We introduce the notions of lacunary ℐ-convergence and lacunary ℐ-Cauchy in the topology induced by random n-normed spaces and prove some important results.

1. Introduction

Menger [1] generalized the metric axioms by associating a distribution function with each pair of points of a set. This system, called a probabilistic metric space, originally a statistical metric space, has been developed extensively by Schweizer and Sklar [2, 3]. The idea of Menger was to use distribution function instead of nonnegative real numbers as values of the metric, which was further developed by several other authors. In this theory, the notion of distance has a probabilistic nature. Namely, the distance between two points x and y is represented by a distribution function Fxy, and for t>0, the value Fxy(t) is interpreted as the probability that the distance from x to y is less than t. Using this concept, Šerstnev [4] introduced the concept of probabilistic normed spaces. It provides an important area into which many deterministic results of linear normed spaces can be generalized. The studies of continuity properties, linear operators, statistical convergence, and ideal convergence in probabilistic normed spaces have gained many attractions, and such studies have diverse applications into various fields [5–13].

In [14], Gähler introduced an attractive theory of 2-normed and n-normed spaces in the 1960s. Since then, many researchers have studied these subjects and obtained various results [15–19].

Since the introduction of the notion of statistical convergence for sequences of real numbers by Steinhaus [20] and Fast [21] independently, the theory has been investigated and developed by several authors. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One very important generalization of this notion was introduced by Kostyrko et al. [22] by using an ideal ℐ of subsets of the set of natural numbers, which they called ℐ-convergence.

Another important variant of statistical convergence is the notion of lacunary statistical convergence introduced by Fridy and Orhan [23]. Recently, Mohiuddine and Aiyub [24] studied lacunary statistical convergence by introducing the concept θ-statistical convergence in random 2-normed space. Their work can be considered as a particular generalization of the statistical convergence. In [25], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, and Debnath [26] investigated lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Also, lacunary statistically convergent double sequences in probabilistic normed space were studied by Mohiuddine and Savaş in [27].

The notion of lacunary ideal convergence has not been studied previously in the setting of n-normed linear spaces. Motivated by this fact, in this paper, as a variant of ℐ-convergence, the notion of lacunary ideal convergence is introduced in a random n-normed space, and some important results are established. Finally, the notions of lacunary ℐF-Cauchy and lacunary ℐF*-Cauchy sequences are introduced and studied.

Throughout the paper, ℕ will denote the set of all natural numbers. First, we recall some of the basic concepts which will be used in this paper.

Definition 1 (see [<xref ref-type="bibr" rid="B9">15</xref>]).

Let n∈ℕ, and let X be a real vector space of dimension d≥n, where n≤d<∞. A real-valued function ∥·,…,·∥ on X×X×⋯×X=Xn, satisfying the following properties:

∥x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent,

∥x1,x2,…,xn∥ is invariant under any permutation of x1,x2,…,xn,

∥x1,x2,…,αxn∥=|α|∥x1,x2,…,xn∥, where α∈ℝ (set of real numbers),

is called an n-norm on X, and the pair (X,∥·,…,·∥) is called an n-normed space.

Given an n-normed space (X,∥·,…,·∥), one can derive a topology for it via the following definition of the limit of a sequence: a sequence x(k) in X is said to be convergent to x in X if limk→∞∥x1,…,xn-1,x(k)-x∥=0 for every x1,…,xn-1∈X.

All the concepts listed later are studied in depth in the fundamental book by Schweizer and Sklar [3].

Definition 2.

Let ℝ denote the set of real numbers, ℝ+={x∈ℝ:x≥0}, and S=[0,1] the closed unit interval. A mapping f:ℝ→S is called a distribution function if it is nondecreasing and left continuous with inft∈ℝf(t)=0 and supt∈ℝf(t)=1.

We denote the set of all distribution functions by D+ such that f(0)=0. If a∈ℝ+, then Ha∈D+, where
(1)Ha(t)={1ift>a,0ift≤a.
It is obvious that H0≥f for all f∈D+.

Definition 3.

A triangular norm (t-norm) is a continuous mapping *:S×S→S such that (S,*) is an abelian monoid with unit one and c*d≤a*b if c≤a and d≤b for all a,b,c,d∈S. A triangle function τ is a binary operation on D+ which is commutative and associative, and τ(f,H0)=f for every f∈D+.

