1. Introduction

The idea of statistical convergence was introduced by *Steinhaus* (^{12}) and also independently by *Fast* (^{7}) for real or complex sequences. Statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence.

A triple sequence (real or complex) can be defined as a function *x*: **N** × **N** × **N** → **R**(**C**), where **N**,**R** and **C** denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by *Sahiner et al.* (^{10}), (^{11}), *Esi et al.* (^{4})-(^{6}), *Datta et al.* (^{2}), *Subramanian et al.* (^{13}), *Debnath et al.* (^{3}) and many others.

Let *K* be a subset of the set **N** × **N** × **N**, and let us denote the set {(*m,n,k*)∈ *K:m*≤ *u,n*≤ *v ,k*≤ *w*} by _{
Kuvw
} . Then the natural density of *K* is given by δ(*K*)= *lim*
_{uvw→ ∞}
, where |_{
Kuvw
} | denotes the number of elements in _{
Kuvw
} . Clearly, a finite subset has natural density zero, and we have δ(_{
Kc
} )=1-δ(*K*) where _{
Kc
} = **N**\*K* is the complement of *K*. If _{
K1
} ⊆ _{
K2
} , then δ(_{
K1
} )≤ δ (_{
K2
} ).

Consider a triple sequence *x*= (_{
xmnk
} ) such that _{
xmnk
} ∈ **R**, *m,n,k* ∈ **N**.

A triple sequence *x* = (_{
xmnk
} ) is said to be statistically convergent to 0 ∈ **R**, written as *st-lim*
*x* = 0, provided that the set

has natural density zero for any ϵ>0. In this case, 0 is called the statistical limit of the triple sequence *x*.

If a triple sequence is statistically convergent, then for every ϵ>0, infinitely many terms of the sequence may remain outside the ϵ- neighbourhood of the statistical limit, provided that the natural density of the set consisting of the indices of these terms is zero. This is an important property that distinguishes statistical convergence from ordinary convergence. Because the natural density of a finite set is zero, we can say that every ordinary convergent sequence is statistically convergent.

If a triple sequence *x*= (_{
xmnk
} ) satisfies some property *P* for all *m,n,k* except a set of natural density zero, then we say that the triple sequence *x* satisfies *P* for “almost all (*m,n,k*)”and we abbreviate this by “*a.a.*(*m,n,k*)”.

Let (_{
xminjk
}
_{ℓ}) be a sub sequence of *x*=(_{
xmnk
} ). If the natural density of the set K={(_{
mi, nj, kℓ
} )∈ **N**
^{3}: (*i,j,*ℓ) ∈ **N**
^{3}} is different from zero, then (_{
xminjkℓ
} ) is called a non thin sub sequence of a triple sequence *x*.

c ∈ **R** is called a statistical cluster point of a triple sequence *x*= (_{
xmnk
} ) provided that the natural density of the set

is different from zero for every ϵ>0. We denote the set of all statistical cluster points of the sequence *x* by Γ_{
x
} .

A triple sequence *x*= (_{
xmnk
} ) is said to be statistically analytic if there exists a positive number *M* such that

The theory of statistical convergence has been discussed in trigonometric series, summability theory, measure theory, turnpike theory, approximation theory, fuzzy set theory and so on.

The idea of rough convergence was introduced by Phu (^{9}), who also introduced the concepts of rough limit points and roughness degree. The idea of rough convergence occurs very naturally in numerical analysis and has interesting applications. Aytar (^{1}) extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence. Pal et al. (^{8}) extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence.

In this paper, we introduce the notion of rough statistical convergence of triple sequences. Defining the set of rough statistical limit points of a triple sequence, we obtain rough statistical convergence criteria associated with this set.

Throughout the paper *r* be a nonnegative real number.

2. Definitions and Preliminaries

2.1. Definition

A triple sequence *x*= (_{
xmnk
} ) is said to be rough convergent (*r*-*convergent*) to *l* (Pringsheim's sense), denoted as _{
xmnk
} →^{r}
*l*, provided that

or equivalently, if

Here *r* is called the roughness of degree. If we take *r*=0, then we obtain the ordinary convergence of a triple sequence.

2.2. Definition

It is obvious that the *r*- limit set of a triple sequence is not unique. The *r*- limit set of the triple sequence *x*= (_{
xmnk
} ) is defined as LIM_{
rxmnk
}: = {*l* ∈ **R**: _{
xmnk
} →^{r}
*l*}.

2.3. Definition

A triple sequence *x*= (_{
xmnk
} ) is said to be *r*- convergent if LIM_{
rx
} ≠ ϕ. In this case, *r* is called the convergence degree of the triple sequence x= (_{
xmnk
} ). For *r* =0, we get the ordinary convergence.

2.4. Definition

A triple sequence *x*= (_{
xmnk
} ) is said to be *r*- statistically convergent to *l*, denoted by _{
xmnk
} → _{
rst l
} , provided that the set

has natural density zero for every ϵ>0, or equivalently, if the condition

is satisfied.

In addition, we can write _{
xmnk
} → ^{rst}
*l* if and only if the inequality

holds for every ϵ>0 and almost all (*m,n,k*). Here *r* is called the roughness of degree. If we take *r* =0, then we obtain the statistical convergence of triple sequences.

In a similar fashion to the idea of classical rough convergence, the idea of rough statistical convergence of a triple sequence can be interpreted as follows:

Assume that a triple sequence *y*= (_{
ymnk
} ) is statistically convergent and cannot be measured or calculated exactly; one has to do with an approximated (or statistically approximated) triple sequence *x*= (_{
xmnk
} ) satisfying |_{
xmnk
} -_{
ymnk
} |≤ *r* for all *m,n,k* (or for almost all (*m,n,k*), i.e.,

Then the triple sequence *x* is not statistically convergent any more, but as the inclusion

holds and we have

i.e., we get

i.e., the triple sequence spaces *x* is *r*- statistically convergent in the sense of definition (2.3)

In general, the rough statistical limit of a triple sequence may not unique for the roughness degree *r*>0. So we have to consider the so called *r-* statistical limit set of a triple sequence x= (_{
xmnk
} ), which is defined by

The triple sequence *x* is said to be *r-* statistically convergent provided that _{
st-LIMr
}
*x* ≠ ϕ. It is clear that if _{
st-LIMr x
} ≠ ϕ. for a triple sequence *x*= (_{
xmnk
} ) of real numbers, then we have

We know that _{
LIMr x
} = ϕ for an unbounded triple sequence *x*= (_{
xmnk
} ). But such a triple sequence might be rough statistically convergent. For instance, define

in **R**. Because the set {1,64,739, …} has natural density zero, we have

and _{
LIMr x
} =ϕ for all *r* ≥ 0.

As can be seen by the example above, the fact that _{
st-LIMr x
} ≠ ϕ does not imply _{
LIMr x
} ≠ ϕ. Because a finite set of natural numbers has natural density zero, _{
LIMr x
} ≠ ϕ implies _{
st-LIMr x
} ≠ ϕ. Therefore, we get _{
LIMr
}
*x* ⊆ _{
st-LIMr x
} . This obvious fact means {r≥ 0: _{
LIMr x
} ≠ ϕ} ⊆ {r≥ 0: _{
st-LIMr x
} ≠ ϕ} in this language of sets and yields immediately

Moreover, it also yields directly *diam*(_{
LIMr x
} )≤ *diam*\left(_{
st-LIMr x
} ).

3. Main Results

3.1. Theorem

For a triple sequence spaces *x* = (_{
xmnk
} ),we have *diam* (_{
st-LIMr
}
*x*)≤ 2*r*. In general *diam* (_{
st-LIMr
}
*x*) has an upper bound.

**Proof:** Assume that *diam* (_{
st-LIMr
}
*x*)> 2r. Then there exist *w, y* ∈ _{
st-LIMr
}
*x* such that |w-y|> 2r. Take ϵ ∈
. Because *w, y* ∈ _{
st-LIMr x,
} we have δ(_{
K1
} )=0 and δ(_{
K2
} )=0 for every ϵ>0 where

and

Using the properties of natural density, we get δ(*K*
_{1}
^{c} ∩ *K*
_{2}
^{c})=1. Thus we can write

for all (*m,n,k*) ∈ *K*
_{1}
^{c} ∩ *K*
_{2}
^{c}, which is a contradiction.

Now let us prove the second part of the theorem. Consider a triple sequence *x*= (*x*__{mnk}) such that *st-lim*
_{
xmnk
} = *l*. Let ϵ>0. Then we can write

Then we get |*l - y*|<*r*+ϵ for each (*m,n,k*) ∈ {(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk-l
} |<ϵ}. Because the triple sequence spaces *x* is statistically convergent to *l*, we have

Therefore we get *y* ∈ _{
st-LIMr
}
*x*. Hence, we can write

Because *diam*(
(*l*))=2r, this shows that in general, the upper bound 2r of the diameter of the set _{
st-LIMr x
} is not an lower bound.

3.2. Theorem

Let *r*>0. Then a triple sequence *x*= (_{
xmnk
} ) is *r*- statistically convergent to *l* if and only if there exists a triple sequence *y*= (*y*
_{mnk}) such that *st-lim y*= *l* and |_{
xmnk
} - _{
ymnk
} |≤ *r* for each (*m,n,k*) ∈ **N**
^{3}.

**Proof:**
**Necessity:** Assume that _{
xmnk
} →^{rst}
*l*. Then we have

Now, define

Then, we write

We have |_{
ymnk -l
} | ≥ |_{
xmnk
}
*-l*|*-r* ⟹ |x_{
mnk
} - _{
l-ymnk
} + *l*|≤ *r*

for all *m,n,k* ∈ **N**. By equation (3.1) and by definition of _{
ymnk
} , we get *st-limsup* |_{
ymnk
} -*l*|=0.

**Sufficiency:** Because _{
st- lim ymnk
} =*l*, we have

for each ϵ>0. It is easy to see that the inclusion

holds. Because δ({(*m,n,k*) ∈ **N**
^{3}: |_{
ymnk
} - *l*|≥ ϵ})=0, we get δ({(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} - *l*|≥ *r* + ϵ}) = 0.

3.3. Remark

If we replace the condition |_{
xmnk
} - _{
ymnk
} |≤ *r* for all *m,n,k* ∈ **N** in the hypothesis of the Theorem (3.2) with the condition

is valid.

3.4. Theorem

For an arbitrary c ∈ Γ_{x} of triple sequence *x*= (_{
xmnk)
} we have |*l - c*|≤ *r* for all *l* ∈ _{
st-LIMr
}
*x*.

**Proof:** Assume on the contrary that there exist a point c ∈ Γ_{x} and *l* ∈ _{
st-LIMr
} x such that |*l - c*|>*r*. Define ϵ:=
. Then

Since c ∈ Γ_{x}, we have

Hence, by (3.3), we get

which contradicts the fact *l* ∈ _{
st-LIMr x
} .

3.5. Proposition

If a triple sequence *x*= (_{
xmnk
} ) is analytic, then there exists a non-negative real number *r* such that _{
st-LIMr
}
*x*≠ ϕ.

**Proof:** If we take the triple sequence is to be statistically analytic, then the of proposition holds. Thus we have the following theorem.

3.6. Theorem

A triple sequence *x*= (_{
xmnk
} ) is statistically analytic if and only if there exists a non-negative real number *r* such that _{
st-LIMr
} x≠ ϕ.

**Proof:** Since the triple sequence *x* is statistically analytic, there exists a positive real number *M* such that

Define

where

Then the set _{
st-LIMŕ
}
*x* contains the origin of **R**. So we have _{
st-LIMr
}
*x*≠ ϕ.

If _{
st-LIMr
}
*x*≠ ϕ for some *r*≥ 0, then there exists *l* such that *l* ∈ _{
st-LIMr
}
*x*, i.e.,

for each ϵ>0. Then we say that almost all _{
xmnk
} are contained in some ball with any radius greater than *r*. So the triple sequence *x* is statistically analytic.

3.7. Remark

If *x´*. =
is a sub sequence of *x*= (_{
xmnk
} ), then _{
LIMr
}
*x* ⊆ _{
LIMr x´
} . But it is not valid for statistical convergence. For example, define

of real numbers. Then the triple sequence *x´*= (1, 64, 739, …) is a subsequence of *x*. We have _{
st-LIMr
}
*x*=
*-r, r*
abd _{
st-LIMr
}
*x´* = ϕ.

3.8. Theorem

Let *x´* =
is a non-thin subsequence of triple sequence *x*= (_{
xmnk
} ), then _{
st-LIMr
}
*x* ⊆ _{
st-LIMr x
} ´.

Proof: Easy, so omitted.

3.9. Theorem

The *r-* statistical limit set of a triple sequence *x*= (_{
xmnk
} ) is closed.

**Proof:** If _{
st-LIMr
}
*x*≠ ϕ, then it is true. Assume that _{
st-LIMr
}
*x*≠ ϕ, then we can choose a triple sequence spaces (_{
ymnk
} ) ⊆ _{
st-LIMr
}
*x* such that _{
ymnk
} → _{
r l
} as *m,n,k* → ∞. If we prove that *l* ∈ _{
st-LIMr
}
*x*, then the proof will be complete.

