VNU Journal of Science, Mathematics - Physics 23 (2007) 70-74
On thestabilityofthedistributionfunctionofthe composed
random variablesbytheirindexrandom variable
Nguyen Huu Bao
∗
Faculty of Infomation Technology, Water Resources University
175 Tay Son, Dong Da, Hanoi, Vietnam
Received 15 November 2006; received in revised form 2 August 2007
Abstract. Let us consider thecomposedrandomvariable η =
ν
k=1
ξ
k
, where ξ
1
, ξ
2
,
are independent identically distributed randomvariables and ν is a positive value random,
independent of all ξ
k
.
In [1] and [2], we gave some the stabilities ofthedistributionfunctionof η in the following
sense: the small changes in thedistributionfunctionof ξ
k
only lead to the small changes in
the distributionfunctionof η.
In the paper, we investigate thedistributionfunctionof η when we have the small changes of
the distributionof ν.
1. Introduction
Let us consider therandomvariable (r.v):
η =
ν
k=1
ξ
k
(1)
where ξ
1
, ξ
2
, are independent identically distributed randomvariables with thedistribution function
F (x), ν is a positive value r.v independent of all ξ
k
and ν has thedistributionfunction A(x).
In [1] and [2], η is called to be thecomposed r.v and ν is called to be its index r.v. If Ψ(x) is
the distributionfunctionof η with the characteristic function ψ(x) respecrively then (see [1] or [2])
ψ(x) = a[ϕ(t)] (2)
where a(z) is the generating functionof ν and ϕ(t) is the characteristic functionof ξ
k
.
In [1] and [2], we gave some the stabilities of Ψ(x) in the following sence: the small changes
in thedistributionfunction F (x) only lead to the small changes in thedistributionfunction Ψ(x).
In this paper, we shall investigate thestabilityof η’s distributionfunction when we have the
small change ofthedistributionoftheindex r.v ν.
∗
Tel.: 84-4-5634255.
E-mail: nhuubao@yahoo.com
70
Nguyen Huu Bao / VNU Journal of Science, Mathematics - Physics 23 (2007) 70-75 71
2. Stability theorem
Let us consider the r.v now:
η
1
=
ν
1
k=1
ξ
k
(3)
where ν
1
has thedistributionfunction A
1
(x) with the generating function a
1
(z). Suppose ξ
k
have the
stable law with the characteristic function
ϕ(t) = exp{iµt −c|t|
α
[1 −eβ
t
|t|
ω(t; α)]} (4)
where c, µ, α, β are real number, c ≥ 0; |β| 1,
2 ≥ α ≥ α
1
> 1; ω(t; α) = tg
αt
2
. (5)
For every ε > 0 is given, such that
ε < (
π
3c
2
)
3
(6)
where c
2
= (c + c|β||tg
α
1
π
2
+ |µ|).
We have the following theorem:
Theorem 2.1 (Stability Theorem). Assume that
ρ(A; A
1
) = sup
x∈R
1
|A(x) −A
1
(x)| ε
µ
α
A
=
+∞
0
z
α
dA(z) < +∞; µ
α
A
1
=
+∞
0
z
α
dA
1
(z) < +∞, ∀α > 0. (7)
Then we have
ρ(Ψ, Ψ
1
) K
1
ε
1/6
where K
1
is a constant independent of ε, Ψ(x) and Ψ
1
(x) are thedistributionfunctionof η and η
1
respectively.
Lemma 2.1. Let a is a complex number, a = ρe
iθ
, such that |θ|
π
3
; 0 ρ 1. Then we have the
following estimation:
|a
t
− 1|
√
14|a −1|
(1 − |a −1|)
(for every t > 0) (8)
Proof. Since a = ρ(cos θ + i sin θ), it follows that a
t
= ρ
t
(cos tθ + i sintθ).
Hence
|a
t
−1|
2
= (ρ
t
cos tθ − 1)
2
+ (ρ
t
sin tθ)
2
, (9)
we also have
(ρ
t
cos tθ − 1) = (ρ
t
−1) cos tθ + (cos tθ − 1),
Notice that |1 −cos x| |x| for all x, thus
|ρ
t
cos tθ − 1| |ρ
t
− 1|+ |tθ|.
On the other hand, since |sin u| |u| for all u,
|a
t
− 1|
2
2|ρ
t
− 1|
2
+ 2t
2
θ
2
+ ρ
2t
t
2
θ
2
, (10)
72 Nguyen Huu Bao / VNU Journal of Science, Mathematics - Physics 23 (2007) 70-75
we can see
|a −1|
2
= (ρ cos θ −1 )
2
+ (ρ
2
sin
2
θ).