Definition 4 (see [<xref ref-type="bibr" rid="B14">28</xref>]).

Let X be a linear space of dimension greater than one, τ a triangle function, and F a mapping from X×X×⋯×X︸n into D+. Then, F is called a probabilistic n-norm and (X,F,τ) a probabilistic n-normed space if the following conditions are satisfied:

F(x1,x2,…,xn;t)=H0(t) if and only if x1,x2,…,xn∈X are linearly dependent, where F(x1,x2,…,xn;t) denotes the value of F(x1,x2,…,xn) at t∈ℝ;

F(x1,x2,…,xn;t)≠H0(t) if and only if x1,x2,…,xn∈X are linearly independent;

F(x1,x2,…,xn;t) is invariant under any permutation x1,x2,…,xn∈X;

F(x1,x2,…,αxn;t)=F(x1,x2,…,xn;t/|α|) for every t>0, α≠0 and x1,x2,…,xn∈X;

F(x1,x2,…,xn+xn′;t)≥τ(F(x1,x2,…,xn;t),F(x1,x2,…,xn′;t)) whenever x1,x2,…,xn,xn′∈X, and t>0.

If (v) is replaced by

F(x1,x2,…,xn+xn′;t1+t2)≥F(x1,x2,…,xn;t1)*F(x1,x2,…,xn′;t2) for all x1,x2,…,xn,xn′∈X and t1,t2∈ℝ+,

then (X,F,*) is called a random n-normed space (for short, RnN space).
Remark 5.

Note that every n-norm space (X,∥·,…,·∥) can be made a random n-normed space in a natural way, by setting

F(x1,x2,…,xn;t)=H0(t-∥x1,x2,…,xn∥), for every x1,x2,…,xn∈X, t>0 and a*b=min{a,b}, a,b∈S;

F(x1,x2,…,xn;t)=t/(t+∥x1,x2,…,xn∥) for every x1,x2,…,xn∈X, t>0 and a*b=ab for a,b∈S.

Let (X,F,*) be a RnN space. Since * is a continuous t-norm, the system of (ε,λ)-neighborhoods of ϕ (the null vector in X)
(2){𝒩ϕ(ε,λ):ε>0,λ∈(0,1)},
where
(3)𝒩ϕ(ε,λ)={x1,x2,…,xn-1∈X:Fx1,x2,…,xn-1(ε)>1-λ},
determines a first countable Hausdorff topology on X, called the F-topology. Thus, the F-topology can be completely specified by means of F-convergence of sequences. It is clear that x1,x2,…,xn-1-y∈𝒩ϕ means that y∈𝒩x1,x2,…,xn-1 and vice versa.

A sequence x=(xk) in X is said to have F-convergence to L∈X if for every ε>0, λ∈(0,1) and for each nonzero x1,…,xn-1∈X there exists a positive integer N such that
(4)x1,…,xn-1,xk-L∈𝒩ϕ(ε,λ)foreachn≥N,
or equivalently,
(5)x1,…,xn-1,xk∈𝒩L(ε,λ)foreachn≥N.
In this case, we write F-limx1,…,xn-1,xk=L.

Lemma 6.

Let (X,∥·,…,·∥) be a real n-normed space, and let (X,F,*) be RnN space induced by the random norm F(x1,x2,…,xn;t)=t/(t+∥x1,x2,…,xn∥), where x1,x2,…,xn∈X and t>0. Then, for every sequence x=(xk) and nonzero x1,x2,…,xn-1 in X,
(6)lim∥x1,x2,…,xn-1,x-L∥=0⇒F-limx1,x2,…,xn-1,(x-L)=0.

Proof.

Suppose that lim∥x1,x2,…,xn-1,x-L∥=0. Then, for every t>0 and for every x1,x2,…,xn-1∈X, there exists a positive integer N=N(t) such that
(7)∥x1,x2,…,xn-1,xk-L∥<tforeachk>N.
We observe that for any given ε>0,
(8)ε+∥x1,x2,…,xn-1,xk-L∥ε<ε+tε
which is equivalent to
(9)εε+∥x1,x2,…,xn-1,xk-L∥>εε+t=1-tε+t.
Therefore, by letting λ=t/(ε+t)∈(0,1), we have
(10)Fx1,x2,…,xn-1,xk-L(ε)>1-λforeachk>N.
This implies that x1,x2,…,xn-1,xk∈𝒩L(ε,λ) for each k>N as desired.

