. A new approach
based on the idea of studying relations between behavior of a spectral func-
tion in zero and some functional of random field at the infinity was proposed
in [5, 6]. In comparison with classical Tauberian theorems this functional
is used as an equivalent of a correlational function. Representation of the
functional was given in terms of the variance of spherical averages of random
field. It also can be calculated as some integral of the correlation function.
In [7], [8] similar investigations were continued. New results on relations
between local behavior of spectral functions in arbitrary point (not nec-
essarily zero) and asymptotics of some functionals of random fields were
obtained. Representations of these functionals were derived in terms of the
variance of weighted averages of random fields. Various properties of weight
functions in such representations were discussed in [7]-[10]. This paper con-
tinues the investigations.
Let Rn, n ≥ 2 be an n−dimentional Euclidean space, ξ(t), t ∈ Rn be
a real measurable mean-square continuous homogeneous isotropic random
field (see, for example, [11]) with zero mean and the correlation function
Bn(r) = Bn(|t|) = Eξ(0)ξ(t), t ∈ Rn.
2000 Mathematics Subject Classification. Primary 60G60, 62E20, 40E05, 26A12,
44A15, 33C05.
Key words and phrases. Tauberian theorems, random fields, correlation function, spec-
tral function, weight function, speed of convergence, OR, asymptotics.
123
124 ANDRIY OLENKO AND BORIS KLYKAVKA
It is known that there exists a bounded nondecreasing function Φ(x), x ≥ 0,
which is called spectral function of the field ξ(t), t ∈ Rn (see [11]), such that
Bn(r) has the representation
Bn(r) = 2
n−2
2 Γ
(n
2
)∫ ∞
0
Jn−2
2
(rx)
(rx)
n−2
2
dΦ(x),
where Jν(z) is a Bessel function of the first kind, ν > −1
2
(see [12]).
Let
b̃a(r) := (2π)n
∫ ∞
0
J2
n
2
(rx)
(rx)n
dΦa(x),
where for arbitrary a ∈ [0, +∞)
Φa(λ) :=
{
Φ(a + λ) − Φ(a − λ), 0 ≤ λ < a;
Φ(a + λ), λ ≥ a.
In [8], it was shown the existence of a real-valued function fr,a(|t|), for which
b̃a(r) = D
[∫
Rn
fr,a(|t|)ξ(t)dt
]
.
The function can be defined by the formula
(1) fr,a(|t|) =
1
|t|n
2
−1
∫ ∞
0
λn/2
Jn
2
(r(λ − a))
(r(λ − a))n/2
Jn
2
−1(|t|λ)︸ ︷︷ ︸
F (λ)
dλ, |t| �= r.
2. Problem to investigate
In the following C denotes a constant, exact value of which is not impor-
tant and which may be different in different places.
By asymptotic properties of the Bessel function (see §7.21, [12])
(2) Jν(z) ∼
√
2
πz
cos
(
z − π
2
ν − π
4
)
, z → ∞
it follows that the integrand F (λ) in the definition (1) of fr,a(|t|) has the
asymptotic behavior
F (λ) ∼ C
λ
cos
(
r(λ − a) − πn
4
− π
4
)
cos
(
|t|λ − π
2
(n
2
− 1
)
− π
4
)
∼
C
λ
(
cos
(
(r + |t|)λ − ra − πn
2
)
− sin ((r − |t|)λ − ra)
)
, λ → ∞.
Hence, the integral in the representation of the function fr,a(|t|) converges
conditionally.
Therefore there arise various problems of studying properties of the func-
tion fr,a(|t|) and calculating its values with a given accuracy.
SOME PROPERTIES OF WEIGHT FUNCTIONS 125
To calculate the integral (1) well known standard approach implies an
application of the Poisson formula (§3.3, [12])
Jν(z) =
(z/2)ν
Γ(ν + 1/2)Γ(1/2)
∫ 1
−1
(1 − x2)ν−1/2 cos(zx)dx.
Then the function fr,a(|t|) can be rewritten as
fr,a(|t|)= 1
2
n
2 Γ(n+1
2 )
√
π |t|
n
2 −1
∫ ∞
0
λ
n
2 J n
2 −1(|t|λ)
∫ 1
−1
(1−x2)
n−1
2 cos(r(λ−a)x)dxdλ.
