EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, 2016, 464-478 ISSN 1307-5543 – www.ejpam.com On g-Statistical Convergence in Paranormed Spaces Kuldip Raj∗, Renu Anand and Seema Jamwal School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India Abstract. In this paper we construct some spaces of lacunary almost convergent sequences and la- cunary strongly almost convergent sequences via sequence of Orlicz functions over n-normed spaces and established some inclusion relations between these spaces. We also make an effort to define a new concept called g-statistical convergence in paranormed spaces where the base space is a n-normed spaces. 2010 Mathematics Subject Classifications: 40F05, 46A45,40A05, 40A30 Key Words and Phrases: Strongly almost convergence, almost convergence, n-norm, g-statistical con- vergence, strongly p-Cesaro summability, Orlicz function 1. Introduction and Preliminaries In [13] Gähler introduced an attractive theory of 2-normed spaces. The notion was further generalized by Misiak [21] by introducing n-normed spaces. Since then these spaces were studied by Gunawan [14, 15]. In [16] Gunawan and Mashadi gave a simple way to derive an (n− 1)-norm from the n-norm and realized that n-normed space is an (n− 1)-normed space. Definition 1. Let n ∈ N and X be a linear space over the field R of real of dimension d, where d ≥ n≥ 2. A real valued function ||·, . . . , ·|| on X n satisfying the following conditions: (i) ||x1, x2, . . . , xn||= 0 if and only if x1, x2, . . . , xn are linearly dependent in X ; (ii) ||x1, x2, . . . , xn|| is invariant under permutation; (iii) ||αx1, x2, . . . , xn||= |α|||x1, x2, . . . , xn|| for any α ∈ R, and (iv) ||x + x ′, x2, . . . , xn|| ≤ ||x , x2, . . . , xn||+ ||x ′, x2, . . . , xn|| is called a n-norm on X and the pair (X , || . . . ||) a n-normed space over the field R. ∗Corresponding author. Email address: kuldipraj68@gmail.com (Kuldip Raj) http://www.ejpam.com 464 c© 2016 EJPAM All rights reserved. K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 465 Example 1. Let X = Rn being equipped with the Euclidean n-norm ||x1, x2, . . . , xn||E = the volume of the n-dimensional parallelopiped spanned by the vectors x1, x2, . . . , xn which may be given explicitly by the formula ||x1, x2, . . . , xn||E = |det(x i j)|, where x i = (x i1, x i2, . . . , x in) ∈ Rn for each i = 1, 2, . . . , n. Let (X , || . . . ||) be a n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, . . . , an} be linearly independent set in X . Then the following function || . . . ||∞ on X n−1 defined by ||x1, x2, . . . , xn−1||∞ =max{||x1, x2, . . . , xn−1, ai|| : i = 1,2, . . . , n} defines an (n− 1)-norm on X with respect to {a1, a2, . . . , an}. A sequence (xk) in a n-normed space (X , || . . . ||) is said to converge to some L ∈ X if lim k→∞ ||xk − L, z1, . . . , zn−1||= 0 for every z1, . . . , zn−1 ∈ X . A sequence (xk) in a n-normed space (X , || . . . ||) is said to be Cauchy if lim k,p→∞ ||xk − xp, z1, . . . , zn−1||= 0 for every z1, . . . , zn−1 ∈ X . If every Cauchy sequence in X converges to some L ∈ X , then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space. Definition 2. Let K be a subset of the set of natural number N. Then the asymptotic density of K denoted by δ(K) = limn→∞ 1 n |{ j ≤ n : j ∈ K}|, where vertical bars denote the cardinality of the enclosed