Definition 7 (see [<xref ref-type="bibr" rid="B4">21</xref>, <xref ref-type="bibr" rid="B5">29</xref>]).

A subset E of ℕ is said to have density δ(E) if δ(E)=limn→∞n-1∑k=1nχE(k) exists. A number sequence (xn)n∈ℕ is said to be statistically convergent to L if for every ε>0,δ({n∈ℕ:|xn-L|≥ε})=0. If (xn)n∈ℕ is statistically convergent to L, we write st-limxn=L, which is necessarily unique.

Definition 8 (see [<xref ref-type="bibr" rid="B16">22</xref>]).

A family ℐ⊂2Y of subsets of a nonempty set Y is said to be an ideal in Y if (i) ∅∈ℐ; (ii) A,B∈ℐ imply A∪B∈ℐ; (iii) A∈ℐ,B⊂A imply B∈ℐ. A nontrivial ideal ℐ in Y is called an admissible ideal if it is different from P(ℕ) and it contains all singletons, that is, {x}∈ℐ for each x∈Y.

Let I⊂P(Y) be a nontrivial ideal. A class ℱ(ℐ)={M⊂Y:∃A∈ℐ:M=Y∖A}, called the filter associated with the ideal ℐ, is a filter on Y.

Definition 9 (see [<xref ref-type="bibr" rid="B16">22</xref>]).

Let ℐ⊂2ℕ be a nontrivial ideal in ℕ. Then, a sequence (xn)n∈ℕ in X is said to be ℐ-convergent to L∈X, if for each ε>0 the set A(ε)={n∈ℕ:∥xn-L∥≥ε} belongs to ℐ.

Definition 10.

By a lacunary sequence we mean an increasing integer sequence θ=(kr) such that k0=0 and hr:=kr-kr-1→∞ as r→∞. Throughout this paper, the intervals determined by θ will be denoted by Ir:=(kr-1,kr]. Let K⊆ℕ. The number
(11)δθ(K)=limr1hr|{k∈Ir:k∈K}|
is said to be the θ-density of K, provided the limit exists (see [23]).

Definition 11 (see [<xref ref-type="bibr" rid="B2">30</xref>]).

Let ℐ⊂2ℕ be a nontrivial ideal in ℕ. A sequence x=(xk) of numbers is said to be lacunary ℐ-convergent to a number L if, for every ε>0,
(12){r∈ℕ:1hr∑k∈Ir|xk-L|≥ε}∈ℐ.

In this case, we write ℐθ-limx=L.

2. <bold>Lacunary</bold><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M222"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula><bold>and</bold><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M223"><mml:mrow><mml:msup><mml:mrow><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula><bold>-Convergence for Sequences in RnN Spaces</bold>

In this section, we study the concepts lacunary ℐF and ℐF*-convergence of a sequence in (X,F,*) and prove some important results.

Definition 12.

Let (X,F,*) be RnN space, and let ℐ be a proper ideal in ℕ. A sequence x=(xk) in X is said to be ℐF-convergent to L∈X (ℐF-convergent to L∈X with respect to the random n-norm F-topology) if for each ε>0, λ∈(0,1) and each nonzero x1,x2,…,xn-1∈X,
(13){k∈ℕ:x1,x2,…,xn-1,xk∉𝒩L(ε,λ)}∈ℐ.

In this case, the vector L is called the ℐF-limit of the sequence x=(xk), and we write ℐF-limx1,x2,…,xn-1,x=L.

Definition 13.

Let (X,F,*) be RnN space, and let ℐ be a proper ideal in ℕ. A sequence x=(xk) in X is said to be lacunary ℐF-convergent to L∈X with respect to the random n-norm F-topology if for each ε>0, λ∈(0,1) and each nonzero x1,x2,…,xn-1∈X,
(14){r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L(ε,λ))}∈ℐ.

In this case, the vector L is called the ℐθF-limit of the sequence x=(xk), and we write ℐθF-limx1,x2,…,xn-1,x=L.

Lemma 14.