Changing the order of integration we obtain
fr,a(|t|)= 1
2
n
2 Γ(n+1
2 )√π |t|
n
2 −1
(∫ 1
−1
(1−x2)
n−1
2 cos(rax)
∫ ∞
0
λ
n
2 J n
2 −1(|t|λ) cos(rλx)dλdx+
+
∫ 1
−1
(1−x2)
n−1
2 sin(rax)
∫ ∞
0
λ
n
2 J n
2 −1(|t|λ) sin(rλx)dλdx
)
.
Next step is to calculate the inner integrals.
Unfortunately it is not allowed to change the order in our case. By the
asymptotic formula (2) the inner integrals with respect to λ, n ∈ N do not
converge. That is why we must use another methods to study properties of
the function fr,a(|t|) in (1).
In [8], it was proposed an efficient approach based on the representation
of the function fr,a(|t|) as the series:
(3) fr,a(|t|) =
⎧⎪⎪⎨⎪⎪⎩
(
2
ar3
)n
2
∑∞
m=0 dm(n, r, a, |t|), |t| < r,(
2
ar|t|2
)n
2
Γ
(
n
2
)∑∞
m=0 sm(n, r, a, |t|), |t| > r,
where
(4) dm(n, r, a, |t|) =
(n
2
+m)Cm
n+m−1Γ(n+m
2
)J n
2 +m(ra) 2F1
n+m
2
,−m
2
, n
2
;( |t|
r )
2
Γ(m
2
+1)
,
sm(n, r, a, |t|)= r2m+1C2m+1
n+2mΓ(n
2
+ m + 1
2
)Jn
2
+2m+1(ra)
|t|2m+1Γ(−m − 1
2
)Γ
(
n
2
+ 2m + 1
) ×
(5) ×2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2;
(
r
|t|
)2
)
2F1(a, b, c; z) is a Gauss hypergeometric function (see [13]).
The rate of convergence of the series (3) is important for numerical cal-
culations. We will study this problem in the paper.
126 ANDRIY OLENKO AND BORIS KLYKAVKA
3. Asymptotic properties of the Gauss hypergeometric
function
To obtain the rate of convergence we will need some properties of the
Gauss hypergeometric function 2F1 in (4) and (5).
The function 2F1(a, b, c; z) is defined as
(6) 2F1(a, b, c; z) =
∞∑
l=0
(a)l(b)l zl
(c)l l!
,
(a)0 = 1 and (a)l = a(a + 1)...(a + l − 1), if l ∈ N,
for arguments values for which the series (6) converges, and as analytical
continuation for another arguments values (complex-valued) if such contin-
uation exists.
Note, that in the cases (4) and (5) the function 2F1 can be correctly
defined by (6). Indeed for m = 2k, k ∈ N ∪ 0 :
(7) 2F1
(
n
2
+ k,−k,
n
2
;
( |t|
r
)2
)
=
k∑
l=0
(
n
2
+ k
)
l
(−k)l(
n
2
)
l
l!
zl
becomes a k-degree polynomial.
It can be found in §2.1.1, [14] that for a, b �= {0,−1,−2, ...} :
(a)l(b)l
(c)l l!
=
Γ(c)
Γ(a)Γ(b)
la+b−c−1[1 + O(l−1)],
and the series (6) converges absolutely for |z| < 1. Therefore, for m �= 2k,
k ∈ N ∪ {0}, |t| < r, the function 2F1
(
n+m
2
,−m
2
, n
2
;
(
|t|
r
)2
)
is correctly
defined by (6). Similarly 2F1
(
n
2
+ m + 1
2
, m + 3
2
, n
2
+ 2m + 2;
(
r
|t|
)2
)
is also
correctly defined by (6), when |t| > r.
Let us consider the asymptotic behavior of 2F1
(
n+m
2
,−m
2
, n
2
;
(
|t|
r
)2
)
,
when m → ∞.
In the following we will use Stirling’s formula (see §540, [15]) for the
Gamma function
Γ(k + 1) ∼
√
2πkk+1/2e−ke
θk
k ,
where θk ∈ (0; 1
12
). Particularly
k! ∼
√
2πkk+1/2e−ke
θk
k .
To obtain asymptotical formulas we could use Watson’s results (see §2.3.2
[14])
SOME PROPERTIES OF WEIGHT FUNCTIONS 127
2F1 (a + λ, b − λ, c, 1/2 − z/2) =
Γ(1 − b + λ)Γ(c)
Γ(1/2)Γ(c − b + λ)
2a+b−1(1− e−ξ)−c+1/2×
(8) (1 + e−ξ)c−a−b−1/2λ−1/2
(
e(λ−b)ξ + e±iπ(c−1/2)e−(λ+a)ξ
) (
1 + O(|λ−1|)) ,
where z ±√
z2 − 1 = e±ξ according to Imz ≷ 0.