set. Definition 3. A sequence x = (x j) is said to be statistically convergent to a number λ if for every ε > 0, the set K(ε) = { j ≤ n : |x j −λ| ≥ ε} has asymptotic density zero, i.e, lim n→∞ 1 n |{ j ≤ n : |x j −λ| ≥ ε}|= 0, in case we write S − lim x = λ. Definition 4. Let X be a linear metric space. A function g : X → R is called paranorm, if (i) g(x)≥ 0 for all x ∈ X , (ii) g(−x) = g(x) for all x ∈ X , (iii) g(x + y)≤ g(x) + g(y) for all x , y ∈ X , (iv) if (λn) is a sequence of scalars with λn → λ as n→∞ and (xn) is a sequence of vectors with g(xn − x)→ 0 as n→∞, then g(λn xn −λx)→ 0 as n→∞. K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 466 A paranorm g for which g(x) = 0 implies x = 0 is called total paranorm and the pair (X , g) is called a total paranormed space. Note that each seminorm (norm) g on X is a paranorm (total) but converse need not be true. It is well known that the metric of any linear metric space is given by some total paranorm (see [28, Theorem 10.4.2, pp. 183]). For more details about sequence spaces see [2, 6, 7, 22, 24–26] and references therein. Definition 5. A sequence x = (x j) in (X , g) paranormed space is said to be convergent (or g−convergent) to a number λ in (X , g) if for every ε > 0 there exists a positive integer j0 such that g(x j − λ) < ε whenever j ≥ j0. In case we write g − lim x = λ and λ is called the g−limit of x (see [1]). Definition 6. An Orlicz function M is a function, which is continuous, non-decreasing and convex on [0,+∞) with M(0) = 0, M(x)> 0 for x > 0 and M(x) −→∞ as x −→∞. Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following se- quence space: `M = ¦ x ∈ω : ∞ ∑ k=1 M � |xk| ρ � <∞, for some ρ > 0 © which is called an Orlicz sequence space. The space `M is a Banach space with the norm ||x ||= inf ¦ ρ > 0 : ∞ ∑ k=1 M � |xk| ρ � ≤ 1 © . It is shown in [17] that every Orlicz sequence space `M contains a subspace isomorphic to `p(p ≥ 1). In the later stage different Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [23], Mursaleen [22] and many others. Definition 7. By a lacunary sequence θ = (kr) where k0 = 0, we shall mean an increasing sequence of non-negative integers with kr − kr−1→∞ as r →∞. The intervals determined by θ will be denoted by Ir = (kr−1, kr]. We write hr = kr − kr−1. The ratio kr kr−1 will be denoted by qr . The space of lacunary strongly convergent sequence was defined by Freedman et al. [11] as follows: Nθ = ¦ x = (xk) : lim r→∞ 1 hr ∑ k∈Ir |xk − L|= 0 for some L © . Lorentz [18] and Duran [9] studied the spaces of almost convergent sequences. The con- cept of strongly almost convergent sequences was introduced by Maddox [19]. In [20], Mad- dox defined a generalization of strong almost convergence. Related articles with the topic almost convergence and strong almost convergence can be seen in [3, 18–20]. In order to ex- tend convergence of sequences, the notion of statistical convergence has been introduced by Fast [10] in 1951 and Schoenberg [27] independently for real sequences. Later on developed by Fridy [12]. Recently, Alotaibi and Alroqi [1] extended this notion in paranormed space. We may refer to [4, 5] which are related with this topic. K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 467 Lorentz [18] proved that x is almost convergent to a number λ if and only if lim n→∞ � � � � � 1 n n−1 ∑ i=0 (x i+q −λ) � � � � � = 0,uniformly in q ≥ 1. In other words, he showed that x is almost convergent to a number λ if and only if tnq(x)→ λ as n→∞, uniformly in q ≥ 1, where tnq(x) = xq + xq+1 + . . .