convergence of the fractional parts of the random variables to the ...

Ann. Acad. Rom. Sci.
Ser. Math. Appl.
ISSN 2066 - 6594 Vol. 4, No. 2 / 2012
CONVERGENCE OF THE
FRACTIONAL PARTS OF THE
RANDOM VARIABLES TO THE
TRUNCATED EXPONENTIAL
DISTRIBUTION ∗
Bogdan Gheorghe Munteanu †
Abstract
Using the stochastic approximations, in this paper it was studied
the convergence in distribution of the fractional parts of the sum of random
variables to the truncated exponential distribution with parameter
λ. This fact is feasible by means of the Fourier-Stieltjes sequence (FSS)
of the random variable.
MSC: 62E20, 60F05
keywords: limit theorems, asymptotic distribution, fractional part, truncated
exponential distribution
1 Introduction
The aim of this paper is to extend the results of Wilms [9] about convergence
of the fractional parts of the random variables.
∗ Accepted for publication in revised form on June 14, 2012.
† munteanu.b@afahc.ro Faculty of Aeronautic Management, Department of Fundamental
Sciences, Henri Coanda Air Forces Academy, 160 Mihai Viteazu St., Brasov, Romania
118

Convergence of the fractional parts of the random variables 119
This theory was analysed by Wilms in [9], where the study of convergence
of the fractional parts of the sum of random variables it was directed towards
the uniform distribution on the interval [0, 1]. Moreover, he identifed the
necessary and sufficient conditions for the convergence of the product of the
random variables, not necessary independent and identically distributed,
towards the same uniform distribution on the interval [0, 1]. The Fourier-
Stieltjes sequence, (see Definition 1), play an important role in the study of
fractional parts of random variables. Also, Wilms obtain conditions under
which fractional parts of products of independent and identically distributed
random variables are uniform distribution on the interval [0, 1]. After a
survey of some results by Schatte in [6], Wilms extend the results of Schatte
on sums of independent and identically distributed lattice random variables.
Furthermore, Schatte in [6], gives rates for the convergence of distribution
function of the fractional parts of the sum of random variables to distribution
function of random variables with continouous uniform distribution on the
interval [0, 1].
The novelty of this paper consist in the identification of the conditions
(Theorems 4, 5, 6) when the distribution of the fractional parts of the sum
of random variables converge to the truncated exponential distribution.
2 Notations, definitions and auxiliary results
Let (Ω , F , P) be the probability space and a random variable X, X : Ω → R
measurable function. The distribution of random variable X is the measure
of probability P X defined on B(R) Borel and P X (B) = P (X ∈ B). The
distribution function of the random variable X is F X (x) = P(X < x), x ∈ R,
x∫
or F X (x) = f X (y)dy where f X represents the density of probability of
−∞
the random variable X.
Throughout the paper, for A ⊂ R, F(A) = {F X | P (X ∈ A) = 1}.
For the random variable X, the fractional part of X is defined as follows:
{X} = X − [X], where [X] represents the integer part of X.
The distribution function of the random variable {X} for any x ∈ [0, 1]
is
F {X} (x) =
∞∑
m=−∞
P(m ≤ X < m + x) =
∞∑
m=−∞
(F X (m + x) − F X (m)) .

120 Bogdan Gheorghe Munteanu
where F X is the distribution function of random variable X.
The random variable X has truncated exponential distribution of parameter
λ (denoted by X ∼ Exp ∗ (λ)), if its distribution function F X ∈ F([0, 1))
and
⎧
⎨
F X (x) =
⎩
0 , x < 0
1−e −λx
1−e −λ , x ∈ [0, 1]
1 , x > 1
Moreover, if random variable X has the density f X , then:
F {X} (x) =
∞∑
j=−∞
that is h {X} (x) =
∫
j+x
j
f X (y)dy =
∞ ∑
j=−∞
∞∑
∫ x
j=−∞
0
f X (j + t)dt =
.
∫ x
0
∞∑
j=−∞
f X (j + t)dt ,
f X (j + x), x ∈ [0, 1] is the density of probability of
the random variable {X}. For example, if X ∼ Exp ∗ (λ), then
(
F {X} (x) = 1 − e −λx) (
/ 1 − e −λ) , x ∈ [0, 1].
So, the characteristic function of the random variable X, ϕ X : R → C is
defined by:
∫+∞
ϕ X (t) := Ee itX = e itx dF X (x) , (t ∈ R) .
−∞
Definition 1. The Fourier-Stieltjes sequence (FSS) of the random variable
X is the function c X : Z → C defined by
c X (k) := ϕ X (2πk) , k ∈ Z .
Proposition 1. ([9]) For any random variable X the following relation occurs:
c X (k) = c {X} (k) , ∀ k ∈ Z .
The properties that characterizes the Fourier-Stieltjes sequence, it was
presented a books [1], [3] and [5].
Theorem 1. (of continuity, [5]) Let (F n ) ∈ F([0, 1)) be a sequence of the
random variables, and let (c n ) be FSS respectively.

