On lambda-statistical convergence using f -density

We study the concept of fλ-statistical convergence in a probabilistic normed space, where f is an unbounded modulus function . Also we are trying to investigate some relation between the ordinary convergence and module statistical convergence for every unbounded modulus function. Later on, we study fλstatistical convergence with partial average too.


Introduction and preliminaries
The study of summability theory and convergence of sequences has been one of the most important and active area of research work in Pure mathematics for the last several decades. Its extensive works are also applicable in Topology, Functional Analysis, Fourier Analysis, Measure Theory, Applied Mathematics, Mathematical Modelling, Computer Science etc. In recent years, the concept of statistical convergence of sequences which was first introduced by Fast [8] as an extension of the usual concept of sequential limits, has also been used as a tool by many mathematicians to solve many open problems in the area of sequence spaces and summability theory and some other applications as well. One may refer to ([6], [16], [12], [14], [17], [19], [21], [22], [23], [25], [31], [10]).
In 2014, A. Aizpuru et al [1] introduced a new concept of density for sets of natural numbers with respect to the modulus function. They studied and characterized the generalization of this notion of fdensity with statistical convergence and proved that ordinary convergence is equivalent to the module statistical convergence for every unbounded modulus function. They also worked on double sequence spaces for the results of f -statistical convergence by using unbounded modulus function. Savaş and Borgohain [24] introduced some new spaces of lacunary f -statistical A-convergent sequences of order α.
Menger [13] introduced the notion of metric space under the name of statistical metric space, which is known as probabilistic metric space. The idea of Menger was to use distribution functions instead of nonnegative real numbers. The probabilistic theory has become an area of active research for the last forty years. It has a wide range of applications in functional analysis also [7]. An important family of probabilistic metric spaces are probabilistic normed spaces (briefly, PN-spaces).

Preliminary Concepts
A sequence (x i ) of real numbers is statistically convergent to L if for arbitrary ε > 0, the set K(i) = {i ≤ k : |x i − L| ≥ ε} has natural density zero, .i.e., where χ K(i) denotes the characteristic function of K(i). By a modulus function, we mean a function f : R + → R + which satisfies ( [15]) :  The probabilistic norm or N-norm is a triangular norm satisfying the following conditions: |α| for all α ∈ R\{0} and for all t > 0, 4. N p+q (s + t) ≥ N p (s) * N q (t) for all p, q ∈ X and s, t ∈ R + 0 ; where N p means N(p) and N p (t) is the value of N p at t ∈ R. (X, N, * ) is named as a probabilistic normed space , in short PN-space.
Let λ = (λ i ) be a non-decreasing sequence of positive numbers such that , Note: The collection of all such sequences λ will be denoted by Λ.
If λ i ≤ µ i , ∀λ, µ ∈ Λ, we have A sequence (x i ) of real numbers is said to be λ-statistically convergent to L if for any ε > 0, and |A| denotes the cardinality of A ⊂ N (refer [14]). Savaş ([18]) studied λ-statistical convergence in random 2-normed spaces and I λ -statistical convergence of sequences in topological groups. In [4], Alotaibi studied the notion of λ-statistical convergence for single sequences in probabilistic normed spaces. Also Savas and Mohiuddine [29] studied λ-statistically convergent double sequences in probabilistic normed spaces.
In this paper, we study the f λ -statistical convergence with respect to the probabilistic norm N in the PN-space (X, N, * ). We also investigate some results on the new concept of f λ -statistical convergence with partial average. Moreover, the concepts of f λ -statistical limits, f λ -cluster points and f λ -equivalence are introduced and try to find out the relations among them.