+ xq+n−1 n (n ∈ N= {1,2, 3, . . . , }). Let f be a set of all almost convergent sequences. We write f − lim x = λ if x is almost convergent to λ. Maddox [20] has defined that x is strongly almost convergent to a number λ if and only if tnq(|x −λ|) = 1 n n−1 ∑ i=0 � � �x i+q −λ � � �→ 0 as n→∞, uniformly in q ≥ 1. By [ f ] we denote the set of all strongly almost convergent sequences. If x is strongly almost convergent to λ we write [ f ] − lim x = λ. Let l∞ be the set of all bounded sequences, it is easy to see that [ f ] ⊂ f ⊂ l∞ and each inclusion is proper. In [8] Konca and Başarir defined the almost convergent sequences F and strongly almost convergent sequences [F], in 2-normed spaces for every z ∈ X . They have also introduced the space of lacunary almost convergent sequences Fθ and lacunary strongly almost convergent sequences [Fθ ], respectively in 2-normed spaces. Let M = (Mk) be a sequence of Orlicz functions, (X , ||·, . . . , ·||) is a n-normed space and p = (pk) be a bounded sequence of positive real numbers. By S(n− X ) we denote the space of all sequences defined over (X , ||·, . . . , ·||). In this paper we define the following sequence spaces: � M , F, p,‖·, . . . , ·‖ � = ¦ x ∈ S(n− X ) : lim n→∞ ∞ ∑ k=1 � Mk � tnq(x −λ) ρ , z1, . . . , zn−1 ��pk = 0, uniformly in q ≥ 1, for some ρ > 0 and for every nonzero z1, . . . , zn−1 ∈ X © and � M , [F], p,‖·, . . . , ·‖ � = ¦ x ∈ S(n− X ) : lim n→∞ ∞ ∑ k=1 � Mk � tnq � x −λ ρ , z1, . . . , zn−1 ���pk = 0, uniformly in q ≥ 1, for some ρ > 0 and for every nonzero z1, . . . , zn−1 ∈ X © . We write � M , F, p,‖·, . . . , ·‖ � − lim x = λ if x is almost convergent to λ and � M , [F], p,‖·, . . . , ·‖ � − lim x = λ K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 468 if x is strongly almost convergent to λ. Taking advantage to (iii) and (iv) conditions of n−norm and definitions of � M , F, p,‖·, . . . , ·‖ � and � M , [F], p,‖·, . . . , ·‖ � , we have the inclusion � M , [F], p,‖·, . . . , ·‖ � ⊂ � M , F, p,‖·, . . . , ·‖ � ⊂ � M , l∞, p,‖·, . . . , ·‖ � holds from the following inequality: tnq(x −λ) ρ , z1, . . . , zn−1 = 1 n ∑n−1 i=0 (x i+q −λ) ρ , z1, . . . , zn−1 ≤ 1 n n−1 ∑ i=0 x i+q −λ ρ , z1, . . . , zn−1 = tnq � x −λ ρ , z1, . . . , zn−1 � . Now we define the spaces of lacunary almost convergent sequences � M , Fθ , p,‖·, . . . , ·‖ � and lacunary strongly almost convergent sequences � M , [Fθ ], p,‖·, . . . , ·‖ � in n-normed spaces as follows: � M , Fθ , p,‖·, . . . , ·‖ � = ¦ x ∈ S(n− X ) : lim r→∞ ∞ ∑ k=1 � Mk 1 hr ∑ i∈Ir � x i+q −λ ρ , z1, . . . , zn−1 � �pk = 0, uniformly in q ≥ 1, for some ρ > 0 and for every nonzero z1, . . . , zn−1 ∈ X © and � M , [Fθ ], p,‖·, . . . , ·‖ � = ¦ x ∈ S(n−X ) : lim r→∞ 1 hr ∑ i∈Ir ∞ ∑ k=1 � Mk � x i+q −λ ρ , z1, . . . , zn−1 ��pk = 0, uniformly in q ≥ 1, for some ρ > 0 and for every nonzero z1, . . . , zn−1 ∈ X © . The main purpose of this paper is to study some generalized spaces of lacunary almost con- vergent sequences and lacunary strongly almost convergent sequences via sequence of Orlicz functions over n-normed spaces. We also established some topological properties and prove some inclusion relations between these spaces. Further we introduced a new concept of statis- tical convergence which will be called g-statistical convergence in a paranormed spaces where the base space is a n-normed spaces. We define and study the notion of statistical convergence and statistical Cauchy. 