Let us choose
a = c =
n
2
, b = 0, λ =
m
2
,
1
2
− z
2
=
( |t|
r
)2
then
2F1
(
n+m
2
,−m
2
, n
2
;
(
|t|
r
)2
)
∼ Γ(m/2+1)Γ(n/2)2n/2−1/2
Γ(n/2+m/2)Γ(1/2)m1/2 ×
×( 2|t|
r )
1/2−n/2 |t|
r
+ ( |t|
r )
2−1
1/2−n/2
2−1/2 1−( |t|
r )
2− |t|
r ( |t|
r )
2−1
−1/2
×
× |t|
r
+ ( |t|
r )
2−1
m/2
+(sin(nπ
2 )±i cos(nπ
2 )) |t|
r
+ ( |t|
r )
2−1
m/2+n/2
=
= O
(
Γ(m
2
+1)
Γ(n
2
+ m
2
)m
1
2
)
∼ O
(
Cm
m
n
2 −1
)
.
when m → ∞.
Unfortunately the formula (8) is true only if z ∈ C\(−∞, 1) (see [16],
[17]). Therefore we cannot use it directly in our case. Nevertheless we will
show that the asymptotics O
(
Cm
m
n
2 −1
)
is valid for our parameters case.
Lemma 1 For |t| < r :
2F1
(
n + m
2
,−m
2
,
n
2
;
( |t|
r
)2
)
= O
(
Cm
m
n
2
− 1
2
)
, m → ∞.
Proof. If m = 2k, k ∈ N ∪ {0}, then it follows from (7)
∣∣∣2F1
(n
2
+ k,−k,
n
2
; z
)∣∣∣ =
∣∣∣∣∣
k∑
l=0
Γ(n
2
+ k + l)Γ(n
2
)(−1)lk!zl
Γ(n
2
+ k)Γ(n
2
+ l)(k − l)!l!
∣∣∣∣∣ ≤
(9) ≤ Γ(n
2
)
Γ(n
2
+ k)
k∑
l=0
C l
kz
l Γ(n
2
+ k + l)
Γ(n
2
+ l)
≤ Γ(n
2
+ 2k)Γ(n
2
)
Γ2(n
2
+ k)
(1 + z)k,
because
Γ(n
2
+ k + l)
Γ(n
2
+ l)
≤ Γ(n
2
+ 2k)
Γ(n
2
+ k)
, l = 0, k.
By Stirling’s formula
Γ(n
2
+ 2k)
Γ2(n
2
+ k)
∼ e−
n
2
−2k+1
(
n
2
+ 2k − 1
)n
2
+2k− 1
2
√
2πe−n−2k+2
(
n
2
+ k − 1
)n+2k−1
=
128 ANDRIY OLENKO AND BORIS KLYKAVKA
(10) =
e
n
2
−12
n
2
+2k− 1
2
(
1 +
n
4
− 1
2
k
) k
n
4 −1
2
n
4 − 1
2
k (n
2
+2k− 1
2)
√
2π
(
1 +
n
2
−1
k
) k
n
2 −1
·
n
2 −1
k
·(n+2k−1)
· kn+2k−1
k
n
2
+2k− 1
2 ∼ 2
n
2
+2k−1
√
πk
n
2
− 1
2
.
By (9) and (10) we obtain
(11) 2F1
(
n
2
+ k,−k,
n
2
;
( |t|
r
)2
)
= O
(
Ck
k
n
2
− 1
2
)
= O
(
Cm
m
n
2
− 1
2
)
,
when m = 2k → ∞.
Let us show that 2F1
(
n+m
2
,−m
2
; n
2
;
(
|t|
r
)2
)
has the same asymptotics
when m is odd. Actually
(12) 2F1
(
n+m
2
,−m
2
; n
2
; z
)
=
[m
2 ]+1∑
l=0
(n
2
+ m
2 )
l
(−m
2 )
l
(n
2 )l
l!
zl +
∞∑
l=[m
2 ]+2
(n
2
+ m
2 )
l
(−m
2 )
l
(n
2 )l
l!
zl.
