The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution

The geometric sums have been arisen from the necessity to resolve practical problems in ruin probability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.


INTRODUCTION
During the last several decades, the weak limit theorems for geometric sums have been become one of the most important problems in applied probability and related topics such as insurance risk theory, stochastic finance and queuing theory, etc. Klebanov et al. (1984) introduced the concepts on geometrically infinitely divisible (GID) distributions and geometrically strictly stable (GSS) distributions 1 . Up to now, the geometric random sums have been investigated by many mathematicians such as Kruglov and Korolev (1990), Kalashnikov (1997), Kotz et al. (2001), Kozubowski (2000), Kozubowski and Podrsky (2010), etc [2][3][4][5][6] . It is worth pointing out that, the class of geometrically strictly stable laws are closely related to the heavy tail distributions like exponential distribution, Laplace distribution and Linnik distribution 7 . Recently, some results on the weak limit theorems for geometric sums together with rates of convergence and its applications were published by Hung (2013), Teke and Deshmukh (2014) 8,9 . However, in any situations, results related to the domain of geometric attractions are still restrictive. For a deeper discussion of this problem we refer the reader to Kruglov and Korolev (1990) 2 and Sandhya and Pillai (1999) 10 .
The main purpose of this paper is to show the nec-

PRELIMINARIES
Before stating the main theorems we first recall fundamental notions and some classical results that had been presented in references 4,11,12 . The characteristic function f (t) of the random variable X is defined in form With respect to the characteristic functions, we will recall following result which will useful for proofs of our main results (see 1 ).
where δ (u) denotes a function of u, such that for all u, For p ∈ (0, 1), a random variable v p is said to be a geometric random variable with mean 1/p, denoted by v p ∼ Geo(p), if its probability distribution given as follows Let {X j , j ≥ 1} be a sequence of independent and identically distributed (i.i.d.) random variables, independent of v p . We write and it is called the geometric sums.
According to 6 , a random variable Y is said to be a standard Laplace distributed random variable, denoted by Y ∼ L (0, 1), if its characteristic function is given as

MAIN RESULTS
Let {X j , j ≥ 1} be a sequence of i.i.d. random variables with common distribution function F(x) and corresponding characteristic function f (t). We introduce the following notations.
where v p is a geometric random variable with mean 1/p, p ∈ (0, 1), independent of all X j for all j ≥ 1. The following statements are equivalent: The characteristic function f (t) satisfies Proof. Since v p ∼ Geo(p), let us denote by the generating function of geometric random variable v p . Then, the characteristic function of the geometric Thus, the characteristic function of c(p) Hence, Equivalently, The proof is complete. Additionally, if the sequence of i.i.d. random variables X 1 , X 2 , . . . has the moments E(X 1 ) and E(X 2 1 ) are finite, then its distribution function will belong to the domain of geometric attraction of standard Laplace distribution. This assertion will be evidenced by the following theorem.  Then, F(x) ∈ DGA L (0,1) .
Proof. Let f (t) be the corresponding characteristic function of the distribution function F(x). Using the hypothesis of this theorem and according to Theorem 2.1, we can write where R(w) denotes a bounded function of w such that R(w) → 0 as w → 0.
Thus, for w = c(p)t, we obtain where R(c(p)) → 0 as p ↓ 0, for all t ∈ R.
Using the condition lim According to Theorem 3.1, it finishes the proof.
The following corollary could be considered as second central limit theorem. Corollary 3.1 Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables with the common distribution function F(x)E(X 1 ) = 0 and E(X 2 1 ) = 1. Then, Proof. Applying to Theorem 3.2 with c(p) = p1/2 the proof is straight-forward.

DISCUSSIONS
There are various methods have been used in investigation of domains of attraction in probability theory like method of characteristic functions, method of linear operators or method of probability distances, etc. Especially, the method of characteristic functions is more effective. For this reason we have used the method of characteristic functions in this study and some results on the domain of geometric attraction of standard Laplace distribution in this research were obtained.

CONCLUSIONS
Based on the obtained results of this article, the analogous results for the domains of geometric attraction of exponential and Linnik distributions shall be established. More generally, the results may be extended to the domain of geometric attraction of geometrically strictly stable distributions. The extension or generalization of received results will be considered in near future.