2. Main Results Lemma 1. Let (x j) be a strongly almost convergent sequence, for a given ε > 0 there exist n0 and q0 such that 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk < ε for all pk ≥ 1, n ≥ n0, q ≥ q0, for every nonzero z1, . . . , zn−1 ∈ X and for some ρ > 0. Then x ∈ � M , [F], p,‖·, . . . , ·‖ � . K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 469 Proof. Let ε > 0 be given. Choose n′0, q0 such that 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk < ε 2 (1) for all n≥ n′0, q ≥ q0, It is enough to prove that there exists n′′0 such that for n> n′′0 , 0≤ q ≤ q0 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk < ε. (2) By taking n0 = max(n′0, n′′0 ), (2) will holds for n ≥ n0 and for all q, which gives the result. Once q0 has been chosen fixed, so q0−1 ∑ j=0 ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk = K , (3) for some K . Now taking 0≤ q ≤ q0 and n> q0, we have 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk = 1 n � q0−1 ∑ j=q + q+n−1 ∑ j=q0 � ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk ≤ K n + 1 n q0+n−1 ∑ j=q0 ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk ≤ K n + ε 2 . The penultimate inequality is from (3), with the last following (1). Taking n sufficiently large, we can make K n + ε 2 < ε which gives (2) and hence the result. Theorem 1. Suppose pk ≥ 1 for all k and for every θ , we have � M , [Fθ ], p,‖·, . . . , ·‖ � = � M , [F], p,‖·, . . . , ·‖ � . Proof. Let {x j} ∈ � M , [Fθ ], p,‖·, . . . , ·‖ � , then for given ε > 0, there exist r0 and λ such that 1 hr q+hr−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk < ε (4) for r ≥ r0 and q = Qr−1 + 1+ i, i ≥ 0. Let n ≥ hr , write n = mhr + θ , where m is an integer. Since h≥ hr , m≥ 1. Now 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk ≤ 1 n q+(m+1)hr−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 470 = 1 n + m ∑ u=0 q+(u+1)hr−1 ∑ j=q+uhr ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk ≤ m+ 1 n hrε ≤ 2mhrε n (m≥ 1). For hr n ≤ 1, since mhr n ≤ 1 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 ��pk ≤ 2ε. Then by Lemma 1, � M , [Fθ ], p,‖·, . . . , ·‖ � ⊆ � M , [F], p,‖·, . . . , ·‖ � . It is trivial to show that � M , [F], p,‖·, . . . , ·‖ � ⊆ � M , [Fθ ], p,‖·, . . . , ·‖ � for every θ . Hence we have the result. Lemma 2. Suppose for a given ε > 0 there exist n0 and q0 such that ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk < ε for all n≥ n0, q ≥ q0, for every nonzero z1, . . . , zn−1 ∈ X and for some ρ > 0. Then x ∈ � M , F, p,‖·, . . . , ·‖ � . Proof. Let ε > 0 be given. Choose n′0, q0 such that ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk < ε 2 (5) for all n ≥ n′0, q ≥ q0. As in Lemma 1, it is enough to prove that there exists n′′0 such that for n≥ n′′0 , 0≤ q ≤ q0 ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ i=0 � x j −λ ρ , z1, . . . , zn−1 � �pk < ε. (6) Since q0 is fixed, let q0−1 ∑ j=0 ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 � �pk = K ′, (7) for some K ′. Now taking 0≤ q ≤ q0 and n> q0, we have ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk ≤ ∞ ∑ k=1 � Mk 1 n q0−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 471 + ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q0 � x j −λ ρ , z1, . . . , zn−1 � �pk ≤ K ′ n + ∞ ∑ k=1 � Mk 1 n q0+n+q−q0−1 ∑ j=q0 � x j −λ ρ , z1, . . . , zn−1 � �pk . (8) Let n− q0 > n′0. Then for 0≤ q < q0, we have n+ q− q0 ≥ n′0. From (5) we have ∞ ∑ k=1 � Mk 1 n+ q+ q0 q0+n+q−q0 ∑ j=q0 � x j −λ ρ , z1, . . . , zn−1 � �pk < ε 2 . (9) From equation (8) and (9) we have ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk ≤ K ′ n + n+ q− q0 n ε 2 ≤ K ′ n + ε 2 <ε, for sufficiently large n. Hence the result. Theorem 2. (i) For every θ , we have � M , Fθ , p,‖·, . . . , ·‖ � ∩ � M , l∞p,‖·, . . . , ·‖ � = � M , F, p,‖·, . . . , ·‖ � . (ii) For every θ , we have � M , Fθ , p,‖·, . . . , ·‖ � 6⊂ � M , l∞, p,‖·, . . . , ·‖ � . Proof. (i) Let {x j} ∈ � M , Fθ , p,‖·, . . . , ·‖ � ∩ � M , l∞, p,‖·, . . . , ·‖ � for every ε > 0, there exist r0 and q0 such that ∞ ∑ k=1 � Mk 1 hr q+hr−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk < ε 2 (10) for r ≥ r0, q ≥ q0, q = Qr−1 + 1+ i, i ≥ 0. Now let n ≥ hr , m is an integer greater than equal to 1. Then ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q ∞ ∑ k=1 Mk � x j −λ ρ , z1, . . . , zn−1 � �pk ≤ ∞ ∑ k=1 � Mk 1 n m−1 ∑ µ=0 q+(µ+1)hr−1 ∑ j=q+µhr � x j −λ ρ , z1, . . . , zn−1 � �pk + 1 n K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 472 = ∞ ∑ k=1 � Mk q+n−1 ∑ j=q+mhr � x j −λ ρ , z1, . . . , zn−1 � �pk . (11) Since {x j} ∈ � M , l∞, p,‖·, . . . , ·‖ � for all j, we have ∞ ∑ k=1 � Mk � x j −λ ρ , z1, . . . , zn−1 � �pk < K , for some K . So from (10) and (11) ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk ≤ 1 n m.hr ε 2 + Khr n , for hr n ≤ 1, since mhr n ≤ 1 and Khr n can be made less than ε 2 , taking n sufficiently large so ∞ ∑ k=1 � Mk 1 n q+n−1 ∑ j=q � x j −λ ρ , z1, . . . , zn−1 � �pk < ε for r ≥ r0, q ≥ q0. Hence, by Lemma 2, � M , Fθ , p,‖·, . . . , ·‖ � ∩ � M , l∞, p,‖·, . . . , ·‖ � ⊆ � M , F, p,‖·, . . . , ·‖ � . It is trivial to show that � M , F, p,‖·, . . . , ·‖ � ⊆ � M , Fθ , p,‖·, . . . , ·‖ � ∩ � M , l∞, p,‖·, . . . , ·‖ � . (ii) It is enough to show � M , Fθ , p,‖·, . . . , ·‖ � 6⊂ � M , l∞, p,‖·, . . . , ·‖ � . Let {x j} = (−1) j jµ where µ is constant with 0< µ < 1. Then q+hr−1 ∑ j=q x j , q ≥ 0 will contains an even number of terms. Let us take X = Rn. It is a straightforward matter to verify that {x j} ∈ � M , Fθ , p,‖·, . . . , ·‖ � with λ= 0. But {x j} is not bounded. Now, we define the paranorm g(x) on the sequence space � M , [F], p,‖·, . . . , ·‖ � and shown that the sequence space � M , [F], p,‖·, . . . , ·‖ � is total paranormed space. We also define a new concept of statistical convergence which will be called g-statistical convergence on the paranormed space �� M , [F], p, |·, . . . , ·‖ � , g � . Theorem 3. The sequence space � M , [F], p,‖·, . . . , ·‖ � is a linear topological space total parnormed by g(x) = sup n≥1, q≥1 0 6=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j ρ , z1, . . . , zn−1 ��pk � = sup n≥1, q≥1 06=z1,...,zn−1∈X ∞ ∑ k=1 Mk �� tnq � x j ρ , z1, . . . , zn−1 ���pk . K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 473 Proof. It is easy to see that � M , [F], p,‖·, . . . , ·‖ � is a linear space with coordinate-wise addition and scalar multiplication. Clearly g(x) = 0⇔ x = 0, g(x) = g(−x) and