Let (X,F,*) be RnN space. If a sequence x=(xk) is ℐθF-convergent with respect to the random n-norm F, then ℐθF-limit is unique.

Proof.

Let us assume that
(15)ℐθF-limx1,x2,…,xn-1,x=L1,ℐθF-limx1,x2,…,xn-1,x=L2,
where L1≠L2. Since L1≠L2, select ε>0, λ∈(0,1) and each nonzero x1,x2,…,xn-1∈X such that 𝒩L1(ε,λ) and 𝒩L2(ε,λ) are disjoint neighborhoods of L1 and L2. Since L1 and L2 both are ℐθF-limit of the sequence (xk), we have
(16)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L1(ε,λ))},B={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L2(ε,λ))},
both belonging to ℐ. This implies that the sets
(17)Ac={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∈𝒩L1(ε,λ))}
and Bc={r∈ℕ:(1/hr)∑k∈Ir(x1,x2,…,xn-1,xk∈𝒩L2(ε,λ))} belong to ℱ(ℐ). In this way, we obtain a contradiction to the fact that the neighborhoods 𝒩L1(ε,λ) and 𝒩L2(ε,λ) of L1 and L2 are disjoints. Hence, we have L1=L2. This completes the proof.

Lemma 15.

Let (X,F,*) be RnN space. Then one has

Fθ-limx1,x2,…,xn-1,xk=L, then ℐθF-limx1,x2,…,xn-1,xk=L;

if ℐθF-limx1,x2,…,xn-1,xk=L1 and ℐθF-limx1,x2,…,xn-1,yk=L2, then ℐθF-limx1,x2,…,xn-1,(xk+yk)=L1+L2;

if ℐθF-limx1,x2,…,xn-1,xk=L and α∈ℝ, then ℐθF-limx1,x2,…,xn-1,αxk=αL;

if ℐθF-limx1,x2,…,xn-1,xk=L1, and ℐθF-limx1,x2,…,xn-1,xk=L2, then ℐθF-limx1,x2,…,xn-1,(xk-yk)=L1-L2.

Proof.

(i) Suppose that Fθ-limx1,x2,…,xn-1,xk=L. Let ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X. Then, there exists positive integer N such that (1/hr)∑k∈Ir(x1,x2,…,xn-1,xk∈𝒩L(ε,λ)) for each k>N. Since the set
(18)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L1(ε,λ))}⊆{1,2,3,…,N-1}
and the ideal ℐ is admissible, we have A∈ℐ. This shows that ℐθF-limx1,x2,…,xn-1,xk=L.

(ii) Let ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X. Choose η∈(0,1) such that (1-η)*(1-η)>(1-λ). Since ℐθF-limx1,x2,…,xn-1,xk=L1 and ℐθF-limx1,x2,…,xn-1,yk=L2, the sets
(19)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L1(ε2,λ))},B={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,yk∉𝒩L2(ε2,λ))}
belong to ℐ. Let C={r∈ℕ:(1/hr)∑k∈Ir(x1,x2,…,xn-1,xk+yk∉𝒩L1+L2(ε,λ))}. Since ℐ is an ideal, it is sufficient to show that C⊂A∪B. This is equivalent to show that Cc⊃Ac∩Bc where Ac and Bc belong to ℱ(ℐ). Let r∈Ac∩Bc, that is, r∈Ac and r∈Bc, and we have
(20)Fx1,x2,…,xn-1,(xk+yk)-(L1+L2)(ε)≥Fx1,x2,…,xn-1,xk-L1(ε2)*Fx1,x2,…,xn-1,yk-L2(ε2)>(1-η)*(1-η)>(1-λ).
Since r∈Cc⊃Ac∩Bc∈ℱ(ℐ), we have C⊂A∪B∈ℐ.

(iii) It is trivial for α=0. Now, let α≠0, ε>0, λ∈(0,1), and nonzero x1,x2,…,xn-1∈X. Since ℐθF-limx1,x2,…,xn-1,xk=L, we have
(21)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L(ε,λ))}∈ℐ
This implies that
(22)Ac={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∈𝒩L(ε,λ))}∈ℱ(ℐ).
Let r∈Ac. Then, we have
(23)Fx1,x2,…,xn-1,αxk-αL(ε)=Fx1,x2,…,xn-1,xk-L(ε|α|)≥Fx1,x2,…,xn-1,xk-L(ε)*F0(ε|α|-ε)>(1-λ)*1=(1-λ).
So,
(24){r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,αxk∉𝒩αL(ε,λ))}∈ℐ.
Hence, ℐθF-limx1,x2,…,xn-1,αxk=αL.