Convergence of the fractional parts of the random variables 121
(i). Let F ∈ F([0, 1)) be with c FSS respectively. If F n
n→∞ →
lim n→∞ c n (k) = c(k), pentru k ∈ Z.
F , then
(ii). If lim n→∞ c n (k) = c(k) is for k ∈ Z, then is F ∈ F([0, 1)) so that
n→∞
F n → F . Then the sequence c is FSS of F .
We define the convolution F of distribution functions F 1 , F 2 ∈ F([0, 1))
such that F ∈ F([0, 1)).
Denote by F 1 ≡ F {X1 }, F 2 ≡ F {X2 } and F ≡ F {X1 +X 2 }.
To this end, let F 1 ∗ F 2 denote the convolution in the customary sense,
[2], i.e.
(F 1 ∗ F 2 ) (x) =
∫ ∞
−∞
F 1 (x − y)dF 2 (y) =
∫ ∞
−∞
with F 1 ∗ F 2 ∈ F([0, 2)) if F 1 , F 2 ∈ F([0, 1)) and x ∈ [0, 1].
Definition 2. Let F, F 1 , F 2 ∈ F([0, 1)). The function
(F 1 ⊗ F 2 ) (x) =
is said to be the truncated convolution.
F 2 (x − y)dF 1 (y),
(F 1 ∗ F 2 ) (x)
, x ∈ [0, 1]
(F 1 ∗ F 2 ) (1) − (F 1 ∗ F 2 ) (0)
In particular, if F 1 , F 2 ∈ F([0, 1)) is the distribution functions of random
variables X 1 and X 2 independent and identically exponential distributed,
then the distribution function of the sum of random variables X 1 and X 2
(F
in fractional part is F (x) =
1 ∗F 2 )(x)
(F 1 ∗F 2 )(1)−(F 1 ∗F 2 )(0) = 1−(1+λx)e−λx , with F ∈
1−(1+λ)e −λ
F([0, 1)), ∀ x ∈ [0, 1].
Theorem 2. (of convolution, [9]) Let F, F 1 , F 2 ∈ F([0, 1)) be with FSS
c, c 1 , c 2 . Then
c = c 1 c 2 ⇐⇒ F = F 1 ⊗ F 2 .
Corollary 1. ([9]) Let X and Y be two independent random variables with
F X , F Y ∈ F([0, 1)). Then
c {X+Y } = c X c Y .

122 Bogdan Gheorghe Munteanu
The next result characterizes FSS with the help of the repartition function
F X .
Proposition 2. ([9]) Let F X ∈ F([0, 1)) be. Then
∫ 1
c F (k) = 1 − 2πik
0
F X (x)e 2πikx dx , k ∈ Z .
The next theorem characterizes the convergence in distribution (denoted
by ” d → ”) by means of FSS.
Theorem 3. ([9]) Let (X m ) be a sequence of independent random variables
and S n :=
n ∑
m=1
Then {S n } d → S if and only if
X m , n ∈ N. Let S be the random variable with F S ∈ F([0, 1)).
n∏
m=1
There are a couple of intermediate results.
c {Xm}(k) → c S (k) if n → ∞ for any k ∈ Z.
Proposition 3. ([7]) Let (a n ) be a sequence of real numbers, a n > 0, for all
∏
n ∈ N. Then ∞ ∞∑
a n is convergent if and only if (1 − a n ) is convergent.
n=0
Proposition 4. ([8]) Let X n be a sequence of independent random variables.
∞∑
We assume that V arX m is finite.
(i). Then
n ∑
m=1
m=1
n=0
(X m − EX m ) converges almost certainly for n → ∞ .
(ii). If
∞∑
m=1
n → ∞ .
EX m is convergent, then
n ∑
m=1
X m converges almost certainly if
3 The convergence in the distribution of the fractional
part
In this section we shall give sufficient conditions for the fractional parts of
the independent and identically distributed random variables. In Theorem