g is sub- additive. To prove the continuity of scalar multiplication, assume that (x (k)) be any sequence of the points in � M , [F], p,‖·, . . . , ·‖ � such that g(x (k) − x) → 0 as k →∞ and (µk) be any sequence of scalars such that µk→ µ as k→∞. Since the inequality g(x (k))≤ g(x) + g(x (k) − x) holds by subadditivity of g, g(x (k)) is bounded. Thus, we have g(µk x (k) −µx) = sup n≥1, q≥1 06=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � µk x (k)j −µx j ρ , z1, . . . , zn−1 ��pk � ≤|µk −µ| sup n≥1, q≥1 0 6=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x (k)j ρ , z1, . . . , zn−1 ��pk � + |µ| sup n≥1, q≥1 06=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x (k)j − x j ρ , z1, . . . , zn−1 ��pk � =|µk −µ|g(x (k)) + |µ|g(x (k) − x), which tends to zero as k→∞. This proves the fact that g is a paranorm on � M , [F], p,‖·, . . . , ·‖ � . Definition 8. A sequence x = (x j) is said to be strongly p−Cesaro summable (0 < p <∞) to a limit λ in �� (M , [F], p,‖·, . . . , ·‖ � , g � if limk→∞ 1 k ∑k j=1(g(x j − λe))p = 0 and we write it as x j → λ[C , g]p. In this case λ is called the [C , g]p-limit of x. We denote the set of all strongly p-Cesaro summable sequences in ( � M , [F], p,‖·, . . . , ·‖ � , g) as [C , g]p = {x : lim k→∞ 1 k k ∑ j=1 (g(x j −λe))p = 0}. Definition 9. A sequence x = (x j) is said to be statistically convergent (or g-statistically convergent) to a number λ in �� M , [F], p,‖·, . . . , ·‖ � , g � if for each ε > 0 lim k→∞ 1 k |{ j ≤ k : g(x j −λe)≥ ε}|= 0 where g(x j −λe) = sup n≥1, q≥1 06=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q ∞ ∑ k=1 � Mk � x j −λe ρ , z1, . . . , zn−1 ��pk � . In this case we write g(stat) − lim x = λ. We denote the set of all g-statistically convergent sequences in �� M , [F], p,‖·, . . . , ·‖ � , g � by S�� M ,[F]p,‖·,...,·‖ � ,g �. K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 474 Definition 10. A sequence x = (x j) is said to be a statistically Cauchy sequence in �� M , [F], p,‖·, . . . , ·‖ � , g � (or g(stat)− Cauchy) if for every ε > 0 there exists a number N = N(ε) such that lim n→∞ 1 n |{ j ≤ n : g(x j − xN )≥ ε}|= 0. Theorem 4. If a sequence x = (x j) is statistically convergent in �� M , [F], p,‖·, . . . , ·‖ � , g � , then g(stat)− lim x is unique. Proof. Suppose that g(stat)− lim x = λ1 and g(stat)− lim x = λ2. Given ε > 0, define the following set as: J1(ε) = ¦ j ∈ N : g(x j −λ1)≥ ε 2 © and J2(ε) = ¦ j ∈ N : g(x j −λ2)≥ ε 2 © . Since g(stat) − lim x = λ1 we have δ(J1(ε)) = 0. Similarly g(stat) − lim x = λ2 we have δ(J2(ε)) = 0, now let J(ε) = J1(ε)∪ J2(ε). Then δ(J(ε)) = 0 and hence the compliment J c(ε) is a non-empty set and δ(J c(ε)) = 1. Now if j ∈ N− J(ε), then we have g(λ1 −λ2)≤ g(x j −λ1) + g(x j −λ2)< ε 2 + ε 2 = ε. Since ε > 0 was arbitrary, we get g(λ1 −λ2) = 0 and hence λ1 = λ2. Theorem 5. Let g(stat)− lim x = λ1 and g(stat)− lim y = λ2. Then (i) g(stat)− lim(x ± y) = λ1 ±λ2 (ii) g(stat)− lim(αx) = αλ1,α ∈ R. Proof. It is easy to prove. Theorem 6. A sequence x = (x j) in �� M , [F], p,‖·, . . . , ·‖ � , g � is statistically convergent to λ if and only if there exists a set J = { j1 < j2 < . . . < jn < . . .