For l = 0, ...,
[
m
2
]
+ 1 :∣∣(n
2
+ m
2
)
l
(−m
2
)
l
∣∣ =
∣∣(n
2
+ m
2
) · ... · (n
2
+ m
2
+ l − 1
) · m
2
· ... · (m
2
− l + 1
)∣∣ ≤∣∣(n
2
+ m
2
+ 1
2
) · ... · (n
2
+ m
2
+ l − 1
2
) · m+1
2
· ... · (m+1
2
− l + 1
)∣∣ =∣∣(n
2
+ m+1
2
)
l
(−m+1
2
)
l
∣∣ .
Hence
(13)
∣∣∣∣∣∣∣
[m
2 ]+1∑
l=0
(n
2
+ m
2 )
l
(−m
2 )
l
(n
2 )l
l!
zl
∣∣∣∣∣∣∣ ≤
m+1
2∑
l=0
(n
2
+ m+1
2 )
l
|(−m+1
2 )
l
|
(n
2 )l
l!
zl = O
(
C
m+1
2
(m+1
2 )
n
2 − 1
2
)
,
where the last identity follows from (9) and (10) proven for even m.
Consider the asymptotics of the second term in (12).
∞∑
l=[m
2 ]+2
(n
2 + m
2 )l(−
m
2 )l
(n
2 )l
l!
zl =
∞∑
l=[m
2 ]+2
Γ( n
2 + m
2 +l)Γ(n
2 )·(−m
2 )(−m
2 +1)·...·(−1
2)· 12 ·...·(−m
2 +l−1)zl
Γ(n
2 + m
2 )Γ(n
2 +l)Γ(l+1)
=
(14)
∣∣∣∣ k = l − [
m
2
] − 1
l = k + m
2
+ 1
2
∣∣∣∣ =
(−1)mΓ(m
2
+1)Γ(n
2
)z
m
2 +1
2
πΓ(n
2
+ m
2
)
∞∑
k=1
Γ(n
2
+m+ 1
2
+k)Γ(k+ 1
2)zk
Γ(n
2
+ m
2
+ 1
2
+k)Γ(k+ m
2
+ 3
2
)
.
By Stirling’s formula with θk,j ∈
(
0, 1
12
)
, j = 1, 4 :
∞∑
k=1
Γ(n
2
+m+ 1
2
+k)Γ(k+ 1
2)zk
Γ(n
2
+ m
2
+ 1
2
+k)Γ(k+ m
2
+ 3
2
)
=
∞∑
k=1
e−
n
2 −m−k+ 1
2 (n
2
+m+k− 1
2)
n
2 +m+k
e
θk,1
n
2 +m+k− 1
2
e−
n
2 −m
2 −k+ 1
2 (n
2
+ m
2
+k− 1
2)
n
2 + m
2 +k
e
θk,2
n
2 + m
2 +k− 1
2
×
SOME PROPERTIES OF WEIGHT FUNCTIONS 129
e
−k+1
2 (k− 1
2)
k
e
θk,3
k− 1
2
e
−k−m
2 − 1
2 (k+ m
2 + 1
2)
k+ m
2 +1
e
θk,4
k+ m
2 +1
zk ≤ C ·
∞∑
k=1
(n
2 +m+k− 1
2)
n
2 +m+k
(k− 1
2)
k
zk
(n
2 + m
2 +k−1
2)
n
2 + m
2 +k
(k+ m
2 + 1
2)
k+ m
2 +1
=
= C ·
∞∑
k=1
(
1 +
m
2
n
2
+ m
2
+k− 1
2
) n
2 + m
2 +k− 1
2
m
2
·
m
2 ·(n
2 + m
2 +k)
n
2 + m
2 +k−1
2
(
1 +
m
2
+ n
2
−1
k+ m
2
+ 1
2
)m
2
zk
(
1 +
m
2
+1
k− 1
2
) k− 1
2
m
2 +1
·
m
2 +1
k− 1
2
·k (
k + m
2
+ 1
2
) := Σ.
To estimate the last sum we use:
Lemma 2 For all x ∈ (0,∞) : 0 < a ≤ (
1 + 1
x
)x ≤ b < +∞.
Proof. The function
(
1 + 1
x
)x
is positive, continuous and it is not equal to 0
or +∞ for any x ∈ (0; +∞). Moreover lim
x→+∞
(1+ 1
x
)x = e, lim
x→+0
(1+ 1
x
)x = 1.