It follows that
|ρ sin θ| |a −1|. (11)
Furthermore,
||a| −1| |a −1| ⇒ |ρ −1| |a −1| ⇒ ρ ≥ 1 − |a − 1|.
From (11) we obtain
|sin θ|
|a −1|
ρ
|a − 1|
1 − |a − 1|
. (12)
Since |θ|
π
3
⇒ |sinθ| ≥
|θ|
2
, so that
|θ|
2|a − 1|
(1 −|a −1|)
. (13)
From (10) and (13), we have
|a
t
−1|
2
2|ρ
t
−1|
2
+
8t
2
|a −1|
2
(1 − |a −1|)
2
+ 4
ρ
2t
t
2
|a − 1|
2
(1 − |a −1|)
2
. (14)
For all t ≥ 0, the following inequality holds:
1 −ρ
t
t(1 − ρ)
ρ
. (15)
Using (11) and notice that |1 −ρ| = |1 − |a| | |a − 1|, we shall have
1 −ρ
t
t|a −1|
ρ
. (16)
Hence by (14) we get
|a
t
− 1|
2
14t
2
|a −1|
2
(1 −|a −1|)
2
.
Lemma 2.2. Under the notation in (2), let δ(ε) be sufficiently small postive number such that δ(ε) → 0
when ε → 0 and
|argϕ(t)|
π
3
∀t, |t| δ(ε).
Then
|ψ(t) − ψ
1
(t)| C|t| ∀t, |t| δ(ε)
where C is a constant independent of ε and ψ
1
(t) is the characteristic function with the distribution
function Ψ
1
(t) respectively.
Proof. We have
|ψ(t) − ψ
1
(t)| = |
+∞
0
|ϕ(t)|
z
d[A(z) − A
1
(z)]|
+∞
0
|ϕ
z
(t) − 1|d[A(z) + A
1
(z)]. (17)
Notice that, for all t ∈ R
1
|e
itx
−1| 3|sin(
tx
2
)|
3
2
|tx| < 2|tx|.
Hence, if we put
µ
F
=
+∞
−∞
|x|dF (x) < +∞; ϕ(t) =
+∞
−∞
e
itx
dF (x),
Nguyen Huu Bao / VNU Journal of Science, Mathematics - Physics 23 (2007) 70-75 73
then
|ϕ(t) −1|
|e
itx
− 1|dF < 2|t|µ
F
.
From lemma 2.1, (with a = ϕ(t); |t| δ(ε))
|ϕ(t) −1|
√
14z|ϕ(t) −1 |
(1 − |ϕ(t) − 1|)
. (18)
Because there exits moments (from (7)) and with t, |t| δ(ε) we can see |1 − ϕ(t)|
1
2
, therefore
|ψ(t) − ψ
1
(t)|
+∞
0
√
14z|ϕ(t) −1 |
(1 − |ϕ(t) − 1|)
d[A(z) + A
1
(z)] 4
√
14µ
F
(µ
A
+ µ
A
1
)|t| = C|t|
(do |ϕ(t) −1| µ
F
|t| ∀t)
where C is a constant independent of ε and µ
F
=
+∞
−∞
|x|dF (x) < ∞.
Proof of Theorem 2.1.
For every N > 0 and t ∈ R
1
, we have
|ψ(t) − ψ
1
(t)| = |
+∞
0
ϕ
z
(t)d[A(z) − A
1
(z)]|
|
N
0
ϕ
z
(t)d[A(z) − A
1
(z)]|+ |
+∞
N
ϕ
z
(t)d[A(z) − A
1
(z)]|
|[A(z) − A
1
(z)]|
N
0
| +
N
0
|A(z) − A
1
(z)||ϕ
z
(t)||ln ϕ(t)|dz +
+∞
N
d[A(z) + A
1
(z)]
= I
1
+ I
2
+ I
3
. (19)
First, it easy to see that
I
1
2ε. (20)
In order to estimate I
2
, notice that ϕ(t) has form (4) with the condition (5) so we have
|ln ϕ(t)| |µ||t|+ |t|
α
(c + c|β||tg
απ
2
|) |µ||t| + C
1
|t|
α
(21)
where C
1
= c + c|β||tg
απ
2
| c + c|β||tg
α
1
π
2
|.
