**INTRODUCTION**

The classical central limit theorem is considered as the heart of probability
and statistics theory which has a number of applications (Lehmann,
1999; DasGupta, 2008). The assumption of independence
for a sequence of observations X_{1},X_{2},... is usually a
technical convenience. Real data always exhibit some dependence and at least
some correlations at small lags (Choi, 2004; Omekara,
2008; Dossou-Gbete *et al*., 2009;
Osareh and Shadgar, 2009; Camminatiello and Lucadamo,
2010). One intensively studied class of dependence is the m-dependence case
(Romano and Wolf, 2000; Christofides
and Mavrikiou, 2003; Chaubey and Doosti, 2005; Francq
and Zakoian, 2005) in which random variables are independent as long as
they are m-step apart. More general measures of dependence are referred to as
mixing conditions which are derived from the estimation of the difference between
distribution functions of averages of dependent and independent random variables.
Various mixing conditions have been proposed by Bradley
(2005), Balan and Zamfirescu (2006), Kaminski
(2007), Ould-Said and Tatachak (2009). The recent
text book of DasGupta (2008) contains a survey of some
related topics with an eye on statistics.

Recently, the exact value of difference between characteristic functions of
sums of dependent and independent random variables is computed to derive central
limit theorems for certain class of dependent random variables by Kaminski
(2007). This class has some appealing physical interpretations and can be
used to describe systems which are globally determined but locally random. His
two main results are restated as follows.

**Theorem 1:** Let{X_{i}}_{i≥1} be a sequence of identically
distributed random variables such that E|X_{1}|^{2+ε}<+∞
for some ε>0. Let Var(X_{1}) = σ^{2} and ε_{1}
be a positive number such that ε_{1}<ε/(2(1+ε)). Denote
by
the partial sum. Suppose that for sufficiently large k, the inequality:

holds, where v_{1},...,v_{j} is any choice of indices such that k^{ε1}<v_{1}<...<v_{j}≤k. Then:

where
denotes convergence in distribution to standard normal distribution.

**Theorem 2:** Let {X_{i}}_{i≥1} be a sequence of identically
distributed random variables such that E|X_{1}|^{2+ε}<+∞
for some ε>0. Let Var (X_{1}) = σ^{2} and ε_{1}
be a positive number such that ε_{1}<ε/(2(1+ε)) Suppose
that for sufficiently large k:

where v_{1},...,v_{j} is any choice of indices such that k^{ε1}<v_{1}<...v_{j}≤k. Then:

The aim of this study is to extend Theorem 1 and 2 in another direction, that
is, consider the central limit theorem for partial sum of a random number of
{X_{1}}. This question is critical not only in probability theory itself
but in sequential analysis, random walk problems, Monte Carlo methods, etc.
Central limit problems for the sum of a random number of random variables have
been addressed in the study of Korolev (1992), Silvestrov
(2006), Przystalski (2009) and Hung
and Thanh (2010).

**RESULTS**

The main results are stated as follows.

**Theorem 3:** Let {X_{i}}_{i≥1} be a sequence of identically
distributed random variables such that E|X_{1}|^{2+ε}<+∞
for some ε>0. Let Var (X_{1}) = σ^{2} and ε_{1}
be a positive number such that ε_{1}<ε/ (2(1+ε)).
Denote by
the partial sum. Let {N_{n}}_{n≥1} denote a sequence of positive
integer-valued random variables such that

where, {ω_{n}}_{n≥1} is an arbitrary positive sequence tending to +∞ and ω is a positive constant. Suppose that for sufficiently large k, the inequality:

holds, where v_{1},...,v_{j} is any choice of indices such that k^{ε1}<v_{1}<...<v_{j}≤k. If (A1) there exists some k_{0}≥0 and c>0 such that, for any λ>0 and k_{0}:

and (A2) Cov (X_{i},X_{j})≥0 for all i and j, then:

**Theorem 4:** Let {X_{i}}_{i≥1} be a sequence of identically
distributed random variables such that E|X_{1}|^{2+ε}<+∞
for some ε>0. Let Var (X_{1}) = σ^{2} and ε_{1}
be a positive number such that ε_{1}<ε/ (2(1+ε)).
Let {N_{n}}_{n≥1} denote a sequence of positive integer-valued
random variables such that:

where, {ω_{n}}_{n≥1} is an arbitrary positive sequence tending to +∞ and ω is a positive constant. Suppose that for sufficiently large k, the inequality:

holds, where v_{1},...,v_{j} is any choice of indices such that k^{ε1}<v_{1}<...<v_{j}≤k. If (A1) there exists some k_{0}≥0 and c>0 such that, for any λ>0 and n>k_{0}:

and (A2) Cov (X_{i},X_{j})≥0 for all i and j, then:

Here are some remarks for Theorem 3 and similar comments may apply to Theorem
4. Firstly, note that the assumption (A1) is for sufficiently large index of
sequence X_{i}, i.e., {X_{i}}_{i>k0}. Secondly, if
{X_{i}}_{i≥1} is independent, then (A1) automatically holds
for k_{0} = 0 and c=1 by using the Kolmogorov inequality Billingsley
(1995). Therefore, the assumption (A1) may be regarded as a relaxed kolmogorov
inequality. Thirdly, the assumption (A2) says that each pair X_{i},
X_{j} of {X_{n}}_{n≥1} are positively correlated.
In view of the independent case studied in previous work, it seems likely that
the assertion of Theorem 3 still holds when ω is a positive random variable.

**Proof:** In the sequel, the proof of Theorem 3 is provided and that of Theorem 4 is left for the interested reader.

Without loss of generality, assume that X_{i} are centered at 0, i.e., EX_{1} = 0. Let 0<η<1/2. From Eq. 5 it follows that there exists some n_{0}, for any n≥n_{0}:

For any x:

By Eq. 13 and 14, for n≥n_{0}:

Let n_{1} = [ω(1-η)ω_{n}] and n_{2}
= [ω(1+η)ω_{n}]. Since ω_{n} tends to infinity,
then n_{1}≥k_{0} for large enough n. Note that
. Therefore, for |n-ωω_{n}|≤ηωω_{n}:

where:

Likewise:

Involving the assumption (A1) and Eq. 17,

the right-hand side of which is less than 1 when η is small enough.

Denote by E the event that
. By virtue of Eq. 15, 16 and 19:

Similarly, from Eq. 15, 18 and
19 it follows that:

Using (19), (21) and the assumption (A2):

where, the first inequality is due to an application of the FKG inequality
(Alon and Spencer, 2008).

Now by Theorem 1:

where, Φ (x) is the standard normal distribution function. Thus, the proof
of Theorem 3 is completed by combining Eq. 20, 22
and 23.

**CONCLUSION**

The central limit theorem for dependent random variables is one of the most active areas of research over the past decades. In this study, the central limit theorems for the sum of a random number of certain classes of dependent random variables are treated. The dependency structure may be reflected in some physical phenomena. Other issues such as the convergence rates and other dependent structures are possible future research.