The statement of the lemma immediately follows from these properties. �
By application of Lemma 2
Σ ≤ C ·
∞∑
k=1
b
m
2 (n
2 + m
2 +k)
n
2 + m
2 +k−1
2
(
1 +
m
2
+ n
2
−1
k+ m
2
+ 1
2
)m
2
zk
a
m
2 +1
k− 1
2
·k (
k + m
2
+ 1
2
) ≤ C ·
∞∑
k=1
C
m
2
1 C
m
2
3 zk
C
m
2
+1
2
(
k + m
2
+ 1
2
) ,
because of(
1 +
m
2
+ n
2
−1
k+ m
2
+ 1
2
)m
2 ≤
(
1 +
m
2
+ n
2
−1
m
2
+ 3
2
)m
2
=
(
2 +
n
2
− 5
2
m
2
+ 3
2
)m
2 ≤ (
n+3
4
)m
2 .
Hence
Σ ≤ C
m
2
∞∑
k=1
zk
k + m
2
+ 1
2
< C
m
2
∞∑
k=1
zk
k + 1
.
By (14) and Stirling’s formula
(15)
∣∣∣∣∣∣∣
∞∑
l=[m
2 ]+2
(n
2
+ m
2 )
l
(−m
2 )
l
(n
2 )l
l!
zl
∣∣∣∣∣∣∣ = O
(
C
m
2 Γ
(
m
2
+ 1
)
Γ
(
n
2
+ m
2
) )
= O
(
Cm
m
n
2
−1
)
,
when m → ∞.
For odd m by (12), (13) and (15) we obtain
2F1
(
n + m
2
,−m
2
,
n
2
;
( |t|
r
)2
)
= O
(
Cm
m
n
2
− 1
2
)
, m → ∞.
Statement of Lemma 1 follows from the last asymptotics and (11). �
To investigate asymptotic properties of
2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2;
(
r
|t|
)2
)
, |t| > r, m → ∞
130 ANDRIY OLENKO AND BORIS KLYKAVKA
we could also use Watson’s results (see §2.3.2 [14])(
z
2
− 1
2
)−a−λ
2F1 (a + λ, a − c + 1 + λ, a − b + 1 + 2λ; 2(1 − z)−1) =
2a+bΓ(a − b + 1 + 2λ)Γ
(
1
2
)
λ− 1
2
Γ(a − c + 1 + λ)Γ(c − b + λ)
e−(a+λ)ξ(1 − e−ξ)−c+ 1
2×
(16) (1 + e−ξ)c−a−b− 1
2
(
1 + O
(|λ|−1
))
,
where ξ is defined as in (8).
Choosing
a =
n
2
+
1
2
, b = −1
2
, c =
n
2
, λ = m,
2
1 − z
=
(
r
|t|
)2
we obtain
2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2;
(
r
|t|
)2
)
∼
∼ 2
n
2
+1Γ(n
2
+ 2m + 2)
√
πm− 1
2
Γ
(
m + 3
2
)
Γ
(
n
2
+ 1
2
+ m
) O (Cm)
(
r
|t|
)n+1+2m
∼ O(Cm) ,
when m → ∞. Unfortunately, similarly to the case (8) we cannot apply (16)
directly. Nevertheless we will show that the asymptotics O (Cm) is valid for
our parameters case.
Lemma 3 For |t| > r :
2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2;
(
r
|t|
)2
)
= O (Cm) , m → ∞.
Proof. By (2) we have
2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2; z
)
=
∞∑
l=0
(
n
2
+ m + 1
2
)
l
(
m + 3
2
)
l(
n
2
+ 2m + 2
)
l
l!
zl =
∞∑
l=0
Γ
(
n
2
+ m + l + 1
2
)
Γ
(
n
2
+ 2m + 2
)
Γ
(
m + l + 3
2
)
zl
Γ
(
n
2
+ m + 1
2
)
Γ
(
n
2
+ 2m + l + 2
)
Γ
(
m + 3
2
)
Γ (l + 1)
=
(17) 1 +
Γ
(
n
2
+ 2m + 2
)
Γ
(
n
2
+ m + 1
2
)
Γ
(
m + 3
2
) ∞∑
l=1
Γ
(
n
2
+ m + l + 1
2
)
Γ
(
m + l + 3
2
)
zl
Γ
(
n
2
+ 2m + l + 2
)
Γ (l + 1)
By Stirling’s formula
Γ(n
2
+2m+2) ∼
√
2π(n
2
+2m+1)
n
2
+2m+ 3
2 e−
n
2
−2m−1 ∼
√
2π (2m)
n
2
+2m+ 3
2 e−2m,
(18) Γ(n
2
+m+ 1
2) ∼
√
2π(n
2
+m− 1
2)
n
2
+me−
n
2
−m+ 1
2 ∼
√
2πm
n
2
+me−m,
Γ(m+ 3
2) ∼
√
2π(m+ 1
2)
m+1e−m− 1
2 ∼
√
2πmm+1e−m, when m → ∞.
