1. Introduction
The concept of statistical convergence was first defined by Steinhaus [3] at a conference held at Wroclaw University, Poland in 1949 and also independently by Fast [4], Buck [5] and Schoenberg [6] for real and complex sequences. Further this concept was studied by Salat [7], Fridy [8], Connor [9] and many others. Statistical convergence is a generalization of the usual notation of convergence that parallels the usual theory of convergence.
A sequence 
is said to be statistically convergent to 
if for a given 
A sequence 
is said to be statistically precauchy if
Connor, Fridy and Klin [1] proved that statistically convergent sequences are statistically pre-cauchy and any bounded statistically pre-cauchy sequence with a nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-cauchy sequences are not necessarily statistically convergent (see [10]).
Throughout a double sequence is denoted by 
A double sequence is a double infinite array of elements 
for all 
The initial works on double sequences is found in Bromwich [11], Tripathy [12], Basarir and Solancan [13] and many others.
Definition 1.1. A double sequence 
is called statistically convergent to 
if
where the vertical bars indicate the number of elements in the set.
Definition 1.2. A double sequence 
is called statistically pre-cauchy if for every 
there exist 
and 
such that
Definition 1.3. An Orlicz Function is a function 
which is continuous, nondecreasing and convex with 
for 
and
, as
.
If convexity of 
is replaced by 
, then it is called a Modulus function (see Maddox [14]). An Orlicz function may be bounded or unbounded. For example,
is unbounded and 
is bounded (see Maddox [14]).
Lindenstrauss and Tzafriri [15] used the idea of Orlicz functions to construct the sequence space,
The space 
is a Banach space with the norm
The space 
is closely related to the space 
which is an Orlicz sequence space with 
for
.
An Orlicz function M is said to satisfy 
condition for all values of 
if there exists a constant 
such that 
for all values of 
The study of Orlicz sequence spaces have been made recently by various authors [1,2,16-20]).
In [1], Connor,Fridy and Klin proved that a bounded sequence 
is statistically pre-cauchy if and only if
The notion of I-convergence is a generalization of statistical convergence. At the initial stage it was studied by Kostyrko, Salat, Wilezynski [21]. Later on it was studied by Salat, Tripathy, Ziman [22] and Demirci [23], Tripathy and Hazarika [24-26]. Here we give some preliminaries about the notion of I-convergence.
Definition 1.4. [20,27] Let X be a non empty set. Then a family of sets 
(
denoting the power set of X) is said to be an ideal in X if
(i) 
(ii) I is additive i.e
.
(iii) I is hereditary i.e
.
An Ideal 
is called non-trivial if
. A non-trivial ideal
is called admissible if 
.
A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal 
containing I as a subset.
For each ideal I, there is a filter 
corresponding to I. i.e.
Definition 1.5. [10,21,28] A double sequence 
is said to be I-convergent to a number L if for every
,
In this case we write 
Definition 1.6. [21] A non-empty family of sets 
is said to be filter on X if and only if
(i)
(ii) For 
we have 
(iii) For each 
and 
implies
.
2. Main Results
In this article we establish the criterion for any arbitrary double sequence to be I-pre-cauchy.
Theorem 2.1. Let 
be a double sequence and let M be a bounded Orlicz function then 
is I-preCauchy if and only if
Proof: Suppose that
For each 
and 
we have that
(1)
(2)
Now by (1) and (2) we have
thus 
is I-pre-Cauchy.
Now conversely suppose that 
is I-pre-Cauchy, and that 
has been given.
Then we have
where,
Let 
be such that 
Since M is a bounded Orlicz function there exists an integer B such that 
for all
. Therefore, for each
(3)
Since 
is I-pre-Cauchy, there is an 
such that the right hand side of (3) is less than 
for all
. Hence
Theorem 2.2. Let 
be a double sequence and let M be a bounded Orlicz function then x is I-convergent to L if and only if
Proof: Suppose that
with an Orlicz function M, then 
is I-convergent to L (See [1])
Conversely suppose that 
is I-convergent to L. We can prove this in similar manner as in Theorem 2.1 assuming that
and M being a bounded Orlicz function.
Corollary 2.3. A sequence 
is I-convergent if and only if
Proof: Let 
Then
Let
(4)
and
(5)
Therefore from (4) and (5) we have,
Hence
if and only if
By an immediate application of Theorem 2.1 we get the desired result.
Corollary 2.4. A sequence 
is I-convergent to L if and only if
Proof: Let 
We can prove this in the similar manner as in the proof of Corollary 2.3.
3. Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.