Convergence of the fractional parts of the random variables 123
n∑
4, supposing that V arX m is convergent, we show that the existence
{ m=1
n∑
}
of limit lim EX m is necessary and sufficient for the convergence of
n→∞ m=1 { n∑
}
the distribution of EX m if n → ∞. Theorem 5 states necessary
m=1 { n∑
}
and sufficient conditions, using FSS for the convergence X m to the
m=1
distribution Exp ∗ (λ) if n → ∞. We also neet conditions of convergence in
Teorema 6.
The Fourier-Stieltjes sequence of the random variable X, X ∼ Exp ∗ (λ),
is presented in the following theoretical result:
Proposition 5. If X ∼ Exp ∗ (λ) and the distribution function F X ∈ F([0, 1)),
then
λ
( )
c Exp
∗ (λ)
(k) =
e 2πikλ − 1 , ∀ k ∈ Z 0 .
2πik − λ
Proof. According to the definition FSS,
c Exp
∗ (λ)
(k) =
=
∫ 1
0
(
e 2πikx d 1 − e −λx) ∫ 1
= λ
λ
2πik − λ
( )
e 2πik−λ − 1 .
0
e (2πik−λ)x dx
Next, we shall present { the original results that inform us under what circumstances
the sum X m converges in distribution towards truncated
n∑
}
m=1
exponential distribution.
Theorem 4. Let (X m ) be a sequence of independent and identically distributed
random variables, so that V arX m is finite and X 1 ∼ Exp ∗ (λ).
∞∑
{ m=1
n∑
}
{ n∑
}
Then X m converges in distribution if and only if lim n→∞ EX m
m=1 m=1
exists.

124 Bogdan Gheorghe Munteanu
Proof. First, we shall proof sufficiency. Since X 1 ∼ Exp ∗ (λ), it results that
EX m = 1 λ .
{ n∑
} { n∑
{
(
On the other hand, X m = Xm − 1 ) n∑
}}
λ +
1
λ
. According
to Proposition 4, Xm − λ) 1 a.s
n∑
m=1 m=1
m=1 {
n∑ (
}
→ X, for n → ∞. Then m
{ m=1
m=1X
X +
1
λ}
.
{ n∑
}
As a necessity, let X m be shall converge for n → ∞. Similarly,
we have
{ m=1
n∑
} { n∑
{
(
1
λ
=
1
λ − X ) n∑
m + X m
}}. From Proposition
4, 1
m=1
m=1
m=1
n∑ (
λ − X )
m converges almost certainly when n → ∞. Therefore,
{ m=1
n∑
}
1
λ
exists.
m=1
The following theorems provide the necessary and sufficient conditions
for the fractional parts of the sums of the independent random variables
identically towards the truncated exponential distribution of parameter λ.
Theorem 5. Let (X m ) be a sequence of independent and identically random
variables with (c m ), the corresponding FSS , and also S n =
(i). {S n } d → Exp ∗ (λ) ⇐⇒
n ∏
m=1
c m (k) →
n ∑
m=1
d
→
X m , n ∈ N.
λ (
2πik−λ e 2πik−λ − 1 ) , n → ∞,
(ii). We suppose c m (k) ≠ 0, ∀ k ∈ Z 0 , m ∈ N, {S n } does not converge to
Exp ∗ (λ) is equivalent to
∞∑
(1 − |c m (k)|) is divergent, ∀ k ∈ Z 0 .
m=1
Proof. [i]. Based on the Theorem 3,
{S n } → d Exp ∗ (λ) ⇐⇒ c {Sn}(k) n→∞ → c Exp
∗ (λ)
(k) Proposition ⇐⇒
5
⇐⇒
n∏
c m (k) →
m=1
λ ( )
e 2πik−λ − 1
2πik − λ
, n → ∞ .