The asymptotic behaviour of the first multiplier in (17) is
SOME PROPERTIES OF WEIGHT FUNCTIONS 131
(19)
Γ
(
n
2
+ 2m + 2
)
Γ
(
n
2
+ m + 1
2
)
Γ
(
m + 3
2
) ∼ 2
n
2
+2m+ 3
2 m
1
2√
2π
.
Using Stirling’s formula with θl,j ∈ (0, 1
12
), j = 1, 4, we transform the series
in (17):
∞∑
l=1
Γ(n
2
+m+l+ 1
2)Γ(m+l+ 3
2)zl
Γ(n
2
+2m+l+2)Γ(l+1)
=
=
∞∑
l=1
(n
2
+m+l− 1
2)
n
2 +m+l
e−
n
2 −m−l+ 1
2 e
θl,1
n
2 +m+l− 1
2 (m+l+ 1
2)
m+l+1
(n
2
+2m+l+1)
n
2 +2m+l+ 3
2 e−
n
2 −2m−l−1e
θl,2
n
2 +2m+l+1
×
e−m−l− 1
2 e
θl,3
m+l+ 1
2
ll+
1
2 e−le
θl,4
l
zl ≤ C
∞∑
l=1
(n
2
+m+l− 1
2)
n
2 +m+l
(m+l+ 1
2)
m+l+1
(n
2
+2m+l+1)
n
2 +2m+l+3
2 ll+
1
2
zl =
= C
∞∑
l=1
(
1 +
m+ 1
2
l
) l
m+ 1
2
·(m+ 1
2)
zl
(
1 +
m+ 3
2
n
2
+m+l− 1
2
) n
2 +m+l−1
2
m+ 3
2
·(
m+ 3
2)(n
2 +m+l)
n
2 +m+l−1
2
(m+l+ 1
2)
m+1
l
1
2 (n
2
+2m+l+1)
m+ 3
2
= Σ1.
By Lemma 2
(20) Σ1 ≤ C
∞∑
l=1
bm+ 1
2
a
(m+ 3
2)( n
2 +2m+l)
n
2 +m+l− 1
2
· zl
l
1
2
(
n
2
+ 2m + l + 1
) 1
2
≤ C
∞∑
l=1
Cm
1 zl
l
1
2 m
1
2
.
By (17), (19) and (20), we get
2F1
(
n
2
+ m +
1
2
, m +
3
2
,
n
2
+ 2m + 2;
(
r
|t|
)2
)
= O (Cm) , m → ∞. �
4. Rate of convergence
Let us estimate the rate of convergence for the series (3). For this purpose
we will study asymptotics of dm(n, r, a, |t|) and sm(n, r, a, |t|), as m → ∞.
4.1 Case |t| < r.
Let us consider the expression (3) for |t| < r and investigate asymptotics
for multipliers in (4) as m → ∞.
Lemma 4 For |t| < r
(21) dm(n, r, a, |t|) = O
(
Cm
mm−n
2
+ 1
2
)
, m → ∞.
Proof. By Stirling’s formula
Cm
m+n−1 ∼
(m + n − 1)m+n−1/2e−m−n+1
mm+1/2e−m(n − 1)!
∼ mn−1
(n − 1)!
.
132 ANDRIY OLENKO AND BORIS KLYKAVKA
For Gamma functions we have
Γ
(m
2
+ 1
)
∼
√
2πe−
m
2 (m/2)
m+1
2 ,
Γ
(m
2
+
n
2
)
∼
√
2πe−
m
2 (m/2)
m+n−1
2 .
Therefore (
n
2
+ m
)
Cm
n+m−1Γ(n+m
2
)
Γ(m
2
+ 1)
∼ mn
(n − 1)!
(m
2
)n/2−1
.