Main Results
Definition 3.1. Let (X, N, * ) be a PN-space. Then a sequence x = (x i ) is said to be f λ -statistically convergent to L with respect to the probabilistic norm N provided that, for every t > 0 and ε > 0, Definition 3.2. Let (X, N, * ) be a PN-space. An element L ∈ X is said to be an ordinary limit point of the sequence (x i ) ∈ X with respect to the probabilistic norm N if there is a subsequence of the sequence (x i ) which also converges to L with respect to N. We denote, Ω N (x) = set of all ordinary limit points of the sequence w.r.t. N.
is statistically convergent to L with respect to the probabilistic norm N. We denote it as, In the following results , we investigate the relations between f -limit points, f -cluster points and ordinary limit points of (X, N, * ) with respect to the non-decresing sequences of positive numbers (λ i ) and (µ i ) where λ i ≤ µ i . Also in Theorem 3.2. , we establish the result of f λ -statistical convergence of (x i ) with respect to f λ -limit points and f λ -cluster points of (X, N, * ).
Proof. Let L be a f µ -limit point of (x i ). Then for R = {i m : i 1 < i 2 < ...} such that δ f µ (R) = 1 and lim m x i m = L with respect to N. For each t > 0, ε > 0, there exists i 0 ∈ N such that for i > i 0 , which means that L is a f λ -cluster point of (x i ) in (X, N, * ).
To prove the second part, let us take L to be a f λ -cluster point of (x i ) in (X, N, * ). Then for given t > 0 and ε > 0, we have, Construct a set I = {i m : i 1 < i 2 < i 3 < ...} such that a subsequence (x i m ) of (x i ) that converges to L with respect to the probabilistic norm N. Thus, L is an ordinary limit point of (x i ) in (X, N, * ).
Hence, f λ -cluster point of (x i ) ⇒ ordinary limit point of (x i ) with respect to the probabilistic norm N.
Theorem 3.6. Let (X, N, * ) be a PN-space. Then the f λ -limit points and f λ -cluster points of a sequence (x i ) are same, say L, if it is f λ -statisically convergent to L.
Proof. Let ρ be a f λ -limit point such that ρ = L. Construct two subsets of N as K = {k m : k 1 < k 2 < k 3 < ...} and L = {l m : l 1 < l 2 < l 3 ...} such that δ f λ (K) = 1 and δ f λ (L) = 1. Also (x k m ) and (x l m ) are convergent to L and ρ respectively with respect to the probabilistic norm N.
We assume that ρ is the f λ -limit point of X, then for given t > 0 and ε > 0, there exists m 0 ∈ N such that N x lm −ρ (t) > 1 − ε, ∀m ≥ m 0 . Let Since (x i ) is convergent to L with respect to the probabilistic norm N, then N x i −L (t) > 1 − ε. So we have for each t > 0 and ε > 0, the set δ f λ ({i ∈ I i : If ρ = L, then Q ∩ R c is empty and Q ⊂ R. Also δ f λ (R) = 0 implies δ f λ (Q) = 0, which leads to a contradiction that δ f λ (Q) = 1. Hence L is also a f λ -limit point of (x i ) in (X, N, * ).
To prove the second part, let us take ρ be a f λ -cluster point such that L = ρ. Also, for each t > 0 and ε > 0, we construct the sets A = {i ∈ I i : N x i −L (t) > 1 − ε} and B = {i ∈ I i : N x i −ρ (t) > 1 − ε} such that δ f λ (A) = 1 and δ f λ (B) = 1 respectively.
We asuume that L = ρ , which implies that A ∩ B = φ and therefore B ⊂ A c where A c = {i ∈ I i : Also, as (x i ) is convergent to L with respect to N, we have δ f λ (A c ) = 0 ⇒ δ f λ (B) = 0, which is a contradiction.
Hence, ρ = L, i.e. L is also a f λ -cluster point. This completes the proof.
In this section, we define the concept of f λ -statistically Cauchy in (X, N, * ) and establish the relation between f -statistical Cauchy and f -statistical convergence of (x i ) with respect to (λ i ) and (µ i ) where λ i ≤ µ i .
which leads to a contradiction that This arrives to the conclusion that (x i ) is f µ -statistically Cauchy implies that it is f λ -statistically con- This completes the proof.
Partial average/means of sequences has an imporatnt role in the theory of ergodic systems [34] . Very recently, Mark Burgin and Oktay Duman [33] studied the statistical convergence of sequences of real numbers in the direction of Statistics. In this section, we define the idea of f -partially statistical convergence and f λ -partially statistical convergence of (x i ) with respect to the probabilistic norm N and investigate some relations between f λ -partial statistical convergence and f λ -statistica convergence.
We denote it as f λ p − stat lim Theorem 4.4. Let (X, N, * ) be a PN-space. If a bounded sequence x = (x i ) is f λ -statistcially convergent to L with repsect to the probabilistic norm N then it is also f λ -partially statistical convergent to L with respect to N. But not conversely.
Proof. Let x = (x i ) be f λ -statistical convergent to L with respect to N. Then for any t > 0 and ε > 0, By taking i → ∞, we get f λ -statistically convergence implies f λ -partially statistical convergence. For the converse part, let f be an unbounded modulus function such as f (x) = log(1 + x). Take (R, N, * ) be a PN-space with a * b = ab, N x (t) = t t+|x| . Now we define a sequence , otherwise.
which shows that (x i ) is f λ -statistically divergent, but it is f λ -partially statistical convergent.
Theorem 4.5. Let (X, N, * ) be a PN-space . Then the f λ -partial statistical limit of (x i ) ∈ X is unique with respect to the probabilistic norm N.
Proof. Proof is easy, so ommitted.
Corollary 4.6. Let (X, N, * ) be a PN-space. If a sequence (x i ) ∈ X is f λ -partial statistical convergence then it has also a partial convergent subsequence with respect to N.
, then a sequence (x i ) is partially statistical convergent to L wih respect to the probabilistic norm N, then it is f λ -statistically convergent to L with respect to N.
Proof. Let (x i ) be a partially convergent to L with respect to the probabilistic norm N, then for ε > 0 and t > 0, we have, We can write, Consequently we get, Since lim i in f f (λ i ) i > 0, so it follows that partial convergence imples f λ -statistical convergence with respect to the probabilistic norm N.This completes the proof.

Conclusion
Convergence of statistical characteristics such as the average/mean and standard deviation are related to statistical convergence as we see in our results. This concept of convergence with respect to means/avearages and standard deviations have been studied by many mathematicians. We have studied the convergence of the average/mean in the propbabilistic normed spaces. It can be further extended in the direction of fuzzy real numbers and can be found very interesting results in intutionistic fuzzy normed linear spaces too using various aspects.