(iv) The result follows from (ii) and (iii).

We introduce the concept of ℐθF*-convergence closely related to ℐθF-convergence of sequence in random n-normed space and show that ℐθF*-convergence implies ℐθF-convergence but not conversely.

Definition 16.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal. We say that a sequence x=(xk) in X is said to be lacunary ℐF*-convergent to L∈X with respect to the random n-norm F if there is a set K={m1<m2<⋯<mk<⋯}⊆ℕ such that K′={r∈ℕ:mk∈Ir}∈ℱ(ℐ) and Fθ-limkx1,x2,…,xn-1,xmk=L for each nonzero x1,x2,…,xn-1∈X.

In this case, we write ℐθF*-limx1,x2,…,xn-1,xk=L, and L is called the ℐθF*-limit of the sequence x=(xk).

Theorem 17.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal. If ℐθF*-limx1,x2,…,xn-1,xk=L, then ℐθF-limx1,x2,…,xn-1,xk=L.

Proof.

Suppose that ℐθF*-limx1,x2,…,xn-1,xk=L. Then, by definition, there exists
(25)K={m1<m2<⋯<mk<⋯}⊆ℕ
such that K′∈ℱ(ℐ) and Fθ-limkx1,x2,…,xn-1,xmk=L for each nonzero x1,x2,…,xn-1∈X. Let ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X be given. Since Fθ-limkx1,x2,…,xn-1,xmk=L, there exists N∈ℕ such that
(26)1hr∑k∈Ir(x1,x2,…,xn-1,xmk∈𝒩L(ε,λ))
for every m≥N. Since
(27)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xmk∉𝒩L(ε,λ))}
is contained in
(28)B={m1,m2,…,mN-1}
and the ideal ℐ is admissible, we have A∈ℐ. Hence,
(29){r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L(ε,λ))}⊆K∪B∈ℐ
for ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X. Therefore, we conclude that ℐθF-limx1,x2,…,xn-1,xk=L.

The following example shows that the converse of Theorem 17 needs not to be true.

Example 18.

Consider X=ℝn with
(30)∥x1,x2,…,xn∥=abs(|x11…x1n⋮xn1…xnn|),
where xi=(xi1,…,xin)∈ℝn for each i=1,2,…,n, and let a*b=ab for all a,b∈S. For all (y1,y2,…,yn-1,x)∈ℝn and t>0, consider
(31)Fy1,y2,…,yn-1,x(t)=tt+∥y1,y2,…,yn-1,x∥.
Then, (ℝn,F,*) is RnN space. Consider a decomposition of ℕ as ℕ=∪iΔi such that for any m∈ℕ, each Δi contains infinitely many (i)’s, where i≥m and Δi∩Δm=∅ for i≠m. Let ℐ be the class of all subsets of ℕ which intersect almost a finite number of Δi’s. Then, ℐ is an admissible ideal. We define a sequence (xm) as follows: xm=(1/i,0,0,…,0)∈ℝn, i=1,2,…, if m∈Δi. Then, for nonzero y1,y2,…,yn-1∈X, we have
(32)Fxm,y1,y2,…,yn-1(t)=tt+∥y1,y2,…,yn-1,xm,∥→1
as m→∞. Hence, ℐθF-limmy1,y2,…,yn-1,xm=0.

Now, we show that ℐθF*-limmy1,y2,…,yn-1,xm≠0. Suppose that
(33)ℐθF*-limmy1,y2,…,yn-1,xm=0.
Then, by definition, there exists a subset
(34)K={mj:m1<m2<⋯}⊂ℕ
such that K′∈ℱ(ℐ) and Fθ-limjy1,y2,…,yn-1,xmj=0. Since K′∈ℱ(ℐ), there exists H∈ℐ such that K=ℕ∖H. Then, there exist positive integers p such that
(35)H⊂(⋃m=1pΔm).
Thus, Δp+1⊂K, and so xmj=1/(p+1)>0 for infinitely many values mj’s in K. This contradicts the assumption that Fθ-limjy1,y2,…,yn-1,xmj=0. Hence, ℐθF*-limmy1,y2,…,yn-1,xm≠0.