For ν > 0, z > 0 due to a representation of the Bessel function as a series
(see §8.1 [12]) it follows that
|Jν(z)| =
∣∣∣∑∞
m=0
(−1)m( 1
2
z)ν+2m
m!Γ(ν+m+1)
∣∣∣ =
∣∣∣ (z/2)ν
Γ(ν+1)
∑∞
m=0
(−1)m(z/2)2m
m!(ν+1)(ν+2)...(ν+m)
∣∣∣ ≤
(22) ≤ (z/2)ν
Γ(ν + 1)
∞∑
m=0
(z/2)2m
m!νm
.
By Stirling’s formula
(23)
(z/2)ν
Γ(ν + 1)
∞∑
m=0
(z/2)2m
m!νm
∼ (z/2)νeν
√
2πνν+1/2
∞∑
m=0
( z2
4ν
)m
m!
=
e
z2
4ν
+ν
√
2πν
( z
2ν
)ν
.
For large values of m
(24) Jn
2
+m(ra) < 1√
(n+2m)π
e
(ra)2+(n+2m)2
2n+4m
(
ra
n+2m
)n
2
+m
= O
(
Cm
mm+ n
2 + 1
2
)
.
Applying Lemma 1 and all previous asymptotics to (4) we obtain
dm(n, r, a, |t|) = O
(
Cm
mm−n
2
+1
)
, m → ∞. �
Consider the series (3) remainder for |t| < r.
Theorem 1 For |t| < r
(25)
∞∑
m=N
dm(n, r, a, |t|) = O
( ∞∑
m=N
Cm
mm−n
2
+1
)
, N → ∞.
For any ε > 0 :
(26)
∞∑
m=N
dm(n, r, a, |t|) = O
(
CN
NN(1−ε)
)
, N → ∞.
Proof. The asymptotic formula (25) is a direct corollary of Lemma 4. The
assertion (26) follows from the the chain of estimates
∞∑
m=N
Cm
mm− n
2 +1 ≤
∞∑
m=N
Cm
mm(1−ε) ≤
∞∑
m=N
(
C
N1−ε
)m
=
SOME PROPERTIES OF WEIGHT FUNCTIONS 133
= CN N1−ε
NN(1−ε)(N1−ε−C)
= O
(
CN
NN(1−ε)
)
which is valid for large N. �
4.2 Case |t| > r.
Let us consider the expression (3) for |t| > r and investigate asymptotic
behavior for multipliers in (5) as m → ∞.
Lemma 5 For |t| > r
(27) sm(n, r, a, |t|) = O
(
Cm
m2m−n
2
+2
)
, m → ∞.
Proof. By Stirling’s formula
C2m+1
n+2m ∼ (n + 2m)n+2m+ 1
2 e2m+1
en+2m(2m + 1)2m+ 3
2 (n − 1)!
∼ (2m)n−1
(n − 1)!
,
Γ (n/2 + 2m + 1) ∼
√
2πe−2m(2m)
n
2
+2m+ 1
2 .
Let us find the asymptotics of (2m + 1)!! :
(2m + 1)!! = 2m+1 1
2
(
1
2
+ 1
) (
1
2
+ 2
)
...
(
1
2
+ m
)
=
= 2m+1 Γ(1/2+m+1)
Γ(1/2)
∼ √
2(2m + 1)m+1e−m−1/2.
Using properties of Gamma functions (see §538, [15]) and Stirling’s formula
we obtain
Γ(−m− 1
2)=(−1)m+1
√
π 2m+1
(2m+1)!!
∼(−1)m+1
√
π
2
e
m+ 1
2
(m+ 1
2)
m+1 ∼(−1)m+1
√
π
2
em
mm+1 .
We will use the formula (23) to study the asymptotics of the Bessel function
Jn
2
+2m+1(ra). For large values of m
Jn
2
+2m+1(ra) < e
(ra)2
2n+8m+4 + n
2 +2m+1
2π(n
2
+2m+1)
(
ra
n+4m+1
)n
2
+2m+1
= O
(
Cm
m
n
2 +2m+ 3
2
)
.
Taking into account (18), all previous asymptotics for (5), and Lemma 3 we
obtain
sm(n, r, a, |t|) = O
(
Cm
m2m−n
2
+2
)
, m → ∞. �
Consider the series (3) remainder for |t| > r.