Convergence of the fractional parts of the random variables 125
∞∏
m=1
[ii]. According to Proposition 3,
∞∑
m=1
(1 − |c m (k)|) divergent ⇐⇒
c m (k) is divergent, that is {S n } d → F, with F ≠ Exp ∗ (λ).
Corollary 2. If the sequence (c m ), of the random variables sequence (X m )
∞∑
meets the condition of (1 − |c m (k)|) to be convergent, ∀ k ∈ Z 0 , then
m=1
{S n } d → Exp ∗ (λ), where S n = n ∑
m=1
X m , n ∈ N.
Theorem 6. Let (X m ) be a sequence of independent and identically distributed
random variables with the characteristic function ϕ, X 1 ∼ Exp ∗ (λ),
and 0 < V arX 1 < ∞. Let (a m ) be a sequence of real numbers so that
lim m→∞ a m = 0. We define V n :=
(i). If
∞∑
m=1
n ∑
m=1
a m X m , n ∈ N.
a 2 m is convergent, then {V n } d → Exp ∗ (λ).
(ii). We suppose that there is k ∈ Z 0 , ϕ(2πka m ) ≠ 0. If
a 2 m is divergent,
then {V n } don’t converge to Exp ∗ (λ).
∞∑
m=1
Proof. [i]. Let k ∈ Z 0 be fixed. By Corollary 2, it is sufficient to show that
∞∑
(1 − |ϕ(2πka m )|) is convergent.
m=1
It is known that if X ∼ Exp ∗ (λ), then ϕ X (t) = λ/ (λ − it), from where
|ϕ X (t)| =
( )
λ
t 2 −
1
2
√
(1
λ 2 + t = +
.
λ) 2
If we consider the development in binomial series (1 + x) − 1 2 = 1 − x 2
(
+
3
8 x2 + ..., then we obtain 1 + ( ) )
t 2 −
1
2
λ
= 1 − 1
2λ t2 + o(t 2 ), from where the
following 1
4λ t2 < 1 − |ϕ X (t)| < 1 λ t2 .
Then
∞∑
∞∑ 1
(1 − |ϕ(2πka m )|) <
λ 4π2 k 2 a 2 m = 4π2 k 2 ∞∑
a 2
λ
m.
m=1
m=1
m=1

126 Bogdan Gheorghe Munteanu
As
(1 − |ϕ(2πka m )|) is convergent.
∞∑
m=1
a 2 m is convergent, it means that
[ii]. Since 1−|ϕ X (t)| > t2
4λ , we have ∞ ∑
Results that
∞∑
m=1
m=1
∞∑
m=1
(1 − |ϕ(2πka m )|) > π2 k 2
λ
(1 − |ϕ(2πka m )|) is divergent because ∞ ∑
a 2 m is divergent.
Next Theorem 5(ii) is taken into account .
Example
m=1
∞∑
m=1
a 2 m.
Let (X m ) be a sequence of independent and identically distributed random
∞∑
variables, X 1 ∼ Exp ∗ (λ) and a m = m −b , b > 0; a 2 ∑
m = ∞ 1
=
m
{ m=1 m=1
2b
= ∞ , b ≤
1
2
< ∞ , b > 1 . We have the following situations for the sequence V n =
2
n∑
m=1
1
m b X m :
(1) b ≤ 1 2 , ∞∑
m=1
(2) 1 2 < b ≤ 1 , ∞∑
a 2 m = ∞ T heorem 6 =⇒ {V n } does not converge to Exp ∗ (λ).
m=1
a 2 m < ∞ T heorem 6 =⇒ {V n } d → Exp ∗ (λ) or
{ 1
1 b X 1 + 1 2 b X 2 + ... + 1 n b X n}
(3) b > 1, according to Theorem 4, lim n→∞
n∑
that is {V n } d → F , with F ≠ Exp ∗ (λ).
References
m=1
n→∞ →
Exp ∗ (λ)
EX m = lim n→∞
n
λ = ∞,
[1] P.D.T.A. Elliott, Probabilistic number theory, vol.1 of Grund-lehren der
mathematische Wissenschaften 239, Springer-Verlag, New York, 1979.
[2] W. Feller, An introduction to probability theory and its applications,
Vol 2, John Wiley&Sons, New York, 1971.

Convergence of the fractional parts of the random variables 127
[3] T. Kawata, Fourier analysis in probability theory, Academic Press, London,
1972.
[4] E. Lukacs, Characteristic functions, Griffin, London, 1970.
[5] K.V. Mardia, Statistics of directional data, Academic Press London,
1972.
[6] P. Schatte, On the asymptotic uniform distribution of the n-fold convolution
mod 1 of a lattice distribution, Matematische Nachrichten, 128:
233-241, 1986.
[7] E.C. Titchmarsh, The theory of functions, Oxfoed University Press,
London, second ed., 1960.
[8] H.G. Tucker,A graduate course in probability, Academic Press, New
York, 1967.
[9] R.J. Wilms Gerardus, Fractional parts of random variables, Wibro Dissertatiedrukkerij,
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