Hence, the converse of the theorem needs not to be true.

The following theorem shows that the converse holds if the ideal ℐ satisfies condition (AP).

Definition 19 (see [<xref ref-type="bibr" rid="B16">22</xref>]).

An admissible ideal ℐ⊂P is said to satisfy the condition (AP) if for every sequence (An)n∈ℕ of pairwise disjoint sets from ℐ there are sets Bn⊂ℕ, n∈ℕ, such that the symmetric difference AnΔBn is a finite set for every n and ∪n∈ℕBn∈ℐ.

Theorem 20.

Let (X,F,*) be RnN space, and let the ideal ℐ satisfy the condition (AP). If x=(xk) is a sequence in X such that ℐθF-limy1,y2,…,yn-1,x=L, then ℐθF*-limy1,y2,…,yn-1,x=L.

Proof.

Since ℐθF-limy1,y2,…,yn-1,x=L, so for every ε>0,λ∈(0,1) and nonzero y1,y2,…,yn-1∈X, the set
(36){r∈ℕ:1hr∑k∈Ir(y1,y2,…,yn-1,xk∉𝒩L(ε,λ))}∈ℐ.
We define the set Ap for p∈ℕ as
(37)Ap={k∈ℕ:1-1p≤Fy1,y2,…,yn-1,xk-L<1-1p+1}.
Then, it is clear that {A1,A2,…} is a countable family of mutually disjoint sets belonging to ℐ, and so by the condition (AP), there is a countable family of sets {B1,B2,…}∈ℐ such that the symmetric difference AiΔBi is a finite set for each i∈ℕ and B=∪i=1∞Bi∈ℐ. Since B∈ℐ, there is a set K∈ℱ(ℐ) such that K=ℕ∖B. Now, we prove that the subsequence (xk)k∈K is convergent to L with respect to the random n-norm F. Let η∈(0,1), ε>0 and nonzero y1,y2,…,yn-1∈X. Choose a positive q such that q-1<η. Then,
(38){r∈ℕ:1hr∑k∈Ir(y1,y2,…,yn-1,xk∉𝒩L(ε,η))}⊂{r∈ℕ:1hr∑k∈Ir(y1,y2,…,yn-1,xk∉𝒩L(ε,1q))}⊂⋃i=1q-1Ai.
Since AiΔBi is a finite set for each i=1,2,…,q-1, there exists j0∈ℕ such that
(39)(⋃i=1q-1Bi)∩{k∈ℕ:k≥k0}=(⋃i=1q-1Ai)∩{k∈ℕ:k≥k0}.
If k≥k0 and k∈K, then k∉∪i=1q-1Bi and k∉∪i=1q-1Ai. Hence, for every k≥k0 and k∈K, we have
(40)1hr∑k∈Ir(y1,y2,…,yn-1,xk∉𝒩L(ε,η)).
Since this holds for every ε>0, η∈(0,1) and nonzero y1,y2,…,yn-1∈X, so we have ℐθF*-limy1,y2,…,yn-1,x=L. This completes the proof of the theorem.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M538"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula><bold> and </bold><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M539"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula><bold>-Cauchy Sequences in RnN Spaces</bold>

In this section, we study the concepts ℐθF-Cauchy and ℐθF*-Cauchy of a sequence in (X,F,*). Also, we will study the relations between these concepts.

Definition 21.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal of ℕ. Then, a sequence x=(xk) of elements in X is called lacunary ℐF-Cauchy sequence in X if for every ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X, there exists m∈ℕ satisfying
(41){r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk-xm∉𝒩ϕ(ε,λ))}∈ℐ.

Definition 22.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal of ℕ. We say that a sequence x=(xk) of elements in X is called lacunary ℐF*-Cauchy sequence in X if for every ε>0,λ∈(0,1) and nonzero x1,x2,…,xn-1∈X, there exists a set
(42)K={m1<m2<⋯<mk<⋯}⊂ℕ
such that K′={r∈ℕ:mk∈Ir}∈ℱ(ℐ), and (xmk) is a lacunary Cauchy with respect to the random n-norm F.