Theorem 2 For |t| > r
(28)
∞∑
m=N
sm(n, r, a, |t|) = O
( ∞∑
m=N
Cm
m2m−n
2
+2
)
, N → ∞
For any ε > 0 :
(29)
∞∑
m=N
sm(n, r, a, |t|) = O
(
CN
N2N(1−ε)
)
, N → ∞.
134 ANDRIY OLENKO AND BORIS KLYKAVKA
Proof. The asymptotic formula (28) is a direct corollary of Lemma 5. The
assertion (29) follows from the chain of estimates
∞∑
m=N
Cm
m2m−n
2
+2
≤
∞∑
m=N
Cm
m2m(1−ε)
≤
∞∑
m=N
(
C
N2(1−ε)
)m
=
CNN2(1−ε)
N2N(1−ε)(N2(1−ε) − C)
= O
(
CN
N2N(1−ε)
)
,
which is valid for large N. �
5. Numerical examples
In this section we give some numerical examples of our results.
Let n = 3. In this case fr,a(|t|) can be written explicitly using functions
Si(z), Ci(z) (see §5, [8]).
Plots of the function fr,a(|t|) for r = 1, a = 1.2, and a = 15 are shown
on Fig.1 and Fig.2. To plot the function we used N = 100 first terms of the
series (3).
0.2 0.4 0.6 0.8 1 1.2 1.4
-1
1
2
3
Fig.1. f1,1.2(|t|)
0.2 0.4 0.6 0.8 1
-50
50
100
150
200
250
Fig.2. f1,15(|t|)
Comparison of Fig.1 and Fig.2 with corresponding plots from §5 [8] shows
their identity. The following table gives some exact numerical values of
fr,a(|t|) calculated by formulae from §5 [8] and its approximations f̂N
r,a(|t|)
by increasing number N of first terms in the series (3).
t f1,1.2(t) f̂ 5
1,1.2(t) f̂ 10
1,1.2(t) f̂ 25
1,1.2(t)
0.1 3.3346 3.33328 3.3346 3.3346
0,5 2.54618 2.54648 2.54618 2.54618
0.99 -0.327282 -0.326829 -0.327282 -0.327282
1.01 -0.759792 -0.759792 -0.759792 -0.759792
3 -0.0012482 -0.0012482 -0.0012482 -0.0012482
10 −9.25 × 10−6 −9.25 × 10−6 −9.25 × 10−6 −9.25 × 10−6
SOME PROPERTIES OF WEIGHT FUNCTIONS 135
Analyzing the table we see that even for N = 10 the values of the function
are calculated with a high accuracy. It is important to mention that the
accuracy of calculations declines, when t tends to r, due to discontinuity of
the function fr,a(|t|) in the point |t| = r.
Figures 3, 4, 5, and 6 show the sequence lg(gN(t)) for different t, where
gN(t) :=
∣∣∣f1, 1.2(t) − f̂N
1, 1.2(t)
∣∣∣ · { N
3N
4 , t < 1,
N
3N
2 , t > 1.
10 20 30 40 50
N
25
50
75
100
125
150
gN �t�
t�0.9999999999
t�0.5
t�0.00000000001
Fig.3. Plot of lg(gN(t)), t < 1
10 20 30 40 50
N
50
100
150
200
250
300
gN �t�
t�3
t�1.00000000001
Fig.4. Plot of lg(gN(t)), t > 1
200 400 600 800 1000
N
1000
2000
3000
4000
gN �t�
t�3
t�0.5
Fig.5. Plot of lg(gN(t))
2000 4000 6000 8000 10000
N
10000
20000
30000
40000
50000
60000
70000
gN �t�
t�0.5
Fig.6. Plot of lg(gN(t))
Numerical results shown on the figures 3-6 are in complete accordance
with estimates for the rates of convergence in Theorems 1 and 2 (ε = 1
4
was
chosen).
Concluding remarks
The rate of convergence for weight functions series in Tauberian theorems
for random fields was obtained. Numerical results show that partial sums of
the series give good approximation for weight functions and have required
rate of convergence.
Some asymptotic properties of hypergeometric functions were obtained
as auxiliary results.
136 ANDRIY OLENKO AND BORIS KLYKAVKA
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Department of Probability Theory and Mathematical Statistics, Math-
ematical Faculty, Kyiv University, Volodymyrska 64, Kyiv, 01033, Ukraine
E-mail address: olenk@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics, Math-
ematical Faculty, Kyiv University, Volodymyrska 64, Kyiv, 01033, Ukraine
E-mail address: bklykavka@yahoo.com
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