} ⊆ N with δ(J) = 1 such that g(x jn −λ)→ 0 as n→∞. Proof. Suppose that g(stat)− lim x = λ. Now write for r = 1,2, . . .. Jr(ε) = ¦ n ∈ N : g(x jn −λ1)≤ 1+ 1 r © and Lr(ε) = ¦ n ∈ N : g(x jn −λ1)> 1 r © . Then δ(Jr) = 0 L1 ⊃ L2 ⊃ . . . ⊃ Li ⊃ Li+1 ⊃ . . . (12) K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 475 and δ(Lr) = 1, r = 1,2, . . . (13) Now we have to show that for n ∈ Lr . Since {x jn} is g-convergent to λ. On contrary suppose that {x jn} is not g−convergent to λ. Therefore, there is ε > 0 such that g(x jn − λ) ≤ ε for infinitely many terms. Let Lε = ¦ n ∈ N : g(x jn −λ)> ε © and ε > 1 r , r ∈ N. Then δ(Lε) = 0 (14) and by (12) Lr ⊂ Lε. Hence δ(Lr) = 0 which contradicts (13) and we get that {x jn} is g- convergent to λ. Conversely, suppose that there exists a set J = { j1 < j2 < . . . < jn < . . .} with δ(J) = 1 such that g − limn→∞ x jn = λ then there exists a positive integer N such that g(x j −λ)< ε for j > N . Put Jε(t) = ¦ n ∈ N : g(x j −λ)≥ ε © and J ′ = {JN+1, JN+2, . . .}. Then δ(J ′) = 1 and Jε ⊆ N\J ′ which implies that δ(Lε) = 0. Hence g(stat)− lim x = λ. Theorem 7. Let ( � M , [F], p,‖·, . . . , ·‖ � , g � be a complete paranormed space. Then a sequence x = (x j) of points in �� M , [F], p,‖·, . . . , ·‖ � , g � is statistically convergent if and only if it is statistically Cauchy. Proof. Suppose that g(stat)− lim x = λ, then we get δ(A(ε)) = 0, where A(ε) = ¦ j ∈ N : g(x j −λ)≥ ε 2 © . This implies δ(Ac(ε)) = δ({ j ∈ N : g(x j −λ))< ε}) = 1. Let l ∈ Ac(ε), then g(x l −λ)< ε 2 . Now let B(ε) = ¦ j ∈ N : g(x l − x j)≥ ε}. We need to show that B(ε) ⊂ A(ε). Let j ∈ B(ε) then g(x l − x j)≥ ε and hence g(x j −λ))≥ ε that j ∈ A(ε). Otherwise if g(x j −λ))< ε then ε ≤ g(x j − x l)≤ g(x j −λ) + g(x l −λ)< ε 2 + ε 2 = ε, which is not possible. Hence B(ε) ⊂ A(ε), implies that x = (x j) is g(stat)-convergent. Conversely, suppose that x = (x j) is g(stat)-Cauchy but not g(stat)-convergent. Then there exists t ∈ N such that δ(G(ε)) = 0. where G(ε) = ¦ j ∈ N : g(x j − x t)≥ ε © K. Raj, R. Anand, S. Jamwal / Eur. J. Pure Appl. Math, 9 (2016), 464-478 476 and δ(D(ε)) = 0, where D(ε) = ¦ j ∈ N : g(x j −λ)< ε 2 © i.e, δ(Dc(ε)) = 1, since g(x j − x l) ≤ 2g(x j − λ) < ε. If g(x j − λ) < ε 2 then δ(Gc(ε)) = 0, i.e, δ(G(ε)) = 1 which leads to a contradiction since x = (x j) was g(stat)-Cauchy. Hence x = (x j) must be g(stat)-convergent. Theorem 8. If 0 < p <∞ and x j → λ[C , g]p, then x = (x j) is g-statistically convergent to λ in �� M , [F], p,‖·, . . . , ·‖ � , g � . Proof. Let x j → λ[C , g]p, then 1 k k ∑ j=1 (g(x j −λe))p ≥ 1 k k ∑ j=1 g(x j−λe)≥ε (g(x j −λe))p ≥ εp k |Kε|. Since limk→∞ 1 k |Kε| = 0 and so δ(Kε) = 0, where Kε = { j ≤ k : g(x j − λe) ≥ ε}. Hence x = (x j) is statistically convergent to λ in �� M , [F], p,‖·, . . . , ·‖ � , g � . Theorem 9. If x = (x j) is g-statistically convergent to λ in �� M , [F], p,‖·, . . . , ·‖ � , g � then x j → λ[C , g]p. Proof. Suppose that x = (x j) is g-statistically convergent to λ in �� M , [F], p,‖·, . . . , ·‖ � , g � . Then for ε > 0, we have δ(Kε) = 0, where Kε = { j ≤ k : g(x j −λe)≥ ε}. Since x = (x j) ∈ l∞(M , p,‖·, . . . , ·‖), then there exists K > 0 such that � M � x j −λe ρ , z1, . . . , zn−1 � �pk ≤ K , for all j. Thus, g(x j −λe) = sup n≥1, q≥1 0 6=z1,...,zn−1∈X � 1 n q+n−1 ∑ j=q � M � x j −λe ρ , z1, . . . , zn−1 � �pk � ≤ K . Hence we have result from the following inequality 1 k k ∑ j=1 (g(x j −λe))p = 1 k k ∑ j=1 j /∈Kε (g(x j −λe))p + 1 k k ∑ j=1 j∈Kε (g(x j −λe))p ≤εp + K p k |Kε|. 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