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

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.