If T = T (ε) is a positive number which will be chosen later (T (ε) → ∞ when ε → 0), we can see
that
|ln ϕ(t)| |µ|T + C
1
T
α
(C
1
+ |µ|)T
α
C
2
T
α
∀t, |t| T (ε)
where C
2
= c + c|β||tg
α
1
π
2
| + |µ|; (α ≥ α
1
> 1).
Then
I
2
ε
N
0
C
2
T
α
dz C
2
εT
α
N. (22)
Finally, with α from condition (5), we have
I
3
µ
α
A
+ µ
α
A
1
N
α
. (23)
By using (19), (20), (21), (23), we conclude that
|ψ(t) − ψ
1
(t)| 2ε + C
2
εT
α
N +
µ
α
A
+ µ
α
A
1
N
α
. (24)
74 Nguyen Huu Bao / VNU Journal of Science, Mathematics - Physics 23 (2007) 70-75
Choosing T = ε
−
1
3
α
and N = T = ε
−
1
3
α
, we can see that
C
2
εT
α
N C
2
ε
1−
1
3
−
1
3
= C
2
ε
1
3
,
(µ
α
A
+ µ
α
A
1
)N
−α
= (µ
α
A
+ µ
α
A
1
)ε
1
3
.
Thus
|ψ(t) − ψ
1
(t)| 2ε + C
2
ε
1
3
+ (µ
α
a
+ µ
α
A
1
)ε
1
3
= C
3
ε
1
3
(25)
for every t with |t| T = ε
−
1
3
α
and C
3
is a constant independent of ε.
For all δ(ε) > 0, we consider now
T
−T
|
ϕ(t) −ϕ
1
(t)
t
|dt =
δ(ε)
−δ(ε)
|
ϕ(t) −ϕ
1
(t)
t
|dt +
δ(ε)|t|T
|
ϕ(t) − ϕ
1
(t)
t
|dt.
Since
ln z = ln |z| + iarg(z) (0 argz 2π),
for all complex number z, letting z = ϕ(t), (|t| δ(ε))
|argϕ(t)| |ln ϕ(t)| C
2
δ(ε)
with δ(ε) = ε
1
3
, we shall get |argϕ(t)| C
2
ε
1
3
and from (6)
C
2
ε
1
3
π
3
⇒ |argϕ(t)|
π
3
for every t, |t| δε.
Hence, using lemma 2.2, we obtain:
δ(ε)
−δ(ε)
|
ϕ(t) − ϕ
1
(t)
t
|dt 2Cδ( ε) = 2Cε
1
3
. (26)
On the other hand, using (25), we get
δ(ε)|t|T
|
ϕ(t) − ϕ
1
(t)
t
|dt C
3
ε
1
3
T
δ(ε)
dt
t
= C
3
ε
1
3
ln
T
δ(ε)
= C
3
ε
1
3
ln(
1
ε
1 + α
3α
) C
4
ε
1
6
. (27)
From (26) and (27)
T
−T
|
ϕ(t) − ϕ
1
(t)
t
|dt 2Cε
1
3
+ C
4
ε
1
6
C
5
ε
1
6
where C
5
is constant independent of ε.
Indeed, by using Essen’s inequality (see [3]) we have
ρ(Ψ; Ψ
1
) C
5
ε
1
6
+ C
6
ε
1
4
K
1
ε
1
6
where K
1
is a constant independent of ε.
Nguyen Huu Bao / VNU Journal of Science, Mathematics - Physics 23 (2007) 70-75 75
Acknowledgements. This paper is based on the talk given at the Conference on Mathematics, Me-
chanics, and Informatics, Hanoi, 7/10/2006, on the occasion of 50th Anniversary of Department of
Mathematics, Mechanics and Informatics, Vietnam National University.
References
[1] Tran Kim Thanh, On the characterization ofthedistributionofthecomposedrandomvariables and their stabilities
Dotor thesis, Hanoi 2000.
[2] Tran Kim Thanh, Nguyen Huu Bao, On the geometric composedvariables and the estimate ofthe stable degree of the
Renyi’s characteristic theorem, Acta Mathemaica Vietnamica 21 (1996) 269.
[3] C. G. Essen, Fourier analysis ofdistribution functions, Acta Math. 77 (1945) 125.
. Journal of Science, Mathematics - Physics 23 (2007) 70-74
On the stability of the distribution function of the composed
random variables by their index random. University.
References
[1] Tran Kim Thanh, On the characterization of the distribution of the composed random variables and their stabilities
Dotor thesis, Hanoi 2000.
[2] Tran