The next theorem gives that ℐθF*-Cauchy sequence implies ℐθF-Cauchy sequence.

Theorem 23.

Let (X,F,*) be RnN space, and let ℐ be a nontrivial ideal of ℕ. If x=(xk) is a ℐθF*-Cauchy sequence, then x=(xk) is a ℐθF-Cauchy sequence too.

Proof.

Let (xk) be a ℐθF*-Cauchy sequence. Then, for ε>0,λ∈(0,1) and nonzero x1,x2,…,xn-1∈X, there exist K′∈ℱ(ℐ) and a number N∈ℕ such that
(43)1hr∑k∈Ir(x1,x2,…,xn-1,xmk-xmp∈𝒩ϕ(ε,λ))
for every k,p≥N. Now, fix p=mN+1. Then, for every ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X, we have
(44)1hr∑k∈Ir(x1,x2,…,xn-1,xmk-xp∈𝒩ϕ(ε,λ))for everyk≥N.
Let H=ℕ∖K. It is obvious that H∈ℐ and
(45)A(ε,λ)={∑k,p∈Irr∈ℕ:1hr∑k,p∈Ir(x1,x2,…,xn-1,xk-xp∉𝒩ϕ(ε,λ))}⊂H∪{m1<m2<⋯<mN}∈ℐ.
Therefore, for every ε>0, λ∈(0,1) and nonzero x1,x2,…,xn-1∈X, we can find p∈ℕ such that A(ε,λ)∈ℐ, that is, (xk) is a ℐθF-Cauchy sequence.

Now, we will prove that ℐθF*-convergence implies ℐθF-Cauchy condition in random n-normed space.

Theorem 24.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal of ℕ. If a sequence x=(xk) is ℐθF*-convergent, then it is a ℐθF-Cauchy sequence.

Proof.

By assumption, there exists a set
(46)K={mk:m1<m2<⋯}⊂ℕ
such that K′={r∈ℕ:mk∈Ir}∈ℱ(ℐ), and (xmk) is a lacunary Cauchy with respect to the random n-norm F. Choose η∈(0,1) such that (1-η)*(1-η)>(1-λ). Since
(47)Fx1,x2,…,xn-1,xmk-xmp(ε)≥Fx1,x2,…,xn-1,xmk-L(ε2)*Fx1,x2,…,xn-1,xmp-L(ε2)>(1-η)*(1-η)>1-λ
for every ε>0, λ∈(0,1), each nonzero x1,x2,…,xn-1 in X and k>N,p>N, we have (1/hr)∑k∈Ir(x1,x2,…,xn-1,xmk-xmp∉𝒩L(ε,λ)) for every k,p>N and each nonzero x1,x2,…,xn-1∈X, that is, (xk) in X is an ℐθF*-Cauchy sequence in X. Then, by Theorem 23(xk) is a ℐθF-Cauchy sequence in RnN space.

Theorem 25.

Let (X,F,*) be RnN space, and let ℐ be an admissible ideal of ℕ. If a sequence x=(xk) of elements in X is ℐθF-convergent, then it is ℐθF-Cauchy sequence.

Proof.

Suppose that (xk) is ℐθF-convergent to L∈X. Let ε>0,λ∈(0,1) and nonzero x1,x2,…,xn-1∈X be given. Then, we have
(48)A={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∉𝒩L(ε2,λ))}∈ℐ.
This implies that
(49)Ac={r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk∈𝒩L(ε2,λ))}∈ℱ(ℐ).
Choose η∈(0,1) such that (1-η)*(1-η)>(1-λ). Then, for every r∈Ac,
(50)Fx1,x2,…,xn-1,xk-xs(ε)≥Fx1,x2,…,xn-1,xk-L(ε2)*Fx1,x2,…,xn-1,xs-L(ε2)>(1-η)*(1-η)>(1-λ).
Hence{r∈ℕ:(1/hr)∑k∈Ir(x1,x2,…,xn-1,xk-xs∈𝒩ϕ(ε,λ))}∈ℱ(ℐ). This implies that
(51){r∈ℕ:1hr∑k∈Ir(x1,x2,…,xn-1,xk-xs∉𝒩ϕ(ε,λ))}∈ℐ.
that is, (xk) is a ℐθF-Cauchy sequence.

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