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}} ).

3.1. Theorem

For a triple sequence spaces x = ( ^_xmnk ),we have diam ( _^st-LIMr x)≤ 2r. 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

K₁ = {(m,n,k) ∈ N ³: | ^_xmnk - ^_y1 |≥ r + ϵ} and

K₂ = {(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.