Let ϵ>0 be given. Because _{
ymnk
} → ^{r}
*l*, ∀ ϵ>0, ∃ *i*
_{ϵ} ∈ **N**: *m,n,k* ≥ *i*
_{ϵ} such that

Now choose an (_{
m0, n0, k0
} ) ∈ **N** such that _{
m0, n0, k0
} ≥ *i*
_{ϵ}. Then we can write

On the other hand, because (_{
ymnk
} ) ⊆ _{
st-LIMr
}
*x*, we have _{
ym0n0k0
} ∈ _{
st-LIMr x
} , namely,

Now let us show that the inclusion

holds. Take (i, j, ℓ) ∈ {(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} , - y_{m0n0k0}|< *r*+
}. Then we have

and hence

i.e., (i, j, ℓ) ∈ {(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} - l|< r + ϵ} which proves the equation (3.5). Hence the natural density of the set on the LHS of equation (3.5) is equal to 1. So we get δ ({(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} - *l*|≥ *r* + ϵ})=0.

3. 10. Theorem

The *r-* statistical limit set of a triple sequence is convex.

**Proof:** Let _{
y1, y2
} ∈ _{
st-LIMr
}
*x* for the triple sequence *x*= (_{
xmnk
} ) and let ϵ>0 be given. Define

K_{1} = {(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} - _{
y1
} |≥ r + ϵ} and

K_{2} = {(*m,n,k*) ∈ **N**
^{3}: |_{
xmnk
} - _{
y2
} |≥ r + ϵ}. Because _{
y1, y2
} ∈ _{
st-LIMr x
} , we have δ(_{
K1
} ) = δ(_{
K2
} ) = 0. Thus we have

|_{
xmnk
} -
(1- λ) _{
y1
} + λ _{
y2
}
| = |(1-λ)(_{
xmnk
} - *y*
_{1}) + λ (_{
xmnk
} - _{
y2
} )|< r + ϵ, for each (*m,n,k*) ∈ (_{
K1c
} ∩ _{
K2c
} )$ and each λ ∈
0,1
. Because δ(_{
K1c ∩ K2c
} )= 1, we get

i.e.,
(1- λ) _{
y1
} + λ _{
y2
}
∈ _{
st-LIMr x
} , which proves the convexity of the set _{
st-LIMr x
} .

3.11. Theorem

A triple sequence *x*= (_{
xmnk
} ) statistically converges to *l* if and only if _{
st-LIMr
}
*x*=
(*l*).

Proof: For the necessity part of this theorem is in proof of the Theorem (3.1).

**Sufficiency:** Because _{
st-LIMr
}
*x*=
(*l*)≠ ϕ, then by Theorem (3.5) we can say that the triple sequence spaces *x* is statistically analytic. Assume on the contrary that the triple sequence spaces *x* has another statistical cluster point *l´* different from *l*. Then the point

satisfies

Because *l´* is a statistical cluster point of the triple sequence spaces *x*, by Theorem (2.4) this inequality implies that
∉ _{
st-LIMr
}
*x*. This contradicts the fact |
- *l*|= *r* and *st-*LIM^{r} x=
(*l*). Therefore, *l* is the unique statistical cluster point of the triple sequence spaces *x*. Hence the statistical cluster point of a statistically analytic triple sequence spaces is unique, then the triple sequence spaces *x* is statistically convergent to *l*.

3.12. Theorem

**Proof: (a)** Assume that *l* ∈ _{
st-LIMr x
} and c ∈ Γ_{
x
} . Then by Theorem 3.4, we have

other wise we get

for ϵ=
. This contradicts the fact *l* ∈ _{
st-LIMr x.
}

**(b)** By the equation (3.9), we can write

Now assume that *y* ∈
Then we have

for all c ∈ Γ_{
x
} , which is equivalent to Γ_{
x
} ⊆
(*y*), i.e.,

Now let *y* ∉ _{
st-LIMr x
} . Then there exists an ϵ>0 such that

the existence of a statistical cluster point *c* of the triple sequence spaces *x* with |*y - c*| ≥ *r* + ϵ, i.e., Γ_{
x
}
(*y*) and *y* ∉ {*l* ∈ **R**: Γ_{
x
} ⊆
(*l*)}.

Hence *y* ∈ _{
st-LIMr
}
*x* follows from *y* ∈ {*l* ∈ **R**: Γ_{
x
} ⊆
(*l*)}., i.e.,

Therefore the inclusions (3.11)-(3.13) ensure that (3.10) holds.

**Competing Interests:** The authors declare that there is not any conflict of interests